DIAGRAM ALGEBRAS, HECKE ALGEBRAS AND

DECOMPOSITION NUMBERS AT ROOTS OF UNITY.

J.J. Graham and G.I. Lehrer

School of Mathematics and Statistics The University of Birmingham Birmingham B15 2TT, United Kingdom

School of Mathematics and Statistics University of Sydney N.S.W. 2006, Australia

6 December 2001

Abstract. We prove that the cell modules of the affine Temperley-Lieb algebra have the same composition factors, when regarded as modules for the affine Hecke algebra of type A, as certain standard modules which are defined homologically. En route, we relate these to the cell modules of the Temperley-Lieb algebra of type B. Applications include the explicit determination of some decomposition numbers of standard modules at roots of unity, which in turn has implications for certain Kazhdan-Lusztig polynomials associated with nilpotent orbit closures. The methods involve the study of the relationships among several algebras defined by concatenation of braid-like diagrams and between these and Hecke algebras. Connections are made with earlier work of Bernstein-Zelevinsky on the “generic case” and of Jones on link invariants.

Contents

1. Introduction 2

2. Some generalised Artin braid groups 3

3. Affine Hecke algebras of type A 8

4. Affine and finite dimensional Temperley-Lieb algebras 10

5. The Temperley-Lieb algebras of type B 12

6. Cell modules for T_{n}^{a}(q) and T LB_{n}(q, Q) 17

7. Eigenvalues of the translation elements X_{i} 25

8. The case of generic parameters 30

Key words and phrases. Representations, braids, affine Hecke algebras, root of unity, diagram algebras.

2000 Mathematics Subject Classification. Primary 20C08, 16G30, 20G05, Secondary 20C35. Both authors thank the Australian Research Council for support. The second author thanks

the Alexander von Humboldt Stiftung for support and the Universität Bielefeld for hospitality during the preparation of this work.

Typeset by AMS-TEX

9. Induced modules and standard modules 32

10. Irreducible modules and decomposition numbers 39

11. Concluding remarks 45

1. Introduction

Let R be a commutative ring with 1 and let q ∈ R^{×}, where for any ring A with 1, A^{×} denotes the group of its invertible elements. In [GL2], we defined a “Temperley-Lieb category” 𝕋^{a}, whose objects are the natural numbers ℕ, and whose morphisms are R-linear combinations of “planar diagrams from t to n” (for t, n ∈ ℕ), with composition depending on the element q. In particular, we have the algebras of endomorphisms T_{n}^{a}(q) := Hom_{𝕋^{a}}(n, n), for n = 0, 1, 2, 3, . . .. These were called in [GL2] the affine Temperley-Lieb algebras. Using a calculus of diagrams, together with the philosophy of cellular algebras, we developed in [op. cit.] a theory of cell modules W_{t,z}(n) (0 ≤ t ≤ n, n − t ∈ 2ℤ, z ∈ R^{×}) for the algebras T_{n}^{a}(q), and gave a complete description of the composition factors of the cell modules, valid for all q, even for q a root of unity, the most complicated case. The description of the irreducible T_{n}^{a}(q)-modules is valid when R is any algebraically closed field. The analysis of the composition factors of the cell modules applies when R is an algebraically closed field whose characteristic is either 0 or p > 0 such that pe > n, where e is the multiplicative order of q^{2}.

Now let Ĥ_{n}^{a}(q) be the extended affine Hecke algebra of type { \atop A_{n−1}}, which corresponds to G = GL_{n}(ℂ). This is the algebra considered in [BZ], [R] and [KL], for which there is a theory of “standard modules” M_{s,N}, which may be constructed by regarding Ĥ_{n}^{a}(q) as a convolution algebra of coherent sheaves acting on the Borel-Moore homology of certain varieties V_{s,N}, where s is a semisimple element of G and N is a nilpotent element of 𝔊 =Lie(G) such that Ad(s)N = q^{2}N. When q is not a root of unity, the structure of the modules M_{s,N} is fairly well understood, while when q is a root of unity, it was conjectured in [LLT] and proved in [A] (cf. also [G]), that the decomposition numbers of the standard modules, i.e. the multiplicities of the irreducibles in M_{s,N}, are given by values of certain Kazhdan-Lusztig polynomials, which are generally not known explicitly (see also [LTV]).

In this work, we show that the cell modules W_{t,z}(n) of the algebra T_{n}^{a}(q) (cf. [GL2]) may be inflated via a family {ψ_{α} | α ∈ R^{×}} of surjections Ĥ_{n}^{a}(q) → T_{n}^{a}(q) (see (5.12) below) to modules for Ĥ_{n}^{a}(q), which we identify explicitly (in the Grothendieck group Γ(Ĥ_{n}^{a}(q)) of finite dimensional Ĥ_{n}^{a}(q)-modules) with the standard modules M_{s,N} where N has just two Jordan blocks. This enables us to use the results of [GL2] to give completely explicit decompositions of these standard modules, and therefore give character formulae for their irreducible heads. We also obtain much detailed information about their internal structure. Among the consequences of our results are the statements that M_{s,N} is always multiplicity free, and that when q is a root of unity, the composition length may be arbitrarily large (as n increases). The key point in this work is the identification of the inflations of our cell modules with the standard modules up to Grothendieck equivalence (Theorem (9.8)).

To achieve this, we show that the inflations ψ_{α}^{∗}W_{t,z}(n) are generically (i.e. for generic q) induced modules from a parabolic subalgebra of Ĥ_{n}^{a}(q). For this we need to understand the action of the “translation elements” X_{i} on the inflations. This in turn depends on the relationship between two (known) ways of viewing Ĥ_{n}^{a}(q); the first as a twisted tensor product of the group ring R[⟨V ⟩] of ℤ ≅ ⟨V ⟩ with the Hecke algebra H_{n}^{a}(q) of the Coxeter system of affine type { \atop A_{n−1}}, the second as the tensor product of the finite dimensional Hecke algebra H_{n}(q) of type A_{n−1} with the R-algebra of Laurent polynomials R[X_{1}^{±1}, ⋯, X_{n}^{±1}]. We approach this relationship via generalised Artin braid groups. In addition, we shall have recourse to the “Temperley-Lieb algebra” of type B_{n}, denoted by T LB_{n}(q, Q) below (it is sometimes referred to as the “blob algebra”), to determine the action of the X_{i}, since T_{n−1}^{a}(q) is not naturally a subalgebra of T_{n}^{a}(q), and therefore one does not have restriction. We circumvent this difficulty by proving that for any Q, there is a pair of natural surjections from T_{n}^{a}(q) to T LB_{n}(q, Q), and studying restriction from T LB_{n}(q, Q) to T LB_{n−1}(q, Q). This could be used to study the “modular representation theory” of Ĥ_{n}^{a}(q), but we do not do this here. In the final section (§11) we interpret our results in terms of a generalisation to the non-generic case of the “multisegments” of Zelevinsky and Bernstein.

Since the “annular algebras” of V. Jones ([J1]) are quotients of the algebras T_{n}^{a}(q), their representation theory may be thought of as a subset of the story below. Hence our work throws light on the connection between the work of Jones on link invariants (cf. [J2]) and affine Hecke algebras.

2. Some generalised Artin braid groups

Let W be the symmetric group Sym_{n}, realised as a Coxeter group generated by the reflections s_{i} in the hyperplanes z_{i} − z_{i+1} = 0 of V = ℂ^{n} (i = 1, 2, …, n − 1). Write s_{n} for reflection in the affine hyperplane z_{1} − z_{n} = 1. Then {s_{1}, …, s_{n}} are Coxeter generators for the affine Weyl group W^{a} ≅ W ⋉ ℤ^{n−1}, which may also be thought of as generated by W together with translations by vectors (a_{1}, …, a_{n}), with a_{i} ∈ ℤ and a_{1} + ⋯ + a_{n} = 0. Write Ŵ^{a} for the semidirect product of W with the group ℤ^{n} of all translations by vectors with integer coordinates. Write v_{0} = (1, 1, …, 1) and denote by p_{0} the orthogonal projection p_{0} : V → v_{0}^{⊥}. Then W and W^{a} act irreducibly as Coxeter groups on V_{0} = v_{0}^{⊥}, and W^{a} is a normal subgroup of Ŵ^{a}, with quotient ℤ.

The reflecting hyperplanes of W^{a} acting on V are the hyperplanes z_{i} − z_{j} = k, for 1 ≤ i < j ≤ n and k ∈ ℤ. Write M^{a} for the complement of these hyperplanes in V and M_{0}^{a} for p_{0}(M^{a}). Thus, explicitly,

M^{a} = {(x_{1}, x_{2}, …, x_{n}) ∈ V = ℂ^{n} | x_{i} − x_{j} ∉ ℤ if i ≠ j}.

(2.1) Proposition (Nguyen [N]). The fundamental group π_{1}(M_{0}^{a}/W^{a}) is isomorphic to the Artin group associated to the Coxeter system (W^{a}, {s_{1}, …, s_{n}}).

In fact Nguyen proves this for any affine type Coxeter group by giving a “cell decomposition” of the space M_{0}^{a}/W^{a}. In our case we have explicitly that π_{1}(M_{0}^{a}/W^{a}), which we denote henceforth by Δ_{n}, is generated by elements {σ_{1}, σ_{2}, …, σ_{n}} subject to the relations

(BR) σ_{i}σ_{j} = σ_{j}σ_{i} if j ≢ i ± 1 (mod n) σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1} for i = 1, 2, …, n,

where the subscripts in (BR) are taken mod n.

(2.2) Lemma. The map p_{0} : M^{a} → M_{0}^{a} is a W^{a}-homotopy equivalence. Hence the quotient spaces M^{a}/W^{a} and M_{0}^{a}/W^{a} are homotopy equivalent. In particular they have the same fundamental group.

Proof. If I is the unit interval and i is the inclusion of M_{0}^{a} in M^{a}, then p_{0}∘i = id_{M_{0}^{a}},

and the map (v, t) ↦ v − (1 − t)\frac{⟨v, v_{0}⟩}{n}v_{0} (M^{a} × I → M^{a}) defines a homotopy from i ∘ p_{0} to id_{M^{a}}, which commutes with the W^{a} action for each t ∈ I. □

(2.3) Lemma. Let Ŵ^{a} ⊃ W^{a} be as described above. The map M^{a}/W^{a} ⟶ M^{a}/Ŵ^{a} is an unramified covering with covering group ℤ.

The proof is easy. Note that the quotient Ŵ^{a}/W^{a} is generated by the element τ′ = (c_{n}, (1, 0, 0, …, 0)) ∈ Sym_{n} ⋉ ℤ^{n}, where c_{n} is the n-cycle (12 … n) ∈ Sym_{n}. This element has the property that τ^{n} lies in the centre Z(Ŵ^{a}). It follows from the lemma that M^{a}/Ŵ^{a} may be thought of as the quotient of M^{a}/W^{a} by the cyclic group ⟨τ′⟩.

(2.4) Corollary. There is an exact sequence

1 ⟶ π_{1}(M^{a}/W^{a}) ⟶ π_{1}(M^{a}/Ŵ^{a}) ⟶ ℤ ⟶ 1.

We now identify the space M^{a}/Ŵ^{a} and its fundamental group. For any topological space Y, let X_{n}(Y) be the space

X_{n}(Y) = {(y_{1}, …, y_{n}) ∈ Y^{n} | y_{i} ≠ y_{j} if i ≠ j}/Sym_{n}.

Then evidently M^{a}/Ŵ^{a} ≅ X_{n}(ℂ/ℤ) ≅ X_{n}(ℂ^{∗}). But if M_{B_{n}} is the complex hyperplane complement of type B_{n} (viz. ℂ^{n} with the hyperplanes x_{i} ±x_{j} = 0 and x_{i} = 0 removed), and W_{B_{n}} is the corresponding Weyl group, then clearly

(2.5)  M_{B_{n}}/W_{B_{n}} ≅ X_{n}(ℂ^{∗})/{±id} ≅ X_{n}(ℂ^{∗}) ≅ M_{0}^{a}/Ŵ^{a}.

But Deligne has shown ([D]) that the space of regular orbits of a finite Coxeter group on its complexified reflection representation space is its associated generalised Artin braid group (this is also proved in [N]). Write Γ_{n} for the braid group of type B_{n}. Then Γ_{n} has generators {ξ_{1}, σ_{1}, σ_{2}, …, σ_{n−1}}, with relations

(BRB)  \begin{aligned}σ_{i}σ_{j} & = σ_{j}σ_{i} if i ≠ j \\ σ_{i}σ_{i+1}σ_{i} & = σ_{i+1}σ_{i}σ_{i+1} for i = 1, 2, …, n − 1 \\ ξ_{1}σ_{1}ξ_{1}σ_{1} & = σ_{1}ξ_{1}σ_{1}ξ_{1}. \end{aligned}

Now paths in X_{n}(ℂ^{∗}) may be regarded a periodic braids, or braids on a thickened cylinder, as follows. Think of ℂ^{∗} as the plane with a large hole at 0; choose n points P_{1}, P_{2}, …, P_{n} in ℂ^{∗}. A path in X_{n}(ℂ^{∗}) may then be regarded as a braid “around the hole”, or on the thickened cylinder, where each string starts at some P_{i} and finishes at P_{i′}, where i ↦ i′ is a permutation of {1, 2, …, n}. If we cut the cylinder open, these braids may be drawn in the plane, and regarded as “periodic braids”, or cylindrical braids. These may be drawn as depicted in the diagrams below, in which the two intervals labelled AB are identified by bending the rectangle in towards the page.

1 2  i i + 1  n • • • • • • • • • • • • • • • • • • A B  A B The cylindrical braid σ_{i}

1 2  i i + 1  n • • • • • • • • • • • • • • • • • • A B  A B The cylindrical braid σ_{n}

1 2  i i + 1  n • • • • • • • • • • • • • • • • • • A B  A B The cylindrical braid ξ_{i}

1 2  i i + 1  n • • • • • • • • • • • • • • • • • • A B  A B The cylindrical braid τ

Now let τ ∈ Γ_{n} be the “twist” as shown (it corresponds to the projection of a path in M_{0}^{a}/W^{a} from the base point P to τ′P, where τ′ is the element defined above, such that M^{a}/Ŵ^{a} ≅ (M^{a}/W^{a})/⟨τ′⟩). Further, let σ_{i} be the generating braids depicted in the diagram (i = 1, 2, …, n). It is clear that Γ_{n} then has a presentation with generators {τ, σ_{1}, σ_{2}, …, σ_{n}}, with relations (BR) for the σ_{i}, together with

τσ_{i}τ^{−1} = σ_{i+1} for i = 1, 2, …, n,

where the subscripts are taken mod n.

(2.6) Proposition. With the above notation, let

\begin{aligned}ξ_{1} & = τσ_{n−1}^{−1}σ_{n−2}^{−1} … σ_{1}^{−1} and \\ ξ_{i+1} & = σ_{i}ξ_{i}σ_{i} for i = 1, 2, …, n − 1. \end{aligned}

Then the family {ξ_{1}, σ_{1}, …, σ_{n−1}} generates Γ_{n} subject to the relations (BRB), and {ξ_{1}, …, ξ_{n}} generates a free abelian group of rank n.

Proof. It is clear that {ξ_{1}, σ_{1}, …, σ_{n−1}} generates Γ_{n}, since {τ, σ_{1}, σ_{2}, …, σ_{n−1}} does, and τ = ξ_{1}σ_{1} … σ_{n−1}. We next show that {ξ_{1}, σ_{1}, …, σ_{n−1}} satisfies the relations BRB. This is easy to see from the braid pictures (see below), but we shall provide an algebraic proof. For i = 1, 2, …, n, we have

ξ_{i} = σ_{i−1}σ_{i−2} … σ_{1}τσ_{n−1}^{−1}σ_{n−2}^{−1} … σ_{i}^{−1}.

We shall first show that

(2.6.1)  σ_{j}ξ_{i} = ξ_{i}σ_{j} if j ≠ i − 1, i.

To see (2.6.1), first take i = 1. Then for j > 1,

σ_{j}ξ_{1} = σ_{j}τσ_{n−1}^{−1}σ_{n−2}^{−1} … σ_{1}^{−1}
= τσ_{j−1}^{−1}σ_{n−1}^{−1} … σ_{1}^{−1}
= τσ_{n−1}^{−1} … σ_{j+1}^{−1}σ_{j−1}σ_{j}^{−1}σ_{j−1}^{−1}σ_{j−2}^{−1} … σ_{1}^{−1}
= τσ_{n−1}^{−1} … σ_{j+1}^{−1}σ_{j}^{−1}σ_{j−1}^{−1}σ_{j}σ_{j−2}^{−1} … σ_{1}^{−1} by (BR)
= ξ_{1}σ_{j}.

This proves (2.6.1) for i = 1. But since ξ_{i} = σ_{i−1}σ_{i−2} … σ_{1}ξ_{1}σ_{1} … σ_{i−1}, it follows that (2.6.1) holds for all j > i. Now take j < i − 1. Then

σ_{j}ξ_{i} = σ_{j}σ_{i−1}σ_{i−2} … σ_{1}τσ_{n−1}^{−1} … σ_{i}^{−1}
= σ_{i−1} … σ_{j}σ_{j+1}σ_{j}σ_{j−1} … σ_{1}τσ_{n−1}^{−1} … σ_{i}^{−1}
= σ_{i−1} … σ_{j+1}σ_{j}σ_{j+1}σ_{j−1} … σ_{1}τσ_{n−1}^{−1} … σ_{i}^{−1} by (BR)
= ξ_{i}σ_{j}.

This proves (2.6.1). We next show that

(2.6.2)  ξ_{1}σ_{1}ξ_{1}σ_{1} = σ_{1}ξ_{1}σ_{1}ξ_{1}.

Let

β_{n} = σ_{n−1}σ_{n−2}^{−1} … σ_{1}^{−1}σ_{n−1}^{−1}σ_{n−2}^{−1} … σ_{1}^{−1}.

We shall show by induction on n that

(2.6.2.1)  β_{n} = σ_{n−2}^{−1}σ_{n−1}^{−1}.σ_{n−3}^{−1}σ_{n−2}^{−1}. · · · .σ_{1}^{−1}σ_{2}^{−1}.

If n = 2, both sides of (2.6.2.1) are equal to 1. In general,

β_{n} = σ_{n−1}σ_{n−2}^{−1}σ_{n−1}^{−1}σ_{n−3}^{−1} … σ_{1}^{−1}σ_{n−2}^{−1}σ_{n−3}^{−1} … σ_{1}^{−1}
= σ_{n−2}^{−1}σ_{n−1}^{−1}σ_{n−2}σ_{n−3}^{−1} … σ_{1}^{−1}σ_{n−1}^{−1}σ_{n−2}^{−1} … σ_{1}^{−1}
= σ_{n−2}^{−1}σ_{n−1}^{−1}β_{n−1},

which proves (2.6.2.1).

Next, observe that if γ_{n} = σ_{n−2}^{−1} … σ_{1}^{−1}σ_{n−1}^{−1}σ_{n−2}^{−1} … σ_{2}^{−1}, then γ_{n} = β_{n}. For n = 3, γ_{n} = β_{n} = σ_{1}^{−1}σ_{2}^{−1}. In general, γ_{n} = σ_{n−2}^{−1}σ_{n−1}^{−1}γ_{n−1}, and so

(2.6.2.2)  γ_{n} = β_{n}

for all n, by induction on n.

To prove (2.6.2), we now have

σ_{1}ξ_{1}σ_{1}ξ_{1} = σ_{1}τσ_{n−1}^{−1}σ_{n−2}^{−1} … σ_{2}^{−1}τσ_{n−1}^{−1}σ_{n−2}^{−1} … σ_{1}^{−1}
= σ_{1}τ^{2}σ_{n−2}^{−1}σ_{n−3}^{−1} … σ_{1}^{−1}σ_{n−1}^{−1}σ_{n−2}^{−1}σ_{n−3}^{−1} … σ_{1}^{−1}
= τ^{2}σ_{n−1}σ_{n−2}^{−1}σ_{n−3}^{−1} … σ_{1}^{−1}σ_{n−1}^{−1}σ_{n−2}^{−1} … σ_{1}^{−1}
= τ^{2}β_{n}.

But

ξ_{1}σ_{1}ξ_{1}σ_{1} = τσ_{n−1}^{−1}σ_{n−2}^{−1} … σ_{2}^{−1}τσ_{n−1}^{−1}σ_{n−2}^{−1} … σ_{2}^{−1}
= τ^{2}σ_{n−2}^{−1}σ_{n−3}^{−1} … σ_{1}^{−1}σ_{n−1}^{−1}σ_{n−2}^{−1} … σ_{2}^{−1}
= τ^{2}γ_{n}
= τ^{2}β_{n} by (2.6.2.2)
= σ_{1}ξ_{1}σ_{1}ξ_{1},

which proves (2.6.2).

It follows immediately from (2.6.2) and (2.6.1) that the family {ξ_{1}, σ_{1}, …, σ_{n−1}} satisfies the relations (BRB). But the relations (BR) and τσ_{i}τ^{−1} = σ_{i+1} are similarly shown to follow from (BRB), whence we have a presentation of Γ_{n}.

It remains to show that the ξ_{i} commute with each other. Note that (2.6.2) says that ξ_{1}ξ_{2} = ξ_{2}ξ_{1}. In general, suppose that we know ξ_{i}ξ_{j} = ξ_{j}ξ_{i} for all j > i, 1 ≤ i < k. Since ξ_{k} = σ_{k−1}ξ_{k−1}σ_{k−1}, if j > k, ξ_{j} commutes with σ_{k−1} by (2.6.1), and with ξ_{k−1} by induction, and hence with ξ_{k}. Hence by induction the ξ_{i} all commute. To see that there are no relations among the ξ_{i}, one may use the braid picture as follows. In any cylindrical braid of the form ξ_{1}^{m_{1}} … ξ_{n}^{m_{n}}, note that each string joins points on the top and bottom of the cylinder which have the same label, since this is true of each ξ_{i}. Given the braid, the indices m_{i} may be recovered as the number of times the relevant string winds around the cylinder. □

(2.7) Corollary. The exact sequence (2.4) is realised as

1 ⟶ Δ_{n} ⟶ Γ_{n} ⟶ ℤ ⟶ 1,

where Δ_{n} ⟶ Γ_{n} is inclusion and Γ_{n} ⟶ ℤ is the map taking τ^{r}σ_{i_{1}}^{n_{1}} … σ_{i_{l}}^{n_{l}} ∈ Γ_{n} to r ∈ ℤ.

Our final result in this section is

(2.8) Lemma. We have the following relation in Γ_{n}.

τ^{n} = ξ_{1}ξ_{2} … ξ_{n}.

Proof. Since ξ_{1} = τσ_{n−1}^{−1}σ_{n−2}^{−1} … σ_{1}^{−1} and ξ_{i+1} = σ_{i}ξ_{i}σ_{i}, one shows easily by induction on i that for 1 ≤ i ≤ n − 1,

ξ_{1}ξ_{2} … ξ_{i} = τ^{i}σ_{n−i}^{−1} … σ_{1}^{−1}σ_{n−i+1}^{−1} … σ_{2}^{−1} … σ_{n−1}^{−1} … σ_{i}^{−1}.

Hence ξ_{1}ξ_{2} … ξ_{n−1} = τ^{n−1}σ_{1}^{−1} … σ_{n−1}^{−1} = τ^{n}ξ_{n}^{−1}. □

3. Affine Hecke algebras of type A

Let R be a commutative ring, denote by R^{×} the group of invertible elements of R, and let q ∈ R^{×}. We maintain the notation of the last section, so that Γ_{n} is the Artin braid group B(B_{n}) of type B_{n} and Δ_{n} is the Artin braid group B({ \atop A_{n−1}}) of type { \atop A_{n−1}}, regarded as the subgroup of Γ_{n} generated by σ_{1}, …, σ_{n}. Denote by RΓ_{n} the group ring of Γ_{n} over R.

(3.1) Definition. Let S_{i} be the element S_{i} = (σ_{i} − q)(σ_{i} + q^{−1}) of RΓ_{n} (i = 1, 2, …, n). The affine Hecke algebra Ĥ_{n}^{a}(q) of GL_{n} over R is defined by

Ĥ_{n}^{a}(q) = RΓ_{n}/⟨S_{1}⟩.

Note that since S_{1}, …, S_{n} are all conjugate in ℝΓ_{n}, the ideal ⟨S_{1}⟩ is equal to ⟨S_{1}, …, S_{n}⟩. Let η : RΓ_{n} ⟶ Ĥ_{n}^{a}(q) be the natural map. We then write

(3.2)  \begin{aligned}η(σ_{i}) & = T_{i} for i = 1, …, n \\ η(ξ_{i}) & = X_{i} for i = 1, …, n \\ η(τ) & = V. \end{aligned}

The next proposition collects some well known facts concerning Ĥ_{n}^{a}(q), many of which may be found in §3 of [L3].

(3.3) Proposition.

(i) The elements T_{1}, …, T_{n} generate a subalgebra H_{n}^{a}(q) of Ĥ_{n}^{a}(q), which has R-basis {T_{w} | w ∈ W^{a} ≅ Sym_{n} ⋉ ℤ^{n−1}}, where, if w = s_{i_{1}} … s_{i_{ℓ}} is a reduced expression for w ∈ W^{a}, T_{w} = T_{i_{1}} … T_{i_{ℓ}}. We refer to this as the “unextended” Hecke algebra of type { \atop A_{n−1}}.

(ii) The elements T_{1}, …, T_{n−1} generate a subalgebra H_{n}(q) of Ĥ_{n}^{a}(q) which has (finite) R-basis {T_{w} | w ∈ W ≅ Sym_{n}}.

(iii) We have Ĥ_{n}^{a}(q) ≅ Rℤ ⊗ H_{n}^{a}(q) ≅ R⟨V ⟩ ⊗ H_{n}^{a}(q), where the tensor product is twisted, using the action of V on H_{n}^{a}(q): V T_{i}V^{−1} = T_{i+1}, where the subscript is taken mod n.

(iv) We have Ĥ_{n}^{a}(q) ≅ H_{n}(q) ⊗ R[X_{1}^{±1}, …, X_{n}^{±1}] as R-module, and the multiplication is given by the “Bernstein relations”: for i ∈ {1, …, n − 1}, write s_{i} for the corresponding simple reflection in W and ^{s_{i}}f for the image of f ∈ R[X_{1}^{±1}, …, X_{n}^{±1}] under the natural action of W ≅ Sym_{n}. Then

T_{i}f − (^{s_{i}}f)T_{i} = (q − q^{−1})\frac{f − (^{s_{i}}f)}{1 − X_{i}X_{i+1}^{−1}} .

(v) The centre Z(Ĥ_{n}^{a}(q)) is the ring of symmetric functions in the X_{i}^{±1}. Equivalently, Z(Ĥ_{n}^{a}(q)) = R[Σ_{1}^{±1}, …, Σ_{n}^{±1}], where Σ_{i} is the i^{th} elementary symmetric function in the X_{i}.

Note that some authors use notation which results in the denominator of (iv) above being 1 − X_{i+1}X_{i}^{−1}.

We remark that to prove the relation in (iv) from those given in §2 for the braid group, one observes that the relation is linear in f, and hence need only be proved for monomials; moreover one easily shows that if the relation holds for f_{1} and f_{2}, then it holds for f_{1}f_{2}. Thus one is reduced to proving the relation for f = X_{j}, which is easy.

In addition to the algebras above, we shall need to consider the (finite rank) Hecke algebra of type B, which arises as follows. Let W B := Sym_{n} ⋉ (ℤ/2ℤ)^{n} be the hyperoctahedral group. This is generated as Coxeter group by {s_{1}, …, s_{n−1}}, together with another generator s_{0}. The generators {s_{0}, s_{1}, …, s_{n−1}} are involutions, and satisfy the relations analogous to (BRB) above. Let Q ∈ R^{×}. The Hecke algebra HB_{n}(q, Q) of type B_{n} with parameters (q, Q) is defined as

HB_{n}(q, Q) =Ĥ_{n}^{a}(q)/⟨(X_{1} − Q)(X_{1} + Q^{−1})⟩
=RΓ_{n}/⟨(ξ_{1} − Q)(ξ_{1} + Q^{−1}), (σ_{1} − q)(σ_{1} + q^{−1})⟩.

(3.4) Proposition. Let η_{Q} : Ĥ_{n}^{a}(q) ⟶ HB_{n}(q, Q) be the natural map. Write T_{i} ∈ HB_{n}(q, Q) for the image of T_{i} ∈ Ĥ_{n}^{a}(q) under η_{Q} (i = 1, …, n−1) (relying on the context to distinguish between them), and write T_{0} = η_{Q}(X_{1}). Then HB_{n}(q, Q) has R-basis {T_{w} | w ∈ W B}, where, if w = s_{i_{1}} … s_{i_{ℓ}} is a reduced expression for w ∈ W B, T_{w} = T_{i_{1}} … T_{i_{ℓ}}.

The relationship among the various algebras introduced so far is illustrated in the commutative diagram below.

(3.5)

H_{n}(q)
↓^{incl}
RΔ_{n} ⟶^{η} H_{n}^{a}(q)
↓^{incl}  ↓^{incl}
RΓ_{n} ⟶^{η}  Ĥ_{n}^{a}(q)
↓^{η_{Q}}
HB_{n}(q, Q)

The relations discussed in §2 for the braid groups Γ_{n} and Δ_{n} may be interpreted in the Hecke algebras as follows.

(3.6) Lemma. The following relations hold in Ĥ_{n}^{a}(q).

(3.6.1)  V^{n} = X_{1}X_{2} … X_{n},
(3.6.2)  V = X_{1}T_{1}T_{2} … T_{n−1}.

If we write V for η_{Q}(V) and adopt the notation of (3.4), we also have (in HB_{n}(q, Q))

(3.6.3)  V = T_{0}T_{1} … T_{n−1}.

4. Affine and finite dimensional Temperley-Lieb algebras

Let W = ⟨s_{1}, …, s_{n−1}⟩ ≅ Sym_{n} as above and write W_{i} = ⟨s_{i}, s_{i+1}⟩ ≅ Sym_{3} for i = 1, 2, …, n − 2. Define the element E_{i} ∈ H_{n}(q) ⊂ H_{n}^{a}(q) ⊂ Ĥ_{n}^{a}(q) by

E_{i} = ∑_{w∈W_{i}} q^{ℓ(w)}T_{w}

where ℓ(w) denotes the usual length function. Let I (resp. Î) denote the ideal of H_{n}^{a}(q) (resp. Ĥ_{n}^{a}(q)) generated by E_{1}. Note that since the E_{i} are all conjugate, this is the same as the ideal generated by all the E_{i}.

(4.1) Definition. The affine Temperley-Lieb algebras T L_{n}^{a}(q) and T̂ L_{n}^{a}(q) are defined by

T L_{n}^{a}(q) = H_{n}^{a}(q)/I
T̂ L_{n}^{a}(q) = Ĥ_{n}^{a}(q)/Î

It is known (cf. [GL2, GL3]) that if C_{i} = −(T_{i} + q^{−1}) ∈ H_{n}^{a}(q) (i = 1, …, n), then in H_{n}^{a}(q), C_{i}C_{i+1}C_{i} − C_{i} = C_{i+1}C_{i}C_{i+1} − C_{i+1} = −q^{3}E_{i}, where the indices are taken mod n. If we abuse notation by writing C_{i} ∈ T L_{n}^{a}(q) for the image of C_{i} ∈ H_{n}^{a}(q) under the natural map, it follows easily that T L_{n}^{a}(q) is generated by {C_{1}, …, C_{n}} subject to the relations

(TL)  \begin{aligned}C_{i}^{2} & = δ_{q}C_{i} \\ C_{i}C_{i±1}C_{i} & = C_{i} \\ C_{i}C_{j} & = C_{j}C_{i} if |i − j| ≥ 2 and \{i, j\} ≠ \{1, n\}, \end{aligned}

where, for any element x ∈ R^{×}, δ_{x} := −(x + x^{−1}).

Moreover it is easy to see that (cf. (3.3)(iii) above, or [GL3, §2])

(4.2)  T̂ L_{n}^{a}(q) ≅ R⟨V ⟩ ⊗ T L_{n}^{a}(q),

where V permutes the C_{i} cyclically.

Now in addition to the algebras T L_{n}^{a}(q) and T̂ L_{n}^{a}(q), we shall require the algebra T_{n}^{a}(q) which was defined in [GL2, (2.7)] and referred to there (loc. cit.) as “the affine Temperley–Lieb algebra”. This is defined as an algebra of diagrams or, more accurately, as the algebra of morphisms: n ⟶ n in the category 𝕋^{a} (see [GL2, (2.5)]) and T L_{n}^{a}(q) is identified [GL2, (2.9)] as the subalgebra of T_{n}^{a}(q) spanned by the “non-monic diagrams : n ⟶ n of even rank”, together with the identity. It also occurs independently in the work of Green [Gr] and Fan–Green [FG]. We shall need to make some use of the diagrammatic description in this work; details may be found in [GL2], but a good approximation to the picture is obtained if one thinks of affine diagrams as arcs drawn on the surface of a cylinder joining 2n marked points, n on each circle component of the boundary, in pairs. The arcs must not intersect, and diagrams are multiplied by concatenation in the usual way. These diagrams are represented by periodic diagrams drawn between two horizontal lines, each diagram being determined by the “fundamental rectangle”, from which the cylinder is obtained by identifying vertical edges. In this interpretation, the generators {f_{1}, …, f_{n}, τ} of T_{n}^{a}(q) are represented by the diagrams shown below.

• •  • • •  •  • • • • • • • • •  • 1  n 1 2  n − 1 n 2  n − 1  i i + 1 i i + 1 f_{n} : n → n  f_{i} : n → n, 1 ≤ i ≤ n − 1 ..................................................................................................................  .................................................................................................................. ........................................................... ...........................................................  ........................................................... ...........................................................  .........................................................................................  .................................................................................................................. .................................................................................................................. .........................................................................................

• • •  • • •  • • 1 2 3  t 1 2  t − 1 t τ_{t} : t → t  τ_{0} : 0 → 0 ................................................................................................................................................................................................................................................................................................................................  ................................................................................................................................ ................................................................  ..............................................................................................................................................................................................................................................................................................................................................................................

The elements {f_{1}, …, f_{n}} of the algebra T_{n}^{a}(q) satisfy the relations (TL), with C_{i} replaced by f_{i}, and it is noted in [GL2, (2.9)] that these generate an algebra isomorphic to T L_{n}^{a}(q). Further, τf_{i}τ^{−1} = f_{i+1}, where the index is taken mod n.

(4.3) Proposition.

(i) The is a family of surjections ϕ_{α} : T̂ L_{n}^{a}(q) ⟶ T_{n}^{a}(q) (α ∈ R^{×}), defined by ϕ_{α}(C_{i}) = f_{i} and ϕ_{α}(V) = ατ. Each ϕ_{α} restricts to a monomorphism on T L_{n}^{a}(q).

(ii) The kernel of ϕ_{α} is generated by the element ν_{α} ∈ T̂ L_{n}^{a}(q), where ν_{α} = α^{−2}V^{2}C_{n−1} − C_{1}C_{2} … C_{n−1} = α^{−2}C_{1}V^{2} − C_{1}C_{2} … C_{n−1}.

(iii) If R is an algebraically closed field of characteristic prime to n, any irreducible finite dimensional representation of T̂ L_{n}^{a}(q) is the pullback via ϕ_{α} (for some α ∈ R^{×}) of an irreducible representation of T_{n}^{a}(q).

Proof. The first part of (i) follows immediately from the relations above, while the second follows from the fact (cf. [GL2, 2.9]) that (TL) gives a presentation of T L_{n}^{a}(q). Next, one verifies easily that τ^{2}f_{n−1} = f_{1}f_{2} … f_{n−1} in T_{n}^{a}(q) (see [GL3, 1.11]), which shows that ν_{α} ∈ Kerϕ_{α}. The fact that ν_{α} generates the kernel may be found in [Gr] or [FG]. This relation also appears in [J1]. The statement (iii) may be proved using the argument of Theorem 2.6 in [GL3]. □

In [GL2], we defined cell modules W_{t,z}(n) for the algebra T_{n}^{a}(q), (where t ∈ ℤ, 0 ≤ t ≤ n, t + n ∈ 2ℤ, and z ∈ R^{×}) and when R is an algebraically closed field of characteristic zero, completely determined their composition factors and multiplicities. Our purpose here is to interpret these results for the pullbacks ϕ_{α}^{∗}W_{t,z}(n). To identify these pullbacks as standard modules for Ĥ_{n}^{a}(q) up to Grothendieck equivalence (cf. [KL] or [Z]), we need to determine the action of the translation elements X_{i} on the modules, and for this we shall require the Temperley Lieb algebra T LB_{n}(q, Q) of type B_{n}.

5. The Temperley-Lieb algebras of type B

This algebra has been studied by mathematical physicists ([MW], [MS]), where it is referred to as the “blob algebra”, and in [tD] (see also [CGM], and the references there). We shall present here the main facts which we require concerning T LB_{n}(q, Q), relying for general background on op. cit.. Our notation continues from §3 above, and we start with an algebraic definition of T LB_{n}(q, Q).

Note first that the Hecke algebra HB_{n}(q, Q) has an R-algebra homomorphism ϵ : HB_{n}(q, Q) ⟶ R, defined on the generators T_{i} (i = 0, 1, …, n − 1, see (3.4)) by ϵ(T_{0}) = Q, ϵ(T_{i}) = q for 1 ≤ i ≤ n − 1. As above, let s_{0}, s_{1}, …, s_{n−1} be the simple generators of the hyperoctahedral group W B, and write W_{i} = ⟨s_{i}, s_{i+1}⟩ for i = 0, 1, …, n − 2. Then define E_{i} = ∑_{w∈W_{i}} ϵ(T_{w})T_{w}. For i ≠ 0, these E_{i}

coincide with the E_{i} of §4, and they are all conjugate in HB_{n}(q, Q). For i > 0, let C_{i} = −(T_{i} + q^{−1}), and let C_{0} = −(T_{0} + Q^{−1}). The next lemma summarises several relevant relations in HB_{n}(q, Q).

(5.1) Lemma.

(i) For i = 1, …, n − 2, C_{i}C_{i+1}C_{i} − C_{i} = −q^{3}E_{i}.

(ii) For i = 2, …, n − 1, C_{i}C_{i−1}C_{i} − C_{i} = −q^{3}E_{i}.

(iii) Let X = C_{1}C_{0}C_{1} − κC_{1}, where κ = \frac{q}{Q} + \frac{Q}{q}. Then XC_{0} = C_{0}X =

q^{−2}Q^{−2}E_{0}.

(iv) We have C_{i}^{2} = δ_{q}C_{i} for i ≠ 0, while C_{0}^{2} = δ_{Q}C_{0}.

The proofs are simple computations in HB_{n}(q, Q).

(5.2) Definition. The Temperley-Lieb algebra T LB_{n}(q, Q) of type B_{n} with parameters (q, Q) is defined as

T LB_{n}(q, Q) := HB_{n}(q, Q)/⟨E_{1}, X⟩ = HB_{n}(q, Q)/⟨E_{1}, E_{2}, …, E_{n−2}, X⟩

Note that the kernel of the natural map Ĥ_{n}^{a}(q) ⟶ T LB_{n}(q, Q) contains the kernel of the natural map Ĥ_{n}^{a}(q) ⟶ T̂ L_{n}^{a}(q). Hence the former map factors through a surjection T̂ L_{n}^{a}(q) ⟶ T LB_{n}(q, Q). This is reflected in the diagram (5.6) below.

Denote by lower case letters the images in T LB_{n}(q, Q) of the corresponding elements of Ĥ_{n}^{a}(q) (or T̂ L_{n}^{a}(q) or HB_{n}(q, Q)). Thus in particular we write t_{i}, c_{i} respectively for the images in T LB_{n}(q, Q) of T_{i}, C_{i} ∈ HB_{n}(q, Q) under the natural map (i = 0, 1, 2, …, n). Similarly, we have elements x_{i} (i = 1, …, n) and v ∈ T LB_{n}(q, Q). The next statement is easy to verify from the foregoing discussion.

(5.3) Proposition.

(a) The Temperley-Lieb algebra T LB_{n}(q, Q) is generated as R-algebra by the family {c_{0}, c_{1}, …, c_{n−1}} subject to the relations

(TLB)

c_{0}^{2} = δ_{Q}c_{0}
c_{i}^{2} = δ_{q}c_{i} for 1 ≤ i ≤ n − 1
c_{i}c_{i+1}c_{i} = c_{i} for 1 ≤ i ≤ n − 2
c_{i}c_{i−1}c_{i} = c_{i} for 2 ≤ i ≤ n − 1
c_{1}c_{0}c_{1} = κc_{1},

where κ = \frac{q}{Q} + \frac{Q}{q} and δ_{x} = −(x + x^{−1}) for x ∈ R^{×}.

(b) The elements c_{1}, …, c_{n−1} generate a subalgebra of T LB_{n}(q, Q) which is isomorphic to the usual Temperley-Lieb algebra TL_{n}(q) of type { \atop A_{n−1}}.

(c) The following relations hold in T LB_{n}(q, Q).

(i) t_{0} = −(c_{0} + Q^{−1}) and t_{i} = −(c_{i} + q^{−1}) for i = 1, …, n − 1.

(ii) If v ∈ T LB_{n}(q, Q) is the image of V ∈ HB_{n}(q, Q) under the natural map, then v = t_{0}t_{1} … t_{n−1}.

(iii) x_{1}x_{2} … x_{n} = v^{n}.

A diagram of maps relating the algebras we have now introduced is as follows.

(5.6)

H_{n}(q) ⟶^{η_{1}} T L_{n}(q)
↓^{incl}  ↓^{incl}
RΔ_{n} ⟶^{η} H_{n}^{a}(q) ⟶^{η_{2}} T L_{n}^{a}(q)
↓^{incl}  ↓^{incl}  ↓^{incl}
RΓ_{n} ⟶^{η}  Ĥ_{n}^{a}(q) ⟶^{η_{3}}  T̂ L_{n}^{a}(q) ⟶^{ϕ_{α}} T_{n}^{a}(q)
↓^{η_{Q}}  ↓^{γ_{Q}}
HB_{n}(q, Q) ⟶^{η_{4}} T LB_{n}(q, Q)

where the η_{i} are the natural surjections and γ_{Q} exists because the kernel of η_{4} ∘ η_{Q} contains E_{1} ∈ Ĥ_{n}^{a}(q), which generates the kernel of η_{3}. Thus η_{4} ∘η_{Q} factors through T̂ L_{n}^{a}(q).

The algebra T LB_{n}(q, Q) has a decription in terms of marked diagrams (see [MW, MS, tD]) which we now describe, because this description is convenient for the discussion of cell modules. We shall use the language of [GL2] for diagrams. Recall that if t, n are positive integers of the same parity, a finite (planar) diagram μ : t ⟶ n is represented by a set of non-intersecting arcs which are contained in the “fundamental rectangle” (see below). These arcs divide the fundamental rectangle into regions, among which there is a unique “left region” as shown below.

n • • • • • • • • • • • • • • • • ......................................................................................................................................  ......................................................................................................................................................................................................................................................................................................... ...................................................................................................................................................................................................................................................................................................................................................................................... .....................................................................................  ..................................................................................... .............................................................................................................................................................................................................................................................. .....................................................................................  ......................................................................................................................................t left region

A marked diagram is a (finite planar) diagram, where the interior of the boundary arcs of the leftmost region may be marked with one or more • symbols (“marks”) (see below).

n • • • • • • • • • • • • • • • • ......................................................................................................................................  ......................................................................................................................................................................................................................................................................................................... ...................................................................................................................................................................................................................................................................................................................................................................................... .....................................................................................  ..................................................................................... .............................................................................................................................................................................................................................................................. .....................................................................................  ...................................................................................................................................... •  • •  t left region

The R-linear combinations of diagrams from t to n constitute the morphisms in the Temperley-Lieb category 𝕋, where the objects are the non negative integers ℤ_{≥0}. Composition is defined by concatenation of diagrams, with closed loops being deleted and replaced by the scalar δ_{q}. In particular, Hom_{𝕋}(n, n) ≅ T L_{n}(q). Marked diagrams may be similarly concatenated according to rules we shall now state; this produces a new category, 𝕋𝔹, the Temperley-Lieb category of type B. The composition rules are as follows.

A marked diagram is proper if it has no loops and each arc has at most one mark. The following rules reduce the concatenation of any two diagrams to an R linear combination of proper diagrams:

(5.7.1) If μ is a diagram and L is a loop with no marks, μ ⨿ L = δ_{q}μ.

(5.7.2) If, in (i), L has one mark, μ ⨿ L = κμ, where κ = \frac{q}{Q} + \frac{Q}{q}.

(5.7.3) If some arc of μ has more than one mark and μ′ is the diagram obtained by removing a mark from the arc concerned, then μ = δ_{Q}μ′.

Now consider the following marked diagrams from n to n.

• •  • • •  •  • • • • • • • • •  • 1  n 1 2  n − 1 n 2  n − 1  i i + 1 i i + 1 •  c_{0} : n → n  c_{i} : n → n, 1 ≤ i ≤ n − 1 ..................................................................................................................  .................................................................................................................. ..................................................................................................................  ..................................................................................................................  .........................................................................................  .................................................................................................................. .................................................................................................................. .........................................................................................

(5.8) Proposition. The diagrams labelled c_{0}, c_{1}, …, c_{n−1} satisfy the relations for the generators of T LB_{n}(q, Q) given in (5.3). Moreover T LB_{n}(q, Q) is faithfully represented in this way as End_{𝕋𝔹}(n) in the category 𝕋𝔹.

Proof. The relations among diagrams c_{i} are easily checked using the rules (5.7). It follows that there is a homomorphism : T LB_{n}(q, Q) ⟶ End_{𝕋𝔹}(n), which is easily seen to be surjective, since any proper marked diagram n → n can be obtained by composing the c_{i}. The injectivity of this homomorphism follows from arguments in [tD] or [GLa]. □

Let us now focus attention on the bottom right part of the commutative diagram (5.6). Recall that under the map η_{Q}, X_{1} ∈ Ĥ_{n}^{a}(q) is mapped to T_{0} ∈ T LB_{n}(q, Q).

But for any element μ ∈ R^{×}, we may deform both η_{Q} and γ_{Q} to η_{Q,μ} and γ_{Q,μ} respectively, where η_{Q,μ}(X_{1}) = μT_{0}, η_{Q,μ}(T_{i}) = T_{i} for 1 ≤ i ≤ n − 1, γ_{Q,μ}(V) = μγ_{Q}(V) = μt_{0}t_{1} … t_{n−1} and γ_{Q,μ}(C_{i}) = c_{i} for 1 ≤ i ≤ n − 1. We then obtain a “deformed” version of the bottom right part of the commutative diagram (5.6) as follows.

(5.9)  Ĥ_{n}^{a}(q)  T̂ L_{n}^{a}(q)  T_{n}^{a}(q) η_{Q,μ}  γ_{Q,μ} η_{3}  ϕ_{α} HB_{n}(q, Q)  T LB_{n}(q, Q) η_{4}

We shall study the pullbacks to Ĥ_{n}^{a}(q) of cell modules of T_{n}^{a}(q) by realising the latter as pullbacks of cell modules of T LB_{n}(q, Q) via a homomorphism yet to be defined. With the objective of defining a map : T_{n}^{a}(q) → T LB_{n}(q, Q) which makes the diagram (5.9) commute, we shall prove

(5.10) Theorem. Let t_{i}, c_{i} and v be the elements of T LB_{n}(q, Q) defined just before (5.3) above. Then for n ≥ 2 we have

(5.10.1)  c_{1}v^{2} = −q^{−(n−2)}c_{1}c_{2} … c_{n−1}.

Proof. This will be by induction on n. We first verify the statement for n = 2. The left side of (5.10.1) is then

c_{1}(t_{0}t_{1})^{2}

=c_{1}t_{0}t_{1}t_{0}t_{1}
=c_{1}t_{1}t_{0}t_{1}t_{0} since the t_{i} satisfy the braid relations (BRB)
=qc_{1}t_{0}t_{1}t_{0} since c_{i}t_{i} = qc_{i}
= − qc_{1}(c_{0} + Q^{−1})(c_{1} + q^{−1})(c_{0} + Q^{−1})
= − qc_{1} (c_{0}c_{1}c_{0} + q^{−1}δ_{Q}c_{0} + Q^{−1}(c_{0}c_{1} + c_{1}c_{0}) + 2q^{−1}Q^{−1}c_{0} + Q^{−2}c_{1} + q^{−1}Q^{−2})
= − q(κc_{1}c_{0} + q^{−1}δ_{Q}c_{1}c_{0} + Q^{−1}(κc_{1} + δ_{q}c_{1}c_{0})
+ 2q^{−1}Q^{−1}c_{1}c_{0} + Q^{−2}δ_{q}c_{1} + q^{−1}Q^{−2}c_{1})
= − qc_{1}c_{0} (κ + q^{−1}δ_{Q} + Q^{−1}δ_{q} + 2q^{−1}Q^{−1}) − qc_{1} (Q^{−1}κ + Q^{−2}δ_{q} + q^{−1}Q^{−2})
= − c_{1},

which proves (5.10.1) for the case n = 2.

Now suppose n ≥ 3. Then

c_{1}v^{2} = c_{1}t_{0}t_{1} … t_{n−1}t_{0}t_{1} … t_{n−1}
= c_{1}t_{0}t_{1} … t_{n−2}t_{0}t_{1} … t_{n−3}t_{n−1}t_{n−2}t_{n−1}

= c_{1}t_{0}t_{1} … t_{n−2}t_{0}t_{1} … t_{n−3}t_{n−2}t_{n−1}t_{n−2} by the braid relations (BRB)

= c_{1}(t_{0}t_{1} … t_{n−2})^{2}t_{n−1}t_{n−2}
= −q^{−(n−3)}c_{1}c_{2} … c_{n−2}t_{n−1}t_{n−2} by induction on n
= −q^{−(n−3)}c_{1}c_{2} … c_{n−2}(c_{n−1} + q^{−1})(c_{n−2} + q^{−1})
= −q^{−(n−3)}c_{1}c_{2} … c_{n−3}(c_{n−2} + q^{−1}c_{n−2}c_{n−1} + q^{−1}δ_{q}c_{n−2} + q^{−2}c_{n−2})
= −q^{−(n−2)}c_{1}c_{2} … c_{n−1} by (5.3),

which completes the proof. □

(5.11) Corollary. For each element β ∈ R such that β^{2} = −q^{n−2}, there is a (unique) surjective homomorphism g_{β} : T_{n}^{a}(q) ⟶ T LB_{n}(q, Q) such that g_{β}(f_{i}) = c_{i} for i = 1, 2, …, n − 1, and g_{β}(τ) = βv. If α, μ satisfy α^{−1}μ = β, then the following diagram commutes.

(5.11.1)  Ĥ_{n}^{a}(q)  T̂ L_{n}^{a}(q)  T_{n}^{a}(q) η_{Q,μ}  γ_{Q,μ} η_{3}  ϕ_{α} HB_{n}(q, Q)  T LB_{n}(q, Q) η_{4}  g_{β}

Proof. For i = 1, 2, …, n − 1, the surjection γ_{Q,μ} takes C_{i} ∈ T̂ L_{n}^{a}(q) to c_{i} ∈ T LB_{n}(q, Q), while γ_{Q,μ}(V) = μv = μt_{0}t_{1} … t_{n−1}. Hence by Theorem (5.10), C_{1}V^{2} + μ^{2}q^{−(n−2)}C_{1}C_{2} … C_{n−1} ∈ Ker(γ_{Q,μ}). But Ker(ϕ_{α}) is generated by ν_{α} = α^{−2}C_{1}V^{2} − C_{1}C_{2} … C_{n−1}. Hence Ker(γ_{Q,μ}) ⊃ Ker(ϕ_{α}), provided that

(5.11.2)  α^{2} = −μ^{2}q^{−(n−2)}.

If this condition is satisfied, γ_{Q,μ} : T̂ L_{n}^{a}(q) → T LB_{n}(q, Q) factors through ϕ_{α} : T̂ L_{n}^{a}(q) → T_{n}^{a}(q); i.e., there is a map g_{β} : T_{n}^{a}(q) → T LB_{n}(q, Q) which makes the diagram (5.11.1) commute. For this map, we have by commutativity, g_{β}(f_{i}) = γ_{Q,μ}(C_{i}) = c_{i} for i = 1, 2, …, n − 1, and g_{β}(τ) = γ_{Q,μ}(α^{−1}V) = βv, where β = α^{−1}μ satisfies (5.11.2), i.e. β^{2} = −q^{n−2}. □

Note that the homomorphism g_{β} depends of course on Q. We shall write g_{β} = g_{β,Q} when we need to refer to this dependence.

Suppose now that we have a triple (α, μ, β) of elements of R^{×} such that β^{2} = −q^{n−2} and β = α^{−1}μ. By (5.11), we have a commutative diagram

(5.12)  Ĥ_{n}^{a}(q)  T_{n}^{a}(q) ψ_{α} T LB_{n}(q, Q) ξ_{Q,μ}  g_{β}

where, in the notation of (5.11.1), ψ_{α} = ϕ_{α} ∘ η_{3} and ξ_{Q,μ} = γ_{Q,μ} ∘ η_{3} = η_{4} ∘ η_{Q,μ}.

For future reference, we summarise the definitions of the maps in (5.12) in the following equations.

(5.13)

ξ_{Q,μ}(X_{1}) = μt_{0}
ξ_{Q,μ}(T_{i}) = t_{i} for i = 1, 2, …, n − 1
ψ_{α}(V = X_{1}T_{1} … T_{n−1}) = ατ
ψ_{α} (C_{i} = −(T_{i} + q^{−1})) = f_{i} for i = 1, 2, …, n − 1
g_{β}(f_{i}) = c_{i} = −(t_{i} + q^{−1}) for i = 1, 2, …, n − 1
g_{β}(τ) = βv = βt_{0}t_{1} … t_{n−1}.

Our next objective is to identify the pullbacks via ψ_{α} of the cell modules W_{t,z}(n) of T_{n}^{a}(q) as standard modules M_{s,N} of Ĥ_{n}^{a}(q) up to Grothendieck equivalence. We shall do this by identifying W_{t,z}(n) as the pullback of a cell module W_{t}(n) of T LB_{n}(q, Q) via g_{β} for appropriate Q, and using the commutative diagram (5.12). We therefore need to discuss the cellular structure of T LB_{n}(q, Q), which we do in the next section.

6. Cell modules for T_{n}^{a}(q) and T LB_{n}(q, Q)

The cell modules W_{t,z}(n) of T_{n}^{a}(q) were defined and analysed in [GL2]. We shall freely use the notation and language of that work here. There is an entirely analogous “cellular theory” for the algebras T LB_{n}(q, Q) which we shall now sketch. Recall ([GL1]) that to specify a cellular structure for T LB_{n}(q, Q), we require (i) a poset T , (ii) for each t ∈ T , a set M(t), and (iii) an injection ⨿_{t∈T} M(t)×M(t) ⟶^{C} T LB_{n}(q, Q), whose image is an R-basis {C_{S,T}^{t} } of T LB_{n}(q, Q) which satisfies

aC_{S,T}^{t} = ∑_{S′∈M(t)} r_{a}(S′, S)C_{S′,T}^{t} + lower terms (corresponding to t′ < t)
(a ∈ T LB_{n}(q, Q), r_{a}(S′, S) ∈ R).

We now specify the various elements of a cellular structure for T LB_{n}(q, Q). Take T = {t ∈ ℤ | |t| ≤ n, t ≡ n (mod 2)}, partially ordered as follows: t ≤ s if |t| < |s| or |t| = |s| and t ≤ s.

To define M(t), first take t ∈ T , t ≥ 0. Then M(t) is the set of monic diagrams D : t → n with no marked through strings, where “monic” means that there are t through strings, as in [GL2], where it is shown that this is equivalent to D being a monic morphism in the category-theoretic sense. In general, let M(t) = M(|t|). Then C : M(t) × M(t) → T LB_{n}(q, Q) is defined as follows. Let S, T ∈ M(t). For t ≥ 0, define C_{S,T}^{t} = S ∘ T^{∗}, where ^{∗} denotes reflection in a horizontal axis. For t < 0, define C_{S,T}^{t} = S ∘ c_{o} ∘ T^{∗}, where c_{0} = c_{0}(t) : t → t is the generator shown above (5.8). This is the diagram S ∘ T^{∗}, with the leftmost through string marked. The cellular axioms above are easily checked.

The cell modules W_{t}(n) are now defined in complete analogy with the W_{t,z}(n) of [GL2]. For any t ∈ T , W_{t}(n) has basis M(t). If t ≥ 0, T LB_{n}(q, Q) acts via composition in the category 𝕋𝔹; explicitly, if D ∈ M(t) and ω ∈ T LB_{n}(q, Q), then ω.D = ωD (composition in 𝕋𝔹) if ω ∘ D ∈ M(t), and ω.D = 0 otherwise.

For t < 0, one may think of W_{t}(n) as having basis the set {D ∘ c_{0}(t) | D ∈ M(|t|)} of monic diagrams : t → n in 𝕋𝔹 with the leftmost through string marked. Then the action of T LB_{n}(q, Q) is essentially multiplication in 𝕋𝔹, as in the case t ≥ 0. Thus if t < 0, then ω.(D ∘ c_{0}(t)) = 0 if ω ∘ D is not monic, while ω.(D ∘ c_{0}) = ωDc_{0} (composition in 𝕋𝔹) if ω ∘ D is monic.

It is easily seen (cf. [tD]) that the dimension (i.e. rank over R) of W_{t}(n) is

\binom{n}{\frac{n−|t|}{2}}, which is the same as that of W_{t,z}(n). We now wish to identify the inflation

of W_{t}(n) via the surjection g_{β} : T_{n}^{a}(q) → T LB_{n}(q, Q).

(6.1) Theorem. Suppose R is any commutative ring and suppose that q, Q are elements of R^{×}, and that δ_{q} = −(q + q^{−1}) is not a zero-divisor. Let β ∈ R^{×} satisfy β^{2} = −q^{n−2} and let g_{β} : T_{n}^{a}(q) → T LB_{n}(q, Q) be the surjection defined in (5.11). For t ∈ ℤ such that 0 ≤ t ≤ n and t ≡ n (mod 2), write

z_{t} = (−1)^{t}βQ^{−1}q^{− \frac{1}{2} (n+t−2)}.

If W_{t}(n) is the cell module for T LB_{n}(q, Q) described above, then the inflation g_{β}^{∗}W_{t}(n) (i.e. the pullback via g_{β} of W_{t}(n)) is isomorphic to W_{t,z}(n), where z = z_{t}.

The proof will depend on a sequence of lemmas.

(6.2) Remark. Observe first that the finite rank Temperley-Lieb algebra T L_{n}(q) is naturally a subalgebra of both T_{n}^{a}(q) and T LB_{n}(q, Q); in T_{n}^{a}(q) it is generated by f_{1}, …, f_{n−1}, and has R-basis consisting of “finite diagrams” from n to n, i.e., diagrams with no arcs crossing the boundary of the fundamental rectangle (see [GL2, §2]). In T LB_{n}(q, Q), it is generated by c_{1}, …, c_{n−1} and has basis consisting of unmarked diagrams from n to n. These two sets of diagrams may be identified in the obvious way, and the relations (5.13) show that g_{β} maps a diagram in T L_{n}(q) ⊂ T_{n}^{a}(q) to the same diagram, regarded as a diagram in T LB_{n}(q, Q). A similar observation applies more generally to affine and marked diagrams t to n, where finite and unmarked diagrams respectively may be identified with each other.

Recall ([GL2]) that an affine diagram from t to n is monic if it has t through strings. Let X_{t} be the T_{n}^{a}(q)-module with basis all monic affine diagrams:t → n, with T_{n}^{a}(q) action given by composition in the category 𝕋^{a}, modulo diagrams with fewer than t through strings. Thus X_{t} may be thought of as a quotient of the left T_{n}^{a}(q)-module Hom_{𝕋^{a}}(t, n) by the submodule spanned by diagrams with fewer than t through strings. The T_{n}^{a}(q) module W_{t,z}(n) is defined as the quotient of X_{t} by the ideal

(6.3)  I_{χ} := {γτ_{t} − χγ}

for γ ∈ X_{t}, where χ = χ_{z} = z if t ≠ 0 and χ = χ_{z} = z + z^{−1} if t = 0.

We shall define a homomorphism θ : X_{t} → g_{β}^{∗}W_{t}(n) which we shall prove factors through W_{t,z}(n) for appropriate z. To do this, we require the following diagrams, which, being finite, may (and will) be alternately thought of as lying in the categories 𝕋^{a} and 𝕋𝔹.

(6.4) Definition. First suppose t > 0. Define monic finite diagrams D_{t}, D′_{t} : t → n as depicted below.

n • • • • • • • • • • • • ......................................................................................................................................  ......................................................................................................................................  ...................................................................................................................................... ............................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................................. t D_{t}

n • • • • • • • • • • • • ......................................................................................................................................  ......................................................................................................................................  ...................................................................................................................................... .............................................................................................................................................................................................................................................................................................  ............................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................................. t D′_{t}

In D_{t}, (t > 0) the t through strings are joined to the rightmost t top nodes, while in D′_{t} they are joined to the rightmost t − 1 and leftmost top nodes. Note that D_{t} and D′_{t} are finite diagrams and that

(6.4.1)  D_{t}^{∗}D′_{t} = (D′_{t})^{∗}D_{t} = id_{t}.

When t = 0 the corresponding diagrams are as below.

n • • • • • • • • • • ......................................................................................................................................  ......................................................................................................................................  ......................................................................................................................................  ......................................................................................................................................  ...................................................................................................................................... D_{0}

n • • • • • • • • • • ......................................................................................................................................  ......................................................................................................................................  ......................................................................................................................................  ...................................................................................................................................... ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ D′_{0}

In this case

(6.4.2)  (D′_{0})^{∗}D_{0} = D_{0}^{∗}D′_{0} = δ_{q}id_{0},

(6.5) Definition. Let X_{t} be the T_{n}^{a}(q)-module with basis the set of all monic diagrams t → n (as explained above). Define θ : X_{t} → Z_{t} := g_{β}^{∗}W_{t}(n) by the formula

(6.5.1)  θ(γ) = g_{β}(γD_{t}^{∗})D′_{t},

where γ ∈ X_{t}, and D_{t}^{∗} and D′_{t} are regarded as diagrams in 𝕋^{a} and 𝕋𝔹 respectively (see Remark (6.2)).

(6.6) Lemma.

(i) (cf. Remark (6.2)) If γ ∈ X_{t} is a finite diagram, then provided that t > 0, we have θ(γ) = γ; i.e., θ(γ) is the same diagram as γ, regarded as a marked diagram in W_{t}(n). If t = 0, then θ(γ) = δ_{q}γ.

(ii) Suppose that for some element χ ∈ R, we know that θ(γτ_{t}) = χθ(γ) for all finite diagrams γ ∈ X_{t}. Then θ(γτ_{t}) = χθ(γ) for all γ ∈ X_{t}; i.e., Ker(θ) ⊃ I_{χ}.

Proof. Suppose γ ∈ X_{t} is a finite diagram. Then γD_{t}^{∗} is also finite, and hence is in T L_{n}(q) ⊂ T_{n}^{a}(q). By Remark (6.2), any diagram in T L_{n}(q) is mapped by g_{β} to the same diagram, regarded as an element of T LB_{n}(q, Q), whence θ(γ) = γD_{t}^{∗}D′_{t} = γ, which proves (i) for t > 0. The same argument may be used for the case t = 0, taking into account that D_{0}^{∗}D′_{0} = δ_{q}id_{0}.

(ii) Clearly any diagram γ ∈ X_{t} has a factorisation γ = ωγ_{0}, where γ_{0} ∈ X_{t} is finite, and ω is a diagram in T_{n}^{a}(q) (in fact more is true: X_{t} is cyclic as T_{n}^{a}(q)- module, generated by any one of many finite diagrams). Hence

θ(γτ_{t}) =θ(ωγ_{0}τ_{t})
=g_{β}(ω)θ(γ_{0}τ_{t}) since θ is a T_{n}^{a}(q)-module homomorphism
=χg_{β}(ω)θ(γ_{0})
=χθ(ωγ_{0})
=χθ(γ),

as required. □

(6.7) Corollary. The homomorphism θ : X_{t} → g_{β}^{∗}W_{t}(n) = Z_{t} of T_{n}^{a}(q) modules is surjective if t > 0, and has image precisely δ_{q}Z_{t} when t = 0.

Proof. By (6.6), if t > 0, the image of θ contains all finite diagrams t → n. But these generate W_{t,z}(n) as T_{n}^{a}(q)-module, whence the first statement. If t = 0, then again by (6.6), for any finite diagram γ, θ(γ) ∈ δ_{q}Z_{t}. Since the finite diagrams generate Z_{t} as T_{n}^{a}(q)-module, it follows that the image is contained in δ_{q}Z_{t}. But since the image contains δ_{q}γ for each finite diagram, the result is clear. □

We shall show that θ factors through W_{t,z}(n) for appropriate z ∈ R^{×} using the method of (6.6)(ii), i.e. by showing that Ker(θ) ⊃ I_{χ} for χ = χ_{z}, z = z_{t}. We therefore investigate the right action of τ_{t} on X_{t}.

Recall (5.13) that g_{β}(τ_{n}) = βt_{0}t_{1} … t_{n−1}, where τ_{n} is thought of as an element of T_{n}^{a}(q). Thus to carry through the strategy of (6.6), we shall need the following computation in 𝕋𝔹. We adopt the following notation for diagrams in 𝕋𝔹: we say that a diagram is positive if the leftmost through string is unmarked, and negative otherwise. If E is a positive diagram (e.g. in W_{t}(n)), Ė will denote the diagram obtained from E by marking the leftmost through string.

(6.8) Lemma. Suppose t > 0. Then in the category 𝕋𝔹, we have

βt_{n−1}t_{n−2} … t_{1}t_{0}D′_{t} = z_{t} (D_{t} + QḊ_{t} + E)

where D_{t}, D′_{t} are the diagrams defined above and E is a linear combination of diagrams with < t through strings. The right side of this equation is to be thought of as an element of Hom_{𝕋𝔹}(t, n).

Proof. Recall that for 1 ≤ i ≤ n − 1, t_{i} = −(c_{i} + q^{−1}), while t_{0} = −(c_{0} + Q^{−1}). Hence t_{0}D′_{t} = −Ḋ′_{t} − Q^{−1}D′_{t}. Thus we need to evaluate t_{n−1}t_{n−2} … t_{1}D′_{t} and t_{n−1}t_{n−2} … t_{1}Ḋ′_{t}. For the former, we have t_{1}D′_{t} = −D′_{t}(n − 2) − q^{−1}D′_{t}, where D′_{t}(n − 2) is the diagram obtained from D′_{t} by moving the leftmost through string past the leftmost horizontal arc. Now c_{2}D′_{t}(n − 2) = D′_{t} and c_{2}D′_{t} = δ_{q}D′_{t}, from which it follows that

(6.8.1)  t_{2}t_{1}D′_{t} = q^{−1}D′_{t}(n − 2).

Repeating this argument k times, where 2k = n − t, we obtain

(6.8.2)  t_{2k}t_{2k−1} … t_{2}t_{1}D′_{t} = q^{−k}D_{t}.

But for i > 2k one sees easily that t_{i}D_{t} ≡ −q^{−1}D_{t} modulo diagrams with fewer than t through strings. Hence

(6.8.3) t_{n−1}t_{n−2} … t_{1}D′_{t} = (−1)^{t−1}q^{−t−k+1}D_{t} + E_{1} = (−1)^{t−1}q^{− \frac{n+t−2}{2}} D_{t} + E_{1}

where E_{1} is a linear combination of diagrams with < t through strings. The same computation shows that (6.8.3)

t_{n−1}t_{n−2} … t_{1}(Ḋ′_{t}) = (−1)^{t−1}q^{−t−k+1}(Ḋ_{t}) + E_{2} = (−1)^{t−1}q^{− \frac{n+t−2}{2}} (Ḋ_{t}) + E_{2}

where E_{2} is a linear combination of diagrams with < t through strings. It follows that

βt_{n−1}t_{n−2} … t_{1}t_{0}D′_{t} = βt_{n−1}t_{n−2} … t_{1} (t_{0}D′_{t} = −(Ḋ′_{t}) − Q^{−1}D′_{t})
= β(−1)^{t} (q^{− \frac{n+t−2}{2}} (Ḋ_{t}) + Q^{−1}q^{− \frac{n+t−2}{2}} D_{t}) + E
= z_{t}D_{t} + Qz_{t}(Ḋ_{t}) + E,

where E is a combination of diagrams with < t through strings. □

We now turn to the case t = 0

Recall that the diagrams D_{0}, D′_{0} have been defined above, before the statement of Lemma (6.4). The following relations are easily verified.

(6.9)

(D′_{0})^{∗}D_{0} = (D_{0})^{∗}D′_{0} = δ_{q} id_{0}, and
(c_{0}D′_{0})^{∗}D_{0} = (D′_{0})^{∗}c_{0}D_{0} = κ id_{0} = (\frac{Q}{q} + \frac{q}{Q}) id_{0}

where id_{0} is the empty diagram 0 → 0, in either the category 𝕋^{a} or 𝕋𝔹. Note that although in the discussion above it is always assumed that n ≥ 1, from the point of view of diagram algebras, the point 0 is an object in both categories 𝕋^{a} and 𝕋𝔹. The corresponding algebras are TLB_{0}(q, Q) = R.id_{0} ≅ R and T_{0}^{a}(q) = R⟨τ_{0}⟩ ≅ Rℕ.

(6.10) Lemma. Let γ ∈ X_{0} be a finite diagram. Then

θ(γτ_{0}) = βδ_{q}γ(D′_{0})^{∗}t_{0}t_{1} … t_{n−1}D′_{0}.

Proof. First, observe that in the category 𝕋^{a}, we have the following relation.

(6.10.1)  τ_{0} = D_{0}^{∗}τ_{n}D_{0} = (D′_{0})^{∗}τ_{n}D′_{0}.

It follows that for finite γ ∈ X_{0},

θ(γτ_{0}) = g_{β}(γτ_{0}D_{0}^{∗})D′_{0}
= g_{β}(γ(D′_{0})^{∗}τ_{n}D′_{0}D_{0}^{∗})D′_{0} by (6.10.1)
= g_{β}(γ(D′_{0})^{∗}τ_{n})g_{β}(D′_{0}D_{0}^{∗})D′_{0}
= g_{β}(γ(D′_{0})^{∗}τ_{n})D′_{0}D_{0}^{∗}D′_{0} since D′_{0}D_{0}^{∗} is finite
= δ_{q}g_{β}(γ(D′_{0})^{∗}τ_{n})D′_{0} by (6.9)
= δ_{q}g_{β}(γ(D′_{0})^{∗})g_{β}(τ_{n})D′_{0}
= δ_{q}βγ(D′_{0})^{∗}t_{0}t_{1} … t_{n−1}D′_{0} since γ(D′_{0})^{∗} is finite
□

(6.11) Lemma. We have

t_{1}t_{2} … t_{n−1}D′_{0} = q^{− \frac{n−4}{2}} D_{0}.

Proof. Denote by D′_{0}(n − 2) the diagram from 0 to n which looks like D′_{0} : 0 → n − 2, with the rightmost top vertices joined by a horizontal arc. Then clearly c_{n−1}D′_{0} = D′_{0}(n − 2). Further, c_{n−2}D′_{0} = δ_{q}D′_{0}, and c_{n−2}D′_{0}(n − 2) = D′_{0}. Thus t_{n−1}D′_{0} = −(c_{n−1} + q^{−1})D′_{0} = −(D′_{0}(n − 2) + q^{−1}D′_{0}). Hence

t_{n−2}t_{n−1}D′_{0} = (c_{n−2} + q^{−1})(D′_{0}(n − 2) + q^{−1}D′_{0})
= D′_{0} + q^{−1}δ_{q}D′_{0} + q^{−1}D′_{0}(n − 2) + q^{−2}D′_{0}
= q^{−1}D′_{0}(n − 2).

Repeating this argument \frac{n − 2}{2} times, we obtain

t_{1}t_{2} … t_{n−1}D′_{0} = q^{− \frac{n−2}{2}} t_{1}D_{0}.

But c_{1}D_{0} = δ_{q}D_{0}, whence t_{1}D_{0} = −(δ_{q} + q^{−1})D_{0} = qD_{0}, and the Lemma follows. □

Proof of Theorem (6.1). We show first that for any element γ ∈ X_{t} (t ≥ 0), we have

(6.1.1)  θ(γτ_{t}) = χ_{z}θ(γ),

where z = z_{t} = (−1)^{t}βQ^{−1}q^{− \frac{1}{2} (n+t−2)} and

(6.1.2)  χ_{z} = \{ \begin{matrix}z if t > 0 \\ z + z^{−1} if t = 0. \end{matrix}

By (6.6)(ii), it suffices to prove (6.1.1) for any finite diagram γ ∈ X_{t}.

We prove (6.1.1) first for t > 0. Recall that if E is a positive morphism (diagram) in 𝕋𝔹, Ė is obtained from E by marking the leftmost through string. Then observe that in the above notation, D_{t}^{∗}D′_{t} = id_{t}, while Ḋ_{t}^{∗}D′_{t} = i̇d_{t}, and if we regard W_{t}(n) as a quotient of the bimodule Hom_{𝕋𝔹}(t, n) in the obvious way, then in W_{t}(n), γid_{t} = γ, while γi̇d_{t} = 0. Moreover, if E ∈ 𝕋𝔹 is a linear combination of diagrams with fewer that t through strings, then γE = 0 by definition. Hence

θ(γτ_{t}) =g_{β}(γτ_{t}D_{t}^{∗})D′_{t} by (6.5.1)
=g_{β}(γ(D′_{t})^{∗}τ_{n})D′_{t} since τ_{t}D_{t}^{∗}τ_{n}^{−1} = (D′_{t})^{∗}
=g_{β}(γ(D′_{t})^{∗})g_{β}(τ_{n})D′_{t}
=βγ(D′_{t})^{∗}t_{0}t_{1} … t_{n−1}D′_{t} by (6.5.1) again, and (5.13)
=γ(βt_{n−1}t_{n−2} … t_{1}t_{0}D′_{t})^{∗}D′_{t} since ^{∗} is an anti-involution
=γχ_{z} (D_{t} + Q(Ḋ_{t}) + E)^{∗} D′_{t} by (6.8)
=χ_{z}θ(γ),

which proves (6.1.1) for t > 0. Now take t = 0 and repeat the above computation using Lemmas (6.10,6.11) instead of (6.8). One obtains

θ(γτ_{0}) =βδ_{q}γ(D′_{0})^{∗}t_{0}t_{1} … t_{n−1}D′_{0} by (6.10)
=βδ_{q}γ(D′_{0})^{∗}t_{0}q^{− \frac{n−4}{2}} D_{0} by (6.11)
= − βδ_{q}q^{− \frac{n−4}{2}} γ(D′_{0})^{∗}(c_{0} + Q^{−1})D_{0}
= − βδ_{q}q^{− \frac{n−4}{2}} γ (κid_{0} + Q^{−1}δ_{q}id_{0})
= − (\frac{Q}{q} − Q^{−1}q^{−1})βδ_{q}q^{− \frac{n−4}{2}} γ
= − (Q − Q^{−1})βq^{− \frac{n−2}{2}} θ(γ) by (6.7)(ii).

But z_{0} = βQ^{−1}q^{− \frac{n−2}{2}} , and since β^{2} = −q^{n−2}, β^{−1} = −q^{−(n−2)}β. It follows easily that θ(γτ_{0}) = (z_{0} + z_{0}^{−1})θ(γ), which completes the proof of (6.1.1).

It follows from (6.1.1) (cf. (6.7)) that θ induces a homomorphism which we also denote by θ : W_{t,z}(n) → Z_{t}, where z = z_{t}. Note that W_{t,z}(n) and Z_{t} are free R-modules of the same rank. To complete the proof of the theorem, we discuss the cases t > 0 and t = 0 separately.

If t > 0, then by (6.7), θ is surjective. It follows, since W_{t,z}(n) and Z_{t} are free R-modules of the same rank, that θ is an isomorphism. If t = 0, the same argument (using (6.7)) shows that θ defines an isomorphism W_{t,z}(n) → δ_{q}Z_{t}. But since δ_{q} is not a zero-divisor, δZ_{t} ≅ Z_{t} as T_{n}^{a}(q)-modules, which completes the proof. □

Theorem (6.1) shows in particular that any cell module W_{t,z}(n) for the R-algebra T_{n}^{a}(q) may be realised as the pullback of a cell module of the algebra T LB_{n}(q, Q).

(6.12) Corollary. For any (relevant) pair t, z, the cell module W_{t,z}(n) may be realised as g_{β,Q}^{∗}W_{t}(n), where Q = Q(t, z) = (−1)^{t}βz^{−1}q^{− \frac{1}{2} (n+t−2)}.

Recall from the beginning of this section that we also have cell modules W_{t}(n) for T LB_{n}(q, Q), for t < 0. We next identify their lift to T_{n}^{a}(q) via g_{β,Q}.

(6.13) Theorem. Suppose δ_{q} is not a zero divisor in R and let t satisfy 0 ≤ t ≤ n, t ≡ n(mod 2). Then the inflation

g_{β}^{∗}W_{−t}(n) ≅ W_{t,q^{−t}z_{t}^{−1}}(n) = W_{t,−Q^{2}z_{t}}(n)

where z_{t} = (−1)^{t}βQ^{−1}q^{− \frac{1}{2} (n+t−2)} as in (6.1).

Proof. The case t = 0 has been treated above. Note that z_{0} = βQ^{−1}q^{− \frac{1}{2} (n−2)}, so that z_{0}^{2} = −Q^{−2}. Thus −Q^{2}z_{0} = z_{0}^{−1}, and since W_{0,z}(n) = W_{0,z^{−1}}(n), the case t = 0 follows from Theorem (6.1). We therefore take t > 0.

First observe that in analogy with (6.5), we have a homomorphism of T_{n}^{a}(q)- modules θ^{−} : X_{t} → g_{β}^{∗}W_{−t}(n) given by

(6.13.1)  θ^{−}(γ) = g_{β}(γD_{t}^{∗})(Ḋ′_{t}),

where γ ∈ X_{t}, and D_{t}^{∗} and Ḋ′_{t} = D′_{t} ∘ c_{0}(t) are the diagrams defined in the proof of (6.3), regarded as diagrams in 𝕋^{a} and 𝕋𝔹 respectively (recall (see immediately preceding (6.8)) that for a diagram E with unmarked through strings, Ė denotes the same diagram with the leftmost through string marked. One verifies easily that for a finite diagram γ : t → n(∈ X_{t}) we have

(6.13.2)  θ^{−}(γ) = (γ̇).

In accordance with the strategy of the proof of Theorem (6.1), to prove the theorem it will suffice to show that for any finite diagram γ ∈ X_{t}, we have

(6.13.3)  θ^{−}(γτ_{t}) = q^{−t}z_{t}^{−1}θ^{−}(γ) = q^{−t}z_{t}^{−1}(γ̇).

Arguing as in the proof of (6.1), we have

(6.13.4)

θ^{−}(γτ_{t}) = g_{β}(γτ_{t}D_{t}^{∗})(Ḋ′_{t})
=g_{β}(γ(D′_{t})^{∗}τ_{n})(Ḋ′_{t}) since τ_{t}D_{t}^{∗}τ_{n}^{−1} = (D′_{t})^{∗}
=γ(D′_{t})^{∗}g_{β}(τ_{n})(Ḋ′_{t})
=γ(D′_{t})^{∗}βt_{0}t_{1} … t_{n−1}(Ḋ′_{t})
=γ(βt_{n−1}t_{n−2} … t_{1}t_{0}D′_{t})^{∗}(Ḋ′_{t}) since ^{∗} is an anti-involution
=γz_{t} (D_{t} + Q(Ḋ_{t}) + E)^{∗} (Ḋ′_{t}) by (6.8),

where E is a linear combination of diagrams with fewer than t through strings. Now a straightforward computation shows that

(Ḋ_{t}^{∗})(Ḋ′_{t}) = δ_{Q}(i̇d_{t}) and D_{t}^{∗}(Ḋ′_{t}) = i̇d_{t}.

Hence from (6.13.4) we have

θ^{−}(γτ_{t}) = γz_{t} (D_{t} + Q(Ḋ_{t}) + E)^{∗} (Ḋ′_{t})
= z_{t}γ(i̇d_{t})(1 + Qδ_{Q})
= −Q^{2}z_{t}γ(i̇d_{t})
= −Q^{2}z_{t}γ̇.

But z_{t}^{2} = −q^{−t}Q^{−2}, whence q^{−t}z_{t}^{−1} = −Q^{2}z_{t}, and (6.13.3) follows. This completes the proof of Theorem (6.13). □

Notice that for any t, if z_{t} = (−1)^{t}βQ^{−1}q^{− \frac{1}{2} (n+t−2)}, then z_{−t} = q^{t}z_{t}. Hence in particular if t ≤ 0, q^{t}z_{−t}^{−1} = z_{t}^{−1}. Hence Theorem (6.13) may be stated as follows.

(6.14) Corollary. Suppose t ∈ T and t ≤ 0. Then the inflation g_{β}^{∗}W_{t}(n) ≅ W_{−t,z_{t}^{−1}}(n).

We may combine and generalise the statements (6.1) and (6.13) as follows. Note that the condition on δ_{q} is absent from the statement below.

(6.15) Corollary. Suppose R is any commutative ring and suppose that q, Q are elements of R^{×}. Let β ∈ R^{×} satisfy β^{2} = −q^{n−2} and let g_{β} : T_{n}^{a}(q) → T LB_{n}(q, Q) be the surjection defined in (5.11). For t ∈ ℤ such that |t| ≤ n and t ≡ n (mod 2),

define ϵ_{t} := \frac{t}{|t|} for t ≠ 0, and ϵ_{t} = 1 if t = 0.

Then the inflation g_{β}^{∗}W_{t}(n) of the cell module W_{t}(n) of T LB_{n}(q, Q) is isomorphic to the cell module W_{|t|,z_{t}^{ϵ_{t}}} (n) of T_{n}^{a}(q), where z_{t} is as defined in the statement of (6.1).

Proof. Replace R in the statement of (6.1) or (6.13) by the ring R[q, q^{−1}] of Laurent polynomials in the indeterminate q and replace q ∈ R by q ∈ R[q, q^{−1}]. Then all algebras and modules may be considered over R[q, q^{−1}], and the hypotheses of (6.1) or (6.13) apply and we deduce that there is a (unique) isomorphism of R[q, q^{−1}]- modules : g_{β}^{∗}W_{t}(n) → W_{|t|,z_{t}^{ϵ_{t}}} (n) which takes a finite diagram in W_{t}(n) to the same diagram in W_{|t|,z_{t}^{ϵ_{t}}} (n). But R is a module over R[q, q^{−1}] via the homomorphism which takes q to q. Tensoring the above isomorphism with R provides the required isomorphism. □

7. Eigenvalues of the translation elements X_{i}

Recall that we have a 1-parameter family of surjections ψ_{α} : Ĥ_{n}^{a}(q) ⟶ T_{n}^{a}(q) and that we wish to study the inflations ψ_{α}^{∗}W_{t,z}(n). Now by Corollary (6.12), W_{t,z}(n) may be realised as a pullback via g_{β} = g_{β,Q} of some cell module W_{t}(n) (for t ≥ 0) of T LB_{n}(q, Q). But by the commutativity of the diagram (5.12),

(7.1)  ψ_{α}^{∗}W_{t,z}(n) ≅ ψ_{α}^{∗}g_{β}^{∗}W_{t}(n) ≅ ξ_{Q,μ}^{∗}W_{t}(n),

where α, β and μ are related by β^{2} = −q^{n−2} and μα^{−1} = β.

Thus we study the structure of the pullbacks ψ_{α}^{∗}W_{t,z}(n) by means of the pullbacks ξ_{Q,μ}^{∗}W_{t}(n). For this we shall require some easy facts concerning the submodule structure of W_{t}(n) on restriction to T LB_{n−1}(q, Q). For 1 ≤ i ≤ n, we regard T LB_{i}(q, Q) as the subalgebra generated by {c_{0}, c_{1}, …, c_{i−1}}; in terms of diagrams, it is the subalgebra spanned by diagrams with the rightmost n − i top and bottom vertices joined by unmarked vertical through strings. Throughout this section we assume that δ_{q} is not a divisor of zero in R, since we shall require Theorems (6.1) and (6.13).

(7.2) Proposition. Let t ∈ ℤ, 0 ≤ |t| ≤ n, n + t ∈ 2ℤ.

(i) If t ≥ 0, we have a short exact sequence

(7.2.1)  0 → W_{t−1}(n − 1) → Res_{T LB_{n−1}(q,Q)}^{T LB_{n}(q,Q)}W_{t}(n) → W_{t+1}(n − 1) → 0.

(ii) If t < 0, we have a short exact sequence

(7.2.2)  0 → W_{t+1}(n − 1) → Res_{T LB_{n−1}(q,Q)}^{T LB_{n}(q,Q)}W_{t}(n) → W_{t−1}(n − 1) → 0.

Here we adopt the convention that W_{j}(k) = 0 if |j| > k.

Proof. (i) If t > 0, the diagrams in W_{t}(n) in which the rightmost top and bottom vertices are joined span a T LB_{n−1}(q, Q)- submodule which is clearly isomorphic to W_{t−1}(n − 1). The quotient module is spanned by the images of the other diagrams, which all have the property that the top right vertex is joined to another top vertex. Define a homomorphism W_{t}(n)/W_{t−1}(n − 1) → W_{t+1}(n − 1) by sending a basis diagram ν of W_{t}(n)\W_{t−1}(n − 1) to the diagram t+1 → n−1 obtained by moving the top right vertex to the bottom right without changing any arcs. One checks easily that this is an isomorphism of T LB_{n−1}(q, Q)-modules.

If t = 0, the diagrams in W_{t}(n) = W_{0}(n) such that the arc from the rightmost vertex is marked span a T LB_{n−1}(q, Q)-submodule which is clearly isomorphic to W_{−1}(n−1) (imagine the rightmost vertex as the bottom vertex in a marked diagram 1 → n − 1). The quotient W_{0}(n)/W_{−1}(n − 1) is spanned by the other diagrams, and by a similar argument is isomorphic as T LB_{n−1}(q, Q)-module to W_{1}(n − 1).

(ii) If t < 0, the same construction as in (i) (but with care if t = −1, in which case we take the submodule spanned by diagrams with the rightmost top and bottom vertices joined by a marked string) provides a W_{t−1}(n − 1)-submodule of W_{t}(n) which is isomorphic to W_{−(|t|−1)}(n − 1) = W_{t+1}(n − 1), with corresponding quotient W_{t−1}(n − 1). □

(7.3) Corollary. Let t ∈ ℤ, 0 ≤ |t| ≤ n, n + t ∈ 2ℤ and suppose t ≥ 0. There is a filtration of W_{t}(n) by R-submodules W^{(i)}, i = n, n − 1, n − 2, …, 1 as in (7.3.1) below.

W_{t}(n) ⊃ W_{t−1}(n − 1) ⊃ ⋯ ⊃ W_{0}(n − t) ⊃ W_{−1}(n − t − 1) ⊃ W_{0}(n − t − 2) ⊃ W_{−1}(n − t − 3) ⊃ ⋯ ⊃ W_{0}(2) ⊃ W_{−1}(1). (7.3.1)

Thus

W^{(i)} ≅ \{ \begin{matrix}W_{t−n+i}(i) if n − t ≤ i ≤ n \\ W_{0}(i) if i is even and 0 ≤ i ≤ n − t \\ W_{−1}(i) if i is odd and 0 ≤ i ≤ n − t. \end{matrix}

For each i = 1, 2, 3, …, n, W^{(i)} is a T LB_{i}(q, Q)-submodule of W_{t}(n).

Proof. This is immediate by repeated application of (7.2), which also gives an explicit description of the W^{(i)} in terms of the diagrams of W_{t}(n) which span them. For the reader’s convenience, we give this description here. For 0 ≤ i ≤ n − t, W^{(i)} is the R-submodule spanned by diagrams with the rightmost n − i top vertices joined to the rightmost bottom vertices by through strings. The “remaining” part of the diagram may then be thought of as an element of W_{t−n+i}(i). In particular W^{(n−t)} ≅ W_{0}(n − t).

For i = n − t − 1, we take W^{(i)} to be the R submodule spanned by diagrams in W^{(n−t)} where the (n−t)^{th} top vertex is on a marked arc. By thinking of this (n−t)^{th} vertex as moved to the bottom, it is clear that W^{(n−t−1)} ≅ W_{−1}(n − t − 1). The next term W^{(n−t−2)} is spanned by diagrams in W^{(n−t−1)} such that the top nodes n−t and n−t−1 are joined (with a marked arc). Clearly W^{(n−t−2)} ≅ W_{0}(n−t−2), and we may now repeat the above construction until the sequence terminates. □

To illustate the construction in (7.3), observe that the diagram E_{t} = E_{t,n} below lies in each of the submodules W^{(i)}.

n • • • • • • • • • • • • •  •  • ......................................................................................................................................  ......................................................................................................................................  ...................................................................................................................................... ............................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................................. t E_{t}

We shall now determine the eigenvalues of the translation elements X_{i} ∈ Ĥ_{n}^{a}(q) on the inflation ξ_{Q,μ}^{∗}W_{t}(n) (cf. (7.1)).

(7.4) Proposition. The element X_{1}X_{2} … X_{n} ∈ Ĥ_{n}^{a}(q) acts on the module M = ξ_{Q,μ}^{∗}W_{t}(n) as multiplication by the scalar

(7.4.1)  h(n, t) = (−1)^{\frac{n+t}{2}} μ^{n}Q^{−t}q^{\frac{1}{2} (2n−n^{2}−t^{2})}.

Proof. First take t ≥ 0. By (7.1), ξ_{Q,μ}^{∗}W_{t}(n) ≅ ψ_{α}^{∗}W_{t,z}(n), where z = z_{t}. But by (3.6.1), X_{1}X_{2} … X_{n} = V^{n}, and by (5.13), ψ_{α}(V^{n}) = α^{n}τ^{n} ∈ T_{n}^{a}(q). Moreover τ^{n} is central in T_{n}^{a}(q) and acts on W_{t,z}(n) as multiplication by z^{t}. Hence X_{1}X_{2} … X_{n} acts on ξ_{Q,μ}^{∗}W_{t}(n) as multiplication by the scalar

α^{n}z_{t}^{t} = α^{n} ((−1)^{t}βQ^{−1}q^{− \frac{1}{2} (n+t−2)})^{t} ,

which after simplification using the relations β^{2} = −q^{n−2} and β = α^{−1}μ is easily shown to be equal to h(n, t).

If t < 0, the proof is the same, but we use Theorem (6.13) instead of (6.1). □

For the remainder of this section, we take t ≥ 0.

The inflation ξ_{Q,μ}^{∗}W_{t}(n) defines a structure of Ĥ_{n}^{a}(q)-module on W_{t}(n), and for i = 1, 2, …, n, the submodule W^{(i)} of the filtration (7.3.1) is stable under the action of the subalgebra T LB_{i}(q, Q) which is the image under ξ_{μ,Q} of the subalgebra Ĥ_{i}^{a}(q) of Ĥ_{n}^{a}(q) which is generated by {X_{1}, T_{1}, …, T_{i−1}}. We shall refer to the submodules W^{(i)} as Ĥ_{i}^{a}(q)-modules, with the understanding that the action is induced by ξ_{μ,Q}. Now Ĥ_{i}^{a}(q) contains the elements {X_{1}, X_{2}, …, X_{i}} and it follows from (7.4) that

(7.5) Corollary. The element X_{1}X_{2} … X_{i} ∈ Ĥ_{n}^{a}(q) acts on the submodule W^{(i)} of ξ_{Q,μ}^{∗}W_{t}(n) as multiplication by the scalar β_{i}, where

β_{i} = \{ \begin{matrix}(−1)^{\frac{t−n+2i}{2}} μ^{i}Q^{n−t−i}q^{^{\frac{1}{2}}(2i−i^{2}−(t−n+i)^{2})} if n − t ≤ i ≤ n \\ (−1)^{\frac{i}{2}} μ^{i}q^{\frac{1}{2} (2i−i^{2})} if i is even and 1 ≤ i ≤ n − t \\ (−1)^{\frac{i−1}{2}} μ^{i}Qq^{\frac{1}{2} (2i−i^{2}−1)} if i is odd and 1 ≤ i ≤ n − t. \end{matrix}

Proof. Using the identification of W^{(i)} as a T LB_{i}(q, Q)-module given in (7.3), it follows from (7.4) that X_{1}X_{2} … X_{i} acts as a scalar β_{i} on W^{(i)}, and that in the three cases n − t ≤ i ≤ n, 1 ≤ i ≤ n − t and i even and 1 ≤ i ≤ n − t and i odd, the value of β_{i} is respectively given by h(i, t − n + i), h(i, 0) and h(i, −1). The formulae in the statement are the result. □

We shall determine the action of the X_{i} using the fact that

X_{i} = (X_{1}X_{2} … X_{i}) (X_{1}X_{2} … X_{i−1})^{−1} .

(7.6) Corollary. In the filtration of (7.3)

W_{t}(n) = W^{(n)} ⊃ ⋯ ⊃ W^{(i)} ⊃ W^{(i−1)} ⊃ ⋯ ⊃ W^{(1)} ⊃ 0,

where W^{(i)} is thought of as a Ĥ_{i}^{a}(q)-module, we have, for any pair of integers i, j such that 1 ≤ j < i ≤ n that X_{i} acts on W^{(j)} as multiplication by γ_{i}, where γ_{i} is given by

γ_{i} = \{ \begin{matrix}−μQ^{−1}q^{n+2−t−2i} if n − t ≤ i ≤ n \\ −μQ^{−1}q^{2−i} if i is even and 1 ≤ i ≤ n − t \\ μQq^{1−i} if i is odd and 1 ≤ i ≤ n − t. \end{matrix}

Proof. Since W^{(j)} is a submodule of W^{(i−1)}, it will suffice to show that X_{i} acts on W^{(i−1)} as multiplication by γ_{i}. But W^{(i−1)} is a submodule of W^{(i)}, and hence by (7.5), the elements Y_{i} = X_{1}X_{2} … X_{i} and Y_{i−1} = X_{1}X_{2} … X_{i−1} act as the scalars β_{i}, β_{i−1} respectively on W^{(i−1)} (here we take Y_{0} = 1). It follows that X_{i} = Y_{i}Y_{i−1}^{−1} acts on W^{(i−1)} as the scalar γ_{i} = β_{i}β_{i−1}^{−1}, where β_{0} = 1. An easy calculation now completes the proof. □

(7.7) Proposition. There is an ordering of the diagrams which form an R-basis of W_{t}(n) which is compatible with the filtration (7.3) and with respect to which each element X_{i} (i = 1, 2, …, n) has upper triangular matrix.

Proof. We use induction on n; for n = 1 the result is trivial. Assume the result true for W_{t−1}(n − 1) and W_{t+1}(n − 1). Then there is an ordering of the diagrams in W^{(n)} which is compatible with the filtration W^{(n)} ⊃ W^{(n−1)} ⊃ ⋯ ⊃ 0 such that X_{1}, …, X_{n−1} act in upper triangular fashion. This is because by induction we have such an ordering of the diagrams in W^{(n−1)}, while W^{(n)}/W^{(n−1)} ≅ W_{t−1}(n − 1) or W^{(n)}/W^{(n−1)} ≅ W_{t+1}(n − 1), whence again by induction, there is an appropriate ordering of the other diagrams in W_{t}(n). But by (7.4), Y_{n} := X_{1}X_{2} … X_{n} acts as a scalar on W_{t}(n), whence X_{n} = Y_{n}X_{1}^{−1}X_{2}^{−1} … X_{n−1}^{−1} is also upper triangular. □

(7.8) Definition.

(i) We denote by U(n) the subalgebra of Ĥ_{n}^{a}(q) generated by {X_{1}, …, X_{n}}. By (3.3) this is isomorphic to the ring R[X_{1}^{±1}, …, X_{n}^{±1}] of Laurent polynomials in the X_{i}.

(ii) A character χ : U(n) → R^{×} is the linear extension to U(n) of a multiplicative homomorphism χ : U(n)^{×} → R^{×}, where U(n)^{×} is the group generated by the X_{i}, i = 1, …, n (which is isomorphic to ℤ^{n}). Such a character is clearly determined by the images of the X_{i} and hence the group of characters may be identified with (R^{×})^{n}.

(iii) If χ : U(n) → R^{×} is a character, write σ(χ) = (χ(X_{1}), …, χ(X_{n})) ∈ (R^{×})^{n} and |σ(χ)| for the multiset (i.e. set with multiplicities) {χ(X_{1}), …, χ(X_{n})}. We sometimes identify σ(χ) with χ and refer to the sequence σ(χ) as a character.

Now any diagram D ∈ W_{t}(n) may be considered as an element of an ordered basis of W_{t}(n) as in (7.7). Hence D defines a character χ_{D} of U(n) by

(7.9)  X_{i}D ∈ χ_{D}(X_{i})D + ∑_{E∈W_{t}(n),E≠D} RE,

where the sum is over diagrams in W_{t}(n) which are distinct from (and ordered lower than) D.

(7.10) Theorem. Assume t ≥ 0. For any diagram D ∈ W_{t}(n), the character χ_{D} satisfies |σ(χ_{D})| = {γ_{1}, γ_{2}, …, γ_{n}}, where the γ_{i} are as in (7.6), i.e.,

{γ_{1}, …, γ_{n}} ={μQ, μQq^{−2}, μQq^{−4}, …, μQq^{−(n−t−2)}}
∪ {−μQ^{−1}, −μQ^{−1}q^{−2}, −μQ^{−1}q^{−4}, …, −μQ^{−1}q^{−(n+t−2)}},

where the union is in the sense of multisets.

Proof. This will be by induction on n. For n = 1 the result is trivial. Let D be any diagram in W^{(n)}. Then there is a unique index i such that D ∈ W^{(i)} \ W^{(i−1)}, where … W^{(i)} ⊃ W^{(i−1)} … is the filtration (7.3) of W_{t}(n). If i < n, it follows by induction on n that {χ_{D}(X_{1}), …, χ_{D}(X_{i})} = {γ_{j} = β_{j}β_{j−1}^{−1} | 1 ≤ j ≤ i}. Moreover for j > i, it follows from (7.6) that X_{j}D = γ_{j}D, since X_{j} acts as γ_{j}id on W^{(i)}. Thus the assertion (7.10) is true if i < n. Hence we consider the case i = n.

If t ≠ n, then by (7.2.1), W^{(n)}/W^{(n−1)} ≅ W_{t+1}(n − 1) as T LB_{n−1}(q, Q)-module, and hence

X_{1}X_{2} … X_{n−1}D ≡ h(n − 1, t + 1)D (mod W^{(n−1)}).

Hence χ_{D}(X_{n}) = h(n, t)h(n − 1, t + 1)^{−1} = μQq^{−(n−t−2)}. The set {χ_{D}(X_{i}) | 1 ≤ i ≤ n − 1} may be determined by thinking of the X_{i}, 1 ≤ i ≤ n − 1 as acting on the image of D in W^{(n)}/W^{(n−1)} ≅ W_{t+1}(n − 1). Then by induction we see that {χ_{D}(X_{i}) | 1 ≤ i ≤ n − 1} is the set in the statement of (7.1), with (n, t) replaced by (n − 1, t + 1), i.e.

{χ_{D}(X_{i}) | 1 ≤ i ≤ n − 1} = {γ_{1}, …, γ_{n}} \ {μQq^{−(n−t−2)}},

and hence {χ_{D}(X_{i}) | 1 ≤ i ≤ n − 1} is as stated in (7.1). This proves (7.10) for the case t ≠ n.

If t = n, W_{t}(n) ≅ W_{t−1}(n − 1) as T LB_{n−1}(q, Q)-module, whence

{χ_{D}(X_{i}) | 1 ≤ i ≤ n − 1} = {γ_{1}, …, γ_{n}} \ {−μQ^{−1}q^{−(2n−2)}}.

But X_{1}X_{2} … X_{n} ∈ Ĥ_{n}^{a}(q) acts on the module W_{n}(n) as multiplication by the scalar

h(n, n) = (−1)^{n}μ^{n}Q^{−n}q^{(n−n^{2})}.

Hence X_{n} = −μQ^{−1}q^{−(2n−2)}id on W_{n}(n), which completes the proof for the case t = n. □

8. The case of generic parameters

In this section we prove that the inflations ψ_{α}^{∗}W_{t,z}(n) ≅ ξ_{Q,μ}^{∗}W_{t}(n) are representations of Ĥ_{n}^{a}(q) which are induced from parabolic subalgebras, provided that the pair of parameters q, Q is “generic”. First we make precise the meaning of this term.

(8.1) Definition. We say that the pair (q, Q) of invertible elements is generic in the integral domain R if q is not a root of unity and there is no solution m ∈ ℤ for the equation q^{m} = ±Q^{2}.

(8.2) Proposition. Suppose that the pair (q, Q) is generic.

(i) The elements of the multiset {γ_{1}, …, γ_{n}} of (7.10) are distinct, i.e. have multiplicity one. Moreover if χ_{D}(X_{i}) = χ_{D}(X_{j})q^{−2}, then j < i.

(ii) The characters χ_{D} are distinct; i.e. if χ_{D_{1}} = χ_{D_{2}} for diagrams D_{1}, D_{2} ∈ W_{t}(n), then D_{1} = D_{2}.

Proof. Observe first that by (7.6), the sequence (γ_{1}, …, γ_{n}) is given by

(μQ, −μQ^{−1}, μQq^{−2}, −μQ^{−1}q^{−2}, …, μQq^{−(n−t−2)}, −μQ^{−1}q^{−(n−t−2)},

−μQ^{−1}q^{−(n−t)}, −μQ^{−1}q^{−(n−t+2)}, …, −μQ^{−1}q^{−(n+t−2)}). (8.2.1)

The first assertion of (i) is clear. For the second assertion, we use the argument in the proof of (7.10). If D ∈ W^{(i)} \ W^{(i−1)}, then

σ(χ_{D}) = (χ_{D}^{(i)}(X_{1}), χ_{D}^{(i)}(X_{2}), …, χ_{D}^{(i)}(X_{i}), γ_{i+1}, …, γ_{n}),

where χ_{D}^{(i)} is the character of U(i) = R[X_{1}^{±1}, …, X_{i}^{±1}] on W^{(i)} which corresponds to D ∈ W^{(i)}. Inspection of the sequence (8.2.1) (and induction on n) now yields the second assertion if i < n.

If i = n, the argument of (i) shows that

σ(χ_{D}) = (χ′_{D}(X_{1}), χ′_{D}(X_{2}), …, χ′_{D}(X_{n−1}), μQq^{−(n−t−2)}),

where χ′_{D} is the character of U(n − 1) corresponding to D ∈ W^{(n)}/W^{(n−1)} ≅ W_{t+1}(n − 1). Again by induction, the assertion is true for this sequence, whence the result. This proves (i).

The proof of (ii) is also by induction on n. The result is trivial for n = 1. Let D_{1}, D_{2} be distinct diagrams in W = W^{(n)}. If D_{1}, D_{2} are either both in W^{(n−1)} or both in W^{(n)} \ W^{(n−1)}, the result is immediate by induction. Thus we may take D_{1} ∈ W^{(n−1)} and D_{2} ∈ W^{(n)} \ W^{(n−1)}. But then χ_{D_{1}}(X_{1} … X_{n−1}) ≠ χ_{D_{2}}(X_{1} … X_{n−1}) by Theorem (7.4), since in the generic case h(n − 1, t − 1) ≠ h(n − 1, t + 1). This completes the proof of (8.2). □

(8.3) Corollary. Assume that the pair (q, Q) is generic (see (8.1)). Define sequences Γ_{1}, Γ_{2} as in (8.3.1) below.

(8.3.1)  \begin{aligned}Γ_{1} & = (μQ, μQq^{−2}, μQq^{−4}, …, μQq^{−(n−t−2)}) \\ Γ_{2} & = (−μQ^{−1}, −μQ^{−1}q^{−2}, −μQ^{−1}q^{−4}, …, −μQ^{−1}q^{−(n+t−2)}) \end{aligned}

Then Γ_{1}, Γ_{2} are disjoint and have cardinalities k = \frac{n − t}{2}, n − k respectively. The set of characters (cf. (7.8, (iii)) {σ(χ_{D}) | D a diagram in W_{t}(n)} coincides with the set of all orderings of Γ := Γ_{1} ⨿ Γ_{2} in which Γ_{1}, Γ_{2} appear in the given order.

Proof. By (7.10), each character χ_{D} corresponds to some ordering of Γ. In the generic case, the elements of Γ are distinct, whence the corresponding ordering is unique. But by the second assertion in (8.2)(i), in any ordering corresponding to a character χ_{D}, Γ_{1} and Γ_{2} appear in the given order. But the number of such

orderings of Γ is \binom{n}{\frac{n−t}{2}}, which coincides with the number of distinct diagrams

D ∈ W_{t}(n) (see the paragraph preceding (6.1)). The assertion follows. □

(8.4) Corollary. Suppose that R is any integral domain and let (q, Q) be generic in R. There is a non-zero element Δ ∈ R with the following property. Let R_{Δ} be the localisation R_{Δ} = R[Δ^{−1}]. Write M for the Ĥ_{n}^{a}(q)-module ξ_{Q,μ}^{∗}W_{t}(n) and let M_{Δ} := R_{Δ} ⊗_{R} M. Then M_{Δ} has a unique decomposition M_{Δ} ≅ ⊕_{D}M_{Δ}(D), where the sum is over the diagrams in W_{t}(n) and M_{Δ}(D) is a free R_{Δ}-submodule of rank one, which is invariant under U(n)_{Δ} = R_{Δ}[X_{1}^{±1}, …, X_{n}^{±1}], and on which U(n)_{Δ} acts via the character χ_{D}. Any simultaneous eigenvector E ∈ M_{Δ} for the X_{i} corresponds to one of the characters χ_{D} of U(n).

Proof. By the first statement in (8.2)(i), the characters χ_{D} are distinct. Hence there is an element Y ∈ U(n) whose set {χ_{D}(Y) | D a diagram in W_{t}(n)} of eigenvalues (recall that the elements of U(n) act in triangular fashion on W_{t,z}(n) and the χ_{D}(Y) are the diagonal elements) is distinct. It is easy to show using elementary linear algebra that there is a set of elements x_{D} ∈ M such that Y x_{D} = χ_{D}(Y) (for each diagram D). If Δ is the (non-zero) determinant of the matrix formed by the coefficients of the x_{D} with respect to an R-basis of M, it is clear that M_{Δ} is the direct sum of the rank one submodules R_{Δ}x_{D}, which are each invariant under Y. Since Y has distinct eigenvalues, any linear transformation of M_{Δ} which commutes with Y is diagonal with respect to this eigenspace decomposition, whence we deduce that the eigenspaces of Y are invariant under U(n).

Finally, if E is a simultaneous eigenvector of the X_{i}, it is an eigenvector of Y, whence E ∈ M_{Δ}x_{D} for some diagram D. This proves the last statement. □

Note that if R is a field in (8.4), the statement may be simplified by eliminating any reference to Δ.

We shall refer to an ordering of Γ satisfying the condition of (8.3) as permissible. Note that one permissible ordering is

(Γ) = (−μQ^{−1}, −μQ^{−1}q^{−2}, …, −μQ^{−1}q^{−(n+t−2)}, μQ, μQq^{−2}, … μQq^{−(n−t−2)}).

We write (Γ_{2}, Γ_{1}) = (δ_{1}, …, δ_{n}) for this ordering.

(8.5) Corollary. With notation as in (8.3), let R be a field. Then there is a vector E ∈ M which is a simultaneous eigenvector of X_{1}, X_{2}, …, X_{n} such that χ_{E} = (δ_{1}, …, δ_{n}).

This is clear from (8.4).

We shall require the following elementary result from linear algebra.

(8.6) Lemma. Let V be a vector space over a field R. Let A, B be commuting linear transformations of V and let v ∈ V satisfy

(A + B)v = (a + b)v and ABv = abv for some a, b ∈ R.

Assume that Av ∉ Rv. Then

(i) The plane Π = ⟨v, Av⟩ = ⟨v, Bv⟩ is stable under the semigroup generated by A, B.

(ii) The eigenvalues of A, B on Π are each {a, b}.

(iii) If v_{1} = bv − Av, v_{2} = av − Av then Av_{1} = av_{1}, Bv_{1} = bv_{1}, Av_{2} = bv_{2} and Bv_{2} = av_{2}.

The proof is easy.

(8.7) Proposition. Let E ∈ M be the eigenvector of (8.5). Then T_{j}E = −q^{−1}E

for j ≠ \frac{n + t}{2}, j = 1, 2, …, n − 1.

Proof. Take j ∈ ℤ, 1 ≤ j ≤ n − 1, j ≠ \frac{n + t}{2}, and let v = T_{j}E. Then since X_{i} commutes with T_{j} unless i = j or i = j + 1 (see (2.6.1) or (3.3)(iv)), we have

(8.7.1)  X_{i}v = δ_{i}v if i ≠ j, j + 1.

Moreover since T_{j} commutes with X_{j}X_{j+1} and X_{j} +X_{j+1} (by the Bernstein relation (3.3)(iv)), we have X_{j}X_{j+1}v = δ_{j}δ_{j+1}v and (X_{j} + X_{j+1})v = (δ_{j} + δ_{j+1})v.

Observe that X_{j}v = X_{j}T_{j}E = T_{j}^{−1}X_{j+1}E = δ_{j+1}T_{j}^{−1}E. But since T_{j}^{−1} = T_{j} + q^{−1} − q, we have

(8.7.2)  \begin{matrix}X_{j}v = & δ_{j+1} (T_{j}E + (q^{−1} − q)E) \\ = & δ_{j+1} (v + (q^{−1} − q)E) . \end{matrix}

Suppose X_{j}v is not a scalar multiple of v. Then we may apply (8.6) with A = X_{j}, B = X_{j+1}, a = δ_{j} and b = δ_{j+1}. By (8.6) the vector v_{2} = δ_{j}v−X_{j}v is a simultaneous eigenvector for X_{j}, X_{j+1} with respective eigenvalues δ_{j+1}, δ_{j}. Moreover by (8.7.1), v_{2} is also an eigenvector for each X_{i} (i ≠ j, j + 1) with corresponding eigenvalue δ_{i}. Thus v_{2} corresponds to some character χ_{D} of U(n) by the last statement in (8.4). But our choice of j ensures that δ_{j+1} = δ_{j}q^{−2}, which by the second statement in (8.2)(i) is impossible. It follows that X_{j}v ∈ Rv. Hence by (8.7.2), T_{j}E ∈ RE. By the quadratic relation for T_{j}, we therefore have T_{j}E = qE or T_{j}E = −q^{−1}E; but X_{j+1}E = T_{j}X_{j}T_{j}E = q^{−2}X_{j}E, whence T_{j}E = −q^{−1}E as stated. □

9. Induced modules and standard modules

We shall henceforth assume that R is a field and all modules and algebras will be R-modules and R-algebras.

We shall identify M = ξ_{Q,μ}^{∗}W_{t}(n) as a module induced from a one-dimensional module of a parabolic subalgebra of Ĥ_{n}^{a}(q). Such modules were introduced by Rogawski [R]. The parabolic subalgebras are defined as follows. For any subset J ⊂ {1, 2, …, n − 1} let Ĥ_{J}^{a}(q) be the subalgebra of Ĥ_{n}^{a}(q) which is generated by U(n) = R[X_{1}^{±1}, …, X_{n}^{±1}] and {T_{j} | j ∈ J}. For any partition λ = λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{ℓ} ≥ 0 of n, define Ĥ_{λ}^{a}(q) to be Ĥ_{J}^{a}(q), where J = J_{λ}, defined as follows:

j ∈ J_{λ} ⇔ ∃ k ∈ {1, 2, …, ℓ} with n ≥ ∑_{i=1}^{k} λ_{i} ≥ j + 1 > j > ∑_{i=1}^{k−1} λ_{i}, where, if k = 0,

the smaller sum is interpreted as 0. The subalgebra Ĥ_{λ}^{a}(q) has finite index equal

to the multinomial coefficient \frac{n!}{λ_{1}! … λ_{ℓ}!} in Ĥ_{n}^{a}(q).

Now let R_{λ,χ} be a one-dimensional Ĥ_{λ}^{a}(q)-module on which T_{j} acts as multiplication by q and U(n) = R[X_{1}^{±1}, …, X_{n}^{±1}] acts via the character χ. Then since X_{j+1} = T_{j}X_{j}T_{j}

(9.1)  χ(X_{j+1}) = q^{2}χ(X_{j}) for each j ∈ J_{λ},

and any character satisfying this condition gives rise to a one-dimensional Ĥ_{λ}^{a}(q)- module. A similar definition applies when J_{λ} is replaced by an arbitrary subset J of {1, …, n − 1}.

(9.2) Definition. Let χ be a character of U(n) = R[X_{1}^{±1}, …, X_{n}^{±1}] which satisfies the condition (9.1). Define the Ĥ_{n}^{a}(q)-module K_{λ,χ} as the induced module

K_{λ,χ} = Ĥ_{n}^{a}(q) ⊗_{_{H_{λ}^{a}}(q)} R_{λ,χ},

where R_{λ,χ} is the one-dimensional Ĥ_{λ}^{a}(q)-module defined above. Then the dimen-

sion of K_{λ,χ} is \frac{n!}{λ_{1}! … λ_{ℓ}!}.

Of course this definition applies slightly more generally to the subalgebras Ĥ_{J}^{a}(q), where J_{λ} is replaced by an arbitrary subset J ⊂ {1, 2, …, n − 1} and the corresponding module is denoted by K_{J,χ}.

We have the following easy characterisation of these induced modules.

(9.3) Lemma. Suppose K is a finite dimensional Ĥ_{n}^{a}(q)-module which contains a vector e which generates K as Ĥ_{n}^{a}(q)-module and satisfies the following two conditions for some subset J ⊂ {1, 2, …, n − 1}:

(9.3.1)  T_{i}e = qe for i ∈ J

and

(9.3.2)  X_{i}e = χ(X_{i})e, 1 ≤ i ≤ n, for some character χ of U(n)

Then K is a quotient of K_{J,χ}. In particular, if dim(K)=dim(K_{J,χ}), we have equality.

We wish to identify these modules as “standard modules” in the sense of Kazhdan and Lusztig.

Write G = GL_{n}(R) and 𝔊 = Lie(G). Then U(n) = R[X_{1}^{±1}, …, X_{n}^{±1}] may be thought of as the coordinate ring of the maximal torus S of G consisting of diagonal elements. Any character χ of U(n) = R[X_{1}^{±1}, …, X_{n}^{±1}] is therefore given by evaluation at a (generically) unique diagonal element s ∈ S. It is not difficult to see that in fact the set of pairs (λ, χ), where χ satisfies the condition (9.1), modulo the equivalence defined by isomorphism of the corresponding induced representations K_{λ,χ}, is thus in bijection with the set P̂ of equivalence classes of pairs (s, N) modulo G, where s ∈ G is semisimple, N ∈ 𝔊 is nilpotent and s.N = q^{2}N (the action of G on 𝔊 being the adjoint representation). The correspondence may be realised as follows. Let J_{k} denote the k × k (Jordan) matrix with zeros everywhere except on the super-diagonal, where all entries are 1. Given a partition λ = λ_{1} ≥ ⋯ ≥ λ_{ℓ} > 0 of n, there is a unique block diagonal matrix N_{λ} whose diagonal (Jordan) blocks have size λ_{i}, i = 1, 2, …, ℓ. If R is algebraically closed, then each pair (s, N) such that Ad(g) · N = q^{2}N is G-conjugate to one where

N = N_{λ} = [ \begin{matrix}J_{λ_{1}} & & ⋯ \\ & J_{λ_{2}} & ⋯ \\ & ⋯ \\ & ⋯ & J_{λ_{ℓ}} \end{matrix} ]

and s is also block diagonal, of the form

s = s_{λ} = [ \begin{matrix}s_{λ_{1}} & & ⋯ \\ & s_{λ_{2}} & ⋯ \\ & ⋯ \\ & ⋯ & s_{λ_{ℓ}} \end{matrix} ]

where the matrices s_{λ_{i}} are diagonal of the form

s_{λ_{i}} = [ \begin{matrix}a_{i} \\ & a_{i}q^{−2} \\ & & a_{i}q^{−4} \\ & & & . \\ & & & & . \\ & & & & & a_{i}q^{−2(λ_{i}−1)}. \end{matrix} ]

The matrix s_{λ} depends on the ℓ parameters a_{i} ∈ R^{×}, and the two pairs (s_{λ}, N_{λ}) and (s′_{λ′}, N_{λ′}) are G-conjugate if and only if λ = λ′ and some parts of λ are equal and the corresponding s_{λ_{i}} are permuted to obtain s′.

In view of the correspondence between pairs (s, N) ∈ P̂ and pairs (λ, χ) as in (9.2) we shall use the notation

(9.4)  K_{λ,χ} = K_{s,N}.

The character χ is given in terms of the pair (s, N) by χ(X_{i}) = s_{i}^{−1}, where s_{i} is the i^{th} diagonal entry in s_{i}.

Whenever we use the notation (s, N), we shall assume that R = ℂ unless we specify otherwise, since we aim to integrate our results with the body of literature which relates to this case. Now Chriss-Ginzburg ([CG]) and Kazhdhan-Lusztig ([KL], see [X] for an exposition) have defined “standard modules” M_{s,N} for Ĥ_{n}^{a}(q) (of course as a special case of a general construction) in the case R = ℂ. The space of M_{s,N} is the Borel-Moore homology H_{∗}(B_{N}^{s}) of a certain subvariety B_{N}^{s} of the flag variety of G. We shall need the following result of Ariki. For any finite dimensional Ĥ_{n}^{a}(q)-module M, [M] denotes its class in the Grothendieck group of finite dimensional Ĥ_{n}^{a}(q)-modules.

(9.5) Proposition ([A, Theorem 3.2, p. 798]). We have, in the Grothendieck group of finite dimensional Ĥ_{n}^{a}(q)-modules,

[M_{s,N}] = [K_{s,N}] = [K_{λ,χ}] = [Ĥ_{n}^{a}(q) ⊗_{_{H_{λ}^{a}}(q)} ℂ_{λ,χ}],

for each pair (s, N) ∈ P̂, where N = N_{λ} and χ(X_{i}) = s_{i}^{−1} if s = diag(s_{1}, …, s_{n}).

Next observe that there is an involution ι : Ĥ_{n}^{a}(q) → Ĥ_{n}^{a}(q) which takes T_{i} to −T_{i}^{−1} (i = 1, …, n − 1) and X_{j} to X_{j}^{−1} (j = 1, …, n); this follows by noting that the images of the T_{i} and X_{j} under ι satisfy the the relations (3.3)(iv). It follows that for any Ĥ_{n}^{a}(q)-module M, we may define its inflation ι^{∗}M via ι in the usual way.

(9.6) Theorem. Let R be any field, and suppose the pair (q, z) of invertible elements of R is generic (see (8.1)). Assume there is an element β ∈ R^{×} such that β^{2} = −q^{n−2}. Let ψ_{α} : Ĥ_{n}^{a}(q) ⟶ T_{n}^{a}(q) (α ∈ R^{×}) be the surjection of (5.12) (see also (5.13) and (4.3)(i)) and let W_{t,z}(n) be a cell module for the diagram algebra T_{n}^{a}(q), where t ∈ ℤ, 0 ≤ t ≤ n, n − t ∈ 2ℤ and z ∈ R^{×}. Let M be the Ĥ_{n}^{a}(q)-module ι^{∗}ψ_{α}^{∗}W_{t,z}(n). Then M is isomorphic to the induced module K_{s_{λ},N_{λ}}

(see (9.4)) where λ is the partition (\frac{n + t}{2}, \frac{n − t}{2}) and s_{λ} is the diagonal matrix

diag(s_{1}, …, s_{n}), where the s_{i} are given by

(s_{1}, …, s_{n}) = (a_{1}, a_{1}q^{−2}, …, a_{1}q^{−(n+t−2)}, a_{2}, a_{2}q^{−2}, …, a_{2}q^{−(n−t−2)}),

where a_{1} = (−1)^{t+1}αzq^{\frac{1}{2} (n+t−2)} and a_{2} = (−1)^{t+1}αz^{−1}q^{\frac{1}{2} (n−t−2)}.

Proof. Note first that from the diagram (5.12), we see that if α, β and μ are related by β^{2} = −q^{n−2} and μα^{−1} = β, then (cf. (7.1))

ψ_{α}^{∗}W_{t,z}(n) ≅ ψ_{α}^{∗}g_{β}^{∗}W_{t}(n) ≅ ξ_{Q,μ}^{∗}W_{t}(n)

where W_{t}(n) is the cell module for T LB_{n}(q, Q) and z = z_{t} (see Theorem (6.1)). But given t, z and β, (6.12) shows that W_{t,z}(n) ≅ g_{β}^{∗}W_{t}(n) = g_{β,Q}^{∗}W_{t}(n), where Q = Q(t, z) = (−1)^{t}βz^{−1}q^{− \frac{1}{2} (n+t−2)}. Moreover Q(t, z)^{2} = −z^{−2}q^{−t}, whence if (q, z) is generic, so is (q, Q).

It follows from (8.5) and (8.7) that there is a vector E ∈ M such that X_{i}E = δ_{i}E for i = 1, 2, …, n, where (δ_{1}, …, δ_{n}) is the sequence in (8.5) and T_{j}E = −q^{−1}E for

j = 1, 2, …, n − 1, j ≠ \frac{n + t}{2}. Hence ι(X_{i})E = δ_{i}^{−1}E and ι(T_{j})E = qE for each

relevant i, j. Since M is irreducible because we are in the generic case, E generates M as Ĥ_{n}^{a}(q)-module. It follows from Lemma (9.3) and from the description of the character χ in (9.4) that M is isomorphic to the induced module K_{s,N_{λ}}, where

λ = (\frac{n + t}{2}, \frac{n − t}{2}) and s is the diagonal matrix with diagonal entries

\begin{matrix}(δ_{1}, …, δ_{n}) = (−μQ^{−1}, −μQ^{−1}q^{−2}, …, −μQ^{−1}q^{−(n+t−2)}, μQ, \\ μQq^{−2}, …, μQq^{−(n−t−2)}), \end{matrix}

where Q = Q(t, z) = (−1)^{t}βz^{−1}q^{− \frac{1}{2} (n+t−2)}.

Using the relations among α, β, z and Q, we see that μQ = (−1)^{t+1}αz^{−1}q^{\frac{1}{2} (n−t−2)} and −μQ^{−1} = (−1)^{t+1}αzq^{\frac{1}{2} (n+t−2)}. □

(9.7) Notation. We fix here some notation for the rest of this work. First, if t ∈ ℤ, 0 ≤ t ≤ n, n − t ∈ 2ℤ, we shall consistently write

(9.7.1)  k = \frac{n − t}{2}.

Corresponding to each such t (or k) we have the 2-step partition λ_{t} = (n − k, k). Note that n − k ≥ k.

The diagonal matrix in the statement of (9.6) will be denoted s_{k}(a_{1}, a_{2}) or just s(a_{1}, a_{2}), i.e.

(9.7.2)

s(a_{1}, a_{2}) = s_{k}(a_{1}, a_{2}) = diag(s_{1}, …, s_{n})
= diag(a_{1}, a_{1}q^{−2}, …, a_{1}q^{−(n+t−2)}, a_{2}, a_{2}q^{−2}, …, a_{2}q^{−(n−t−2)}).

Recall the definition of the induced module K_{λ,χ} where λ is a partition of n and χ is a character of U(n) which satisfies the condition (9.1). A special case of this definition is when λ = λ_{t} as above, and χ corresponds to the semisimple element s_{k}(a_{1}, a_{2}) of (9.7.2), i.e. χ(X_{j}) = s_{j}^{−1}. We denote the corresponding induced module of the statement of (9.6) in this case by K_{s,N} = K_{s(a_{1},a_{2}),N_{k}}, where N_{k} = N_{λ_{t}}.

We next wish to identify the Grothendieck class of the module M of (9.6) (i.e. its set of composition factors) in general, i.e. when (q, z) is not necessarily a generic pair. We shall prove

(9.8) Theorem. Let F be field, and let (q, z) be a pair of invertible elements of F. Assume that there are elements \sqrt{q}, \sqrt{−1} ∈ F such that \sqrt{q}^{2} = q and \sqrt{−1}^{2} = −1. Let M be the Ĥ_{n}^{a}(q)-module ι^{∗}ψ_{α}^{∗}W_{t,z}(n) as in (9.6), where Ĥ_{n}^{a}(q) is now an F - algebra, and α ∈ F^{×}. Then in the Grothendieck group of finite dimensional Ĥ_{n}^{a}(q)- modules, M is equivalent to the induced module K_{s_{λ},N_{λ}} (see (9.4)) where λ, s_{λ} and N_{λ} are as in (9.6).

Proof. Let R_{0} = F [q] be the ring of polynomials over F in the indeterminate q. Let R be the completion of R_{0} at the prime ideal 𝔓 = (q − \sqrt{q}), and let L be the quotient field of R. Thus if we write y = q − \sqrt{q} ∈ R_{0}, R is identified with the ring of power series F [[y]], and L = F [[y]]_{y} = F [[y]][y^{−1}]. The ring R is then a complete rank one discrete valuation ring with residue field F. Under the residue class map R → F, q^{2} ↦ q.

Consider the affine Hecke algebra Ĥ_{n}^{a}(q^{2}) over the ring R, corresponding to the invertible element q^{2} ∈ R. The pair (q^{2}, z) is generic in R, and we may therefore interpret W_{t,z}(n) as a module for the R-algebra T_{n}^{a}(q^{2}). Write { \atop W_{t,z}(n)} for this R-module. Let 𝔟 ∈ R satisfy 𝔟^{2} = −q^{2(n−2)}. Such an element obviously exists, and under the residue class map 𝔟 ↦ β ∈ F such that β^{2} = −q^{−(n−2)}.

Write Q = (−1)^{t}𝔟z^{−1}q^{−(n+t−2)}, and denote R-modules by means of a tilde. Then by Theorem (6.1), with the obvious extensions of our standard notation,

(9.8.1)  ψ_{α}^{∗}{ \atop W_{t,z}(n)} ≅ ψ_{α}^{∗}g_{𝔟}^{∗}{ \atop W_{t}(n)} ≅ ξ_{Q,𝔪}^{∗}{ \atop W_{t}(n)},

where 𝔪 = α𝔟.

Since the pair (q^{2}, z) is generic in R, we may invoke Theorem (9.6) to deduce that

(9.8.2)  L ⊗_{R} M̃ = L ⊗_{R} ι^{∗}ψ_{α}^{∗}{ \atop W_{t,z}(n)} ≅ L ⊗_{R} { \atop K_{s_{λ},N_{λ}}}.

The isomorphism (9.8.2) is an isomorphism of L ⊗_{R} Ĥ_{n}^{a}(q^{2})-modules. We shall show how to interpret it in terms of L⊗_{R}T LB_{n}(q^{2}, Q)-modules, where T LB_{n}(q^{2}, Q) is the R-algebra defined in (5.3), corresponding to the parameters q^{2}, Q.

The left side of (9.8.2) is by (9.8.1) the pullback to Ĥ_{n}^{a}(q^{2}) of a T LB_{n}(q^{2}, Q)- module. Hence the kernel of ξ_{Q,𝔪} acts trivially on the right side of (9.8.2) and it follows easily that the kernel of ξ_{Q,𝔪} acts trivially on the R-module { \atop K_{s_{λ},N_{λ}}}, whence the latter may be thought of as a T LB_{n}(q^{2}, Q)-module. Thus the isomorphism (9.8.2) may be interpreted as an isomorphism of L ⊗_{R} T LB_{n}(q^{2}, Q) modules. Moreover we may think of the space of either side as an L-vector space, in which there are two R-lattices invariant under T LB_{n}(q^{2}, Q), which correspond respectively to the representations M̃ and { \atop K_{s_{λ},N_{λ}}} of the R-algebra T LB_{n}(q^{2}, Q).

Now since the pair (q^{2}, z) is generic in R, and therefore generic in L, the L ⊗_{R} T LB_{n}(q^{2}, Q)-module L ⊗_{R} { \atop W_{t}(n)} is absolutely irreducible, whence L is a splitting

field for L ⊗_{R} M̃ ≅ L ⊗_{R} { \atop K_{s_{λ},N_{λ}}}.

The reductions mod 𝔓 of M̃ and { \atop K_{s_{λ},N_{λ}}} are respectively isomorphic to the T LB_{n}(q, Q)-modules M and K_{s_{λ},N_{λ}}, and since F is a splitting field for T LB_{n}(q, Q) (cf. [GL2, §5]), the triple (L, R, F) is a splitting system for T LB_{n}(q^{2}, Q) in the sense of [Be, p. 17]. Moreover since (q^{2}, z) is generic in L, L ⊗_{R} T LB_{n}(q^{2}, Q) is semisimple. It follows from [op. cit., Proposition 1.9.6, p. 19], that the T LB_{n}(q, Q) modules (and hence the Ĥ_{n}^{a}(q)-modules) M and K_{s_{λ},N_{λ}} have the same composition factors, counting multiplicity, which is the required statement. □

The above result enables us to determine the composition factors of the standard modules M_{s,N} for any pair s, N, where N has at most two Jordan blocks (we refer to such N as “two-step nilpotent”) and when q is arbitrary. It has the following consequence.

(9.9) Corollary. Let Ĥ_{n}^{a}(q) be the affine Hecke algebra over the complex numbers ℂ and let Γ(Ĥ_{n}^{a}(q)) be the Grothendieck group of finite dimensional Ĥ_{n}^{a}(q)-modules. Let λ = (n − k, k) be a 2-step partition of n (2k ≤ n), let t = n − 2k and let s be the diagonal matrix of (9.7.2). Write M_{s,N} for the Kazhdan-Lusztig standard module of Ĥ_{n}^{a}(q) corresponding to the pair s = s_{k}(a, b), N = N_{k} (cf. (9.7)). Then

[M_{s,N}] = [θ_{α}^{∗}W_{t,z}(n)],

where [V] denotes the class in Γ(Ĥ_{n}^{a}(q)) of an Ĥ_{n}^{a}(q)-module V, W_{t,z}(n) is a cell module for the diagram algebra T_{n}^{a}(q), θ_{α} : Ĥ_{n}^{a}(q) → T_{n}^{a}(q) is the surjection ψ_{α} of (5.12) composed with the involution ι of (9.6), and α, z, t satisfy the equations

(9.9.1)  \begin{matrix}k = & \frac{n − t}{2} \\ a = & (−1)^{n+1}αzq^{\frac{1}{2} (n+t−2)} \\ b = & (−1)^{n+1}αz^{−1}q^{\frac{1}{2} (n−t−2)} \end{matrix}

Proof. This follows easily from (9.5) and (9.8). the former asserts that [M_{s,N}] = [K_{s,N}], while the latter asserts that [K_{s,N}] = [θ_{α}^{∗}W_{t,z}(n)]. The formulae for a and b are in the statement of (9.6), taking account of the fact that t and n have the same parity. □

We shall use (9.9) to study the composition factors of the standard modules M_{s,N} through our knowledge of those of the cell modules W_{t,z}(n). First we need to understand the correspondence between cell modules and standard modules a little better. Assume henceforth that the ground ring R is ℂ.

(9.10) Corollary. Given the pair (s(a, b), N_{k}) (see (9.7)) with 2k ≤ n, there exist (t, α, z) such that [M_{s(a,b),N_{k}}] = [θ_{α}^{∗}W_{t,z}(n)]. The parameters (t, α, z) are given by

(9.10.1)  \begin{matrix}t = & n − 2k \\ α^{2} = & abq^{−(n−2)} \\ z = & (−1)^{n+1}α^{−1}aq^{− \frac{1}{2} (n+t−2)} \\ = & (−1)^{n+1}αb^{−1}q^{\frac{1}{2} (n−t−2)}, \end{matrix}

where, if t = n (or k = 0), the second equation (for α^{2}) is omitted.

(9.11) Lemma. We have ψ_{α}^{∗}W_{t,z}(n) ≅ ψ_{−α}^{∗}W_{t,−z}(n).

Proof. For any element β ∈ ℂ such that β^{2} = −q^{n−2}, and α, μ ∈ ℂ such that μ = αβ, we have a commutative diagram (5.12) for any Q ∈ ℂ, and by (6.12), W_{t,z}(n) = g_{β,Q}^{∗}W_{t}(n), where Q = Q(t, z) = (−1)^{t}βz^{−1}q^{− \frac{1}{2} (n+t−2)}. Hence ψ_{α}^{∗}W_{t,z}(n) ≅ ψ_{α}^{∗}g_{β,Q}^{∗}W_{t}(n) ≅ ξ_{Q,μ}^{∗}W_{t}(n). If we replace β by −β and α by −α, μ is unchanged. Thus replacing z by −z leaves Q unchanged, and it follows that we also have ξ_{Q,μ}^{∗}W_{t}(n) ≅ ψ_{−α}^{∗}W_{t,−z}(n), as stated. □

(9.12) Corollary. Let P be the set equivalence classes of pairs (s, N) ∈ P̂ where N ∈ 𝔊 is two-step nilpotent, i.e. N ∼ N_{k} for some k with 0 ≤ 2k ≤ n. Let Ω̃ be the set of triples (t, α, z) (t ∈ ℤ, 0 ≤ t ≤ n, n−t ∈ 2ℤ; α, z ∈ ℂ^{×}) and let Ω be the set of equivalence classes of triples in Ω̃ under the equivalence generated by the relations (t, α, z) ∼ (t, −α, −z), (n, α, z) ∼ (n, y^{−1}zα, y) and (0, α, z) ∼ (0, α, z^{−1}). Then (with the obvious abuse of notation) we have well defined Ĥ_{n}^{a}(q)-modules M_{s,N}, (s, N) ∈ P and θ_{α}^{∗}W_{t,z}(n) (see (9.9)), (t, α, z) ∈ Ω, and there is a bijection f : P → Ω such that if (s, N) ∈ P corresponds to (t, α, z) ∈ Ω, [M_{s,N}] = [θ_{α}^{∗}W_{t,z}(n)].

Proof. The modules M_{s,N} are the standard modules of Kazhdan-Lusztig. We need to verify that the isomorphism class of θ_{α}^{∗}W_{t,z}(n) depends only on the equivalence class of (t, α, z). But Lemma (9.11) proves this for one type of equivalence, while if t = 0, we observe that W_{0,z}(n) ≅ W_{0,z^{−1}}(n), and if t = n, in the relations (9.9.1) only a occurs, since N_{k} = N_{0} has only one Jordan block, and clearly a depends only on αz, so that for any y, z, α ∈ ℂ^{×}, θ_{α}^{∗}W_{n,z}(n) ≅ θ_{y^{−1}zα}^{∗}W_{n,y}(n), which completes the verification.

Next, let us define the required bijection f : Ω → P. Given (t, α, z) ∈ Ω, define f(t, α, z) to be the class of (s(a, b), N_{k}), where a, b, k are given by the equations (9.9.1). To check that this is well defined, note that if (t, α, z) is replaced by (t, −α, −z), the equations (9.9.1) are unchanged, and therefore have the same solution. If (0, α, z) is replaced by (0, α, z^{−1}), we obtain the solution (s(b, a), N\frac{n}{2} ) of (9.9.1). But the pairs (s(b, a), N\frac{n}{2} ) and (s(a, b), N\frac{n}{2} ) are conjugate under GL_{n}(ℂ). Finally, the triples (n, α, z) and (n, y^{−1}zα, y) give the same values of k(= 0) and a, which proves that f is well defined.

Finally, define f′ : P → Ω by taking f′(s(a, b), N_{k}) to be the equivalence class of any triple (t, α, z) where the parameters satisfy the equations (9.10.1). If t ≠ n there are just two such triples, which are obviously equivalent. If t = n, all the resulting triples are easily seen to be equivalent, since they have a common value of αz. To check that f′ is well defined, we note that the pairs (s(a, b), N_{k}) and (s(a′, b′), N_{k′}) are conjugate under GL_{n}(ℂ) if and only if either they are equal, or k = k′ = \frac{n}{2} and (a′, b′) = (b, a). In the latter case, solving (9.10.1) shows that the

corresponding triples both have t = 0 and z in one is replaced by z^{−1} in the other, whence the triples are equivalent, which shows that f′ is well defined. Since f, f′ are clearly inverses, both are bijections, and the result now follows from Theorem (9.8). □

(9.13) Remark. There is a close analogy between the set P described here and the “multisegments” of Zelevinsky ([Z, §4]). We shall elaborate on this in §11 below.

10. Irreducible modules and decomposition numbers

We begin by reviewing the main points concerning the representation theory of the algebra T_{n}^{a}(q). In this section, R is a field.

(10.1) Definition. Let Λ^{a}(n)^{+} be the set

Λ^{a}(n)^{+} = {(t, z) | t ∈ ℤ_{≥0}, 0 ≤ t ≤ n, n − t ∈ 2ℤ; z ∈ R^{×}}.

Define Λ^{a}(n) by

(10.1.1)  Λ^{a}(n) = \{ \begin{matrix}Λ^{a}(n)^{+} if q^{2} ≠ −1 \\ Λ^{a}(n)^{+} \ \{(0, ±q)\} if q^{2} = −1. \end{matrix}

Define the equivalence relation ≈ on Λ^{a}(n)^{+} as that which identifies (0, z) and (0, z^{−1}) for all z ∈ R^{×}, and write

(10.1.2)  \begin{matrix}Λ^{a}(n)^{0} = Λ^{a}(n)/ ≈ \\ Λ^{a}(n)^{0+} = Λ^{a}(n)^{+}/ ≈ \end{matrix}

Recall (e.g. from (6.6) above) that if X_{t} is the T_{n}^{a}(q)-module with basis all monic affine diagrams:t → n, the cell module W_{t,z}(n) for T_{n}^{a}(q) is defined as the quotient of X_{t} by the ideal I_{χ} := {γτ_{t}−χγ} for γ ∈ X_{t}, where χ = χ_{z} = z if t ≠ 0 and χ = χ_{z} = z+z^{−1} if t = 0. There is a bilinear map ϕ_{t,z} : W_{t,z}(n)×W_{t,z^{−1}}(n)⟶R which ([GL2, (2.7)]) is invariant in the sense that ϕ_{t,z}(wμ, ν) = ϕ_{t,z}(μ, w^{∗}ν), (μ, ν ∈ W_{t,z}(n), w ∈ T_{n}^{a}(q)) where d^{∗} denotes the reflection of a diagram d ∈ T in a horizontal line. Note that w ↦ w^{∗} is an anti-automorphism of T_{n}^{a}(q), which preserves T L_{n}^{a}(q). If R is a field, the irreducible T_{n}^{a}(q)-modules have the following description.

(10.2) Theorem ([GL2, (2.8)]). Let R be an algebraically closed field and maintain the above notation. For (t_{1}, z_{1}), (t_{2}, z_{2}) ∈ Λ^{a}(n)^{+}, W_{t_{1},z_{1}}(n) ≅ W_{t_{2},z_{2}}(n) if and only if (t_{1}, z_{1}) ≈ (t_{2}, z_{2}). For (t, z) ∈ Λ^{a}(n)^{+}, L_{t,z}(n) := W_{t,z}(n)/rad ϕ_{t,z} is either an irreducible T_{n}^{a}(q) module or zero, and L_{t,z}(n) ≠ 0 if and only if (t, z) ∈ Λ^{a}(n) (see (10.1.1)). All irreducible T_{n}^{a}(q) modules are realised thus, and if (t_{1}, z_{1}) ≉ (t_{2}, z_{2}), then L_{t_{1},z_{1}}(n) ≅ L̸_{t_{2},z_{2}}(n).

It follows that the distinct irreducible T_{n}^{a}(q)-modules are parametrised by Λ^{a}(n)^{0}, while the distinct cell modules are parametrised by Λ^{a}(n)^{0+}. Where there is little danger of confusion, we abuse notation by denoting the elements of Λ^{a}(n)^{0+} as pairs (t, z), rather than equivalence classes of pairs. Thus we speak of W_{t,z}(n) and L_{t,z}(n) for (t, z) ∈ Λ^{a}(n)^{0+}. It follows from (e.g.) (9.12) that to understand the composition factors of the Ĥ_{n}^{a}(q)-module M_{s,N} ((s, N) ∈ P), it suffices to understand those of the T_{n}^{a}(q)-module W_{t,z}(n) for appropriate (t, z) ∈ Λ^{a}(n)^{+}. We therefore now turn to a description of these, which is available from [GL2].

Let ⪯ be the partial order on Λ^{a}(n)^{+} which is generated by the preorder ≺^{∘} which stipulates that (t, z) ≺^{∘} (s, y) if

0 ≤ t ≤ s ≤ n, s = t + 2ℓ (ℓ ∈ ℤ, ℓ > 0) and (10.3a)

z^{2} = q^{ϵ(s,z)s} and y = zq^{−ϵ(s,z)ℓ} for ϵ(s, z) = ±1. (10.3b)

Note that (10.3) implies that

(10.4a)  y^{2} = q^{ϵ(s,z)t} and z^{t} = y^{s}

and

(10.4b)  (t, z) ⪯ (t′, z′) ⟹ z^{t} = (z′)^{t′}.

It suffices to verify (10.4b) when (t, z) ≺^{∘} (t′, z′), in which case it follows easily from (10.3).

It is easily verified that ([GL3, 4.1]) the partial order ⪯ on Λ^{a}(n)^{+} induces a partial order, also denoted ⪯, on the set Λ^{a}(n)^{0} = Λ^{a}(n)/ ≈.

The following result is proved in [GL2, Theorem 5.1].

(10.5) Theorem. Let R be a field of characteristic 0 or p > 0, where pe > n and e is the multiplicative order of q^{2}. Then, in the Grothendieck group Γ(T_{n}^{a}(q)), we have for any (t, z) ∈ Λ^{a}(n)^{+},

(10.5.1)  W_{t,z}(n) = ∑_{(t,z)⪯(s,y)^{(s,y)∈Λ^{a}(n)^{0}}} L_{s,y}(n)

Thus the matrix expressing the cell modules in terms of the irreducibles in Γ(T_{n}^{a}(q)) is upper unitriangular, and has entries 0 or 1. Now if (t, z) is confined to Λ^{a}(n), the relation (10.5.1) can clearly be inverted.

We also have (cf. [GL3, Theorem (4.5)])

(10.6) Theorem. In the notation above, if (t, z) ∈ Λ^{a}(n),

L_{t,z}(n) = ∑_{(t,z)⪯(s,y)^{(s,y)∈Λ^{a}(n)^{0}}} n_{t,z}^{s,y}W_{s,y}(n)

where n_{t,z}^{s,y} = 0 or ±1.

We next explore the implications of these results for the standard modules M_{s,N}.

(10.7) Definition. For any triple (t, α, z) where (t, z) ∈ Λ^{a}(n)^{+} and α ∈ ℂ^{×}, define the modules

(10.7.1)  \begin{matrix}W_{t,z}(α) := & θ_{α}^{∗}W_{t,z}(n) \\ L_{t,z}(α) := & θ_{α}^{∗}L_{t,z}(n) \end{matrix}

(10.8) Lemma. All composition factors of the cell module W_{t,z}(α) are of the form θ_{α}^{∗}L_{t′,z′}(n) for some triple (t′, α, z′).

Proof. It is clear that the pullback via θ_{α} of a composition series for W_{t,z}(n) is a composition series for the Ĥ_{n}^{a}(q)-module W_{t,z}(α). The result therefore follows from Theorem (10.2). □

We need to identify which among the L_{t,z}(α) are distinct. For this, we shall regard the modules we are considering as T̂ L_{n}^{a}(q)-modules, via the commutative diagram

(10.9)  Ĥ_{n}^{a}(q)  T̂ L_{n}^{a}(q)  T L_{n}^{a}(q) ϕ_{α} θ  incl T_{n}^{a}(q) θ_{α}  incl

where θ = η_{3} ∘ ι, η_{3} is as in (5.6) or (5.9), ι is the involution of Ĥ_{n}^{a}(q) defined immediately preceding the statement of Theorem (9.6) above and θ_{α} is defined in (9.9).

(10.10) Proposition. The Ĥ_{n}^{a}(q)-modules L_{t,z}(α) = θ_{α}^{∗}L_{t,z}(n) for (t, z) ∈ Λ^{a}(n), α ∈ ℂ^{×}, are precisely those irreducible Ĥ_{n}^{a}(q)-modules which factor through T̂ L_{n}^{a}(q).

Proof. The statement is equivalent to the assertion that any irreducible T̂ L_{n}^{a}(q)- module is of the form ϕ_{α}^{∗}L_{t,z}(n). To prove this, let M be any irreducible finite dimensional T̂ L_{n}^{a}(q)-module. Recall (see (4.2) above) that T L_{n}^{a}(q) = ⟨C_{1}, …, C_{n}⟩ is a subalgebra of T̂ L_{n}^{a}(q) and T̂ L_{n}^{a}(q) ≅ ℂ⟨V ⟩ ⊗_{ℂ} T L_{n}^{a}(q), where V is the automorphism of T L_{n}^{a}(q) which permutes the C_{i} cyclically.

Let M_{1} be an irreducible T L_{n}^{a}(q)-submodule of M. Then by [GL3, (2.3)], M_{1} ≅ L_{t,z}(n) or M_{1} ≅ L_{0,\sqrt{−}1}^{±}(n). If M_{1} ≅ L_{t,z}(n), then by [GL3, (2.5)] the twist M_{1}^{V} of M_{1} by the automorphism V of T L_{n}^{a}(q) is isomorphic to L_{t,z}(n). It follows from [GL3, Theorem (2.6)] that as a vector space, M = L_{t,z}(n). Moreover by Schur’s Lemma, V acts as a scalar multiple of τ_{n}, i.e. as ατ_{n} for some α ∈ ℂ^{×}, so that M ≅ ϕ_{α}^{∗}L_{t,z}(n). In the other case, [op. cit. Theorem (2.6)] shows that as vector space M ≅ L_{0,\sqrt{−}1}^{+}(n) ⊕ L_{0,\sqrt{−}1}^{−}(n), and following the argument in loc. cit. further shows that M ≅ ϕ_{α}^{∗}L_{0,\sqrt{−}1}(n) for some α ∈ ℂ^{×}. □

(10.11) Definition. Denote by Ω̃_{0} the set of triples (t, α, z) in Ω̃ (see the definition in (9.12)) such that (t, z) ∈ Λ^{a}(n). Then Ω̃_{0} is a union of ∼-classes of Ω̃ and we write Ω_{0} = Ω̃_{0}/ ∼ for the corresponding set of equivalence classes.

The triples (t, α, z) in Ω̃ are by Theorem (10.2) precisely those such that the corresponding module L_{t,z}(α) is non-zero.

We are now in a position to determine the coincidences among the irreducible modules L_{t,z}(α) for different values of the parameters.

(10.12) Theorem. Let (t, α, z) and (t′, α′, z′) be two triples in Ω̃_{0} (see Definition (10.11)). The irreducible T̂ L_{n}^{a}(q)-modules L_{t,z}(α) and L_{t′,z′}(α′) are isomorphic if and only if the triples (t, α, z) and (t′, α′, z′) are equivalent, i.e. represent the same element of Ω (see (9.12)).

Proof. We first show that if (t, α, z) ∼ (t′, α′, z′) then L_{t,z}(α) ≅ L_{t′,z′}(α′). For this, observe that W_{t,z}(α) ≅ W_{t,−z}(−α) for all triples (t, α, z) ∈ Ω̃, that W_{0,z}(α) ≅ W_{0,z^{−1}}(α) for all α, z, and that L_{n,z}(α) = W_{n,z}(α) ≅ W_{n,y}(y^{−1}zα) = L_{n,y}(y^{−1}zα). It follows by taking top quotients that if (t, α, z) ∼ (t′, α′, z′) then L_{t,z}(α) ≅ L_{t′,z′}(α′).

To prove the converse, we need to show that for any triple (t, α, z) in Ω̃_{0}, the equivalence class of (t, α, z) is determined by the isomorphism class of ϕ_{α}^{∗}L_{t,z}(n) as T̂ L_{n}^{a}(q)-module. Recall (cf. the diagram (10.9)) that T L_{n}^{a}(q) is a subalgebra of T̂ L_{n}^{a}(q) and that the restriction of L_{t,z}(α) to T L_{n}^{a}(q) is L_{t,z}(n). It follows from [GL2, (1.9)] (or [GL3], or cf. the proof of (10.10) above) that the equivalence class of the pair (t, z) in Λ^{a}(n) under the equivalence relation ≡ is determined by L_{t,z}(α), where ≡ is generated by the relations

(10.12.1)

(t, z) ≡(t, −z) if t ≠ 0, n
(n, z) ≡(n, y) for any y, z ∈ ℂ^{×}
(0, z) ≡(0, z^{−1}) for z ∈ ℂ^{×}.

Further, recall that V ∈ T̂ L_{n}^{a}(q) acts on ϕ_{α}^{∗}L_{t,z}(n) as ατ_{n} ∈ T_{n}^{a}(q) does.

Suppose for the moment that t ≠ 0, n. Then by (10.12.1), (t, z) = (t′, z′) or (t′, −z′). If (t, z) = (t′, z′), then by hypothesis ϕ_{α}^{∗}L_{t,z}(n) ≅ ϕ_{α′}^{∗}L_{t,z}(n), whence there is a linear map λ : D → D, where D is the ℂ-vector space with basis the standard affine diagrams t ⟶ n (see [GL2, §2]), which commutes with the T L_{n}^{a}(q) action, and intertwines the T̂ L_{n}^{a}(q) structures ϕ_{α}^{∗}L_{t,z}(n) and ϕ_{α′}^{∗}L_{t,z}(n). But in this case the restriction L_{t,z}(n) of ϕ_{α}^{∗}L_{t,z}(n) to T L_{n}^{a}(q) is irreducible, whence λ must be multiplication by a scalar, say a ∈ ℂ. Moreover we have for E ∈ L_{t,z}(n),

(10.12.2)  λ(V.E) = λ(ατ_{n}.E) = aατ_{n}.E = V.λ(E) = α′τ_{n}.aE = aα′τ_{n}.E,

whence α = α′. If (t, z) = (t′, −z′), then since the restrictions of L_{t,z}(α) and L_{t,−z}(α′) to T L_{n}^{a}(q) are isomorphic irreducible representations, any linear map λ : D → D (as above) which intertwines the L_{t,z}(α) and L_{t,−z}(α′)-actions on D is of the form aU, where U is the map defined in the proof of Theorem (2.3) of [GL3], and a ∈ ℂ (for any standard diagram E, U.E = (−1)^{rank(E)}E). Computation of U(V.E) in two different ways as in (10.12.2) then yields that α′ = −α. This completes the proof that (t, α, z) ∼ (t′, α′, z′) if t ≠ 0, n.

If t = n, W_{n,z}(n) is one-dimensional, and hence irreducible. Moreover V acts on ϕ_{α}^{∗}L_{n,z}(n) as multiplication by αz. Hence if t = n, the isomorphism class of ϕ_{α}^{∗}L_{t,z}(n) is determined by αz, whence the result for this case.

Now suppose that t = 0. If z^{2} ≠ −1 (i.e. z + z^{−1} ≠ 0), then the restriction to T L_{n}^{a}(q) of ϕ_{α}^{∗}L_{t,z}(n) is irreducible, and in both cases z′ = z and z′ = z^{−1} an argument like (10.12.2) shows that α = α′, which shows that (t, α, z) ∼ (t′, α′, z′).

Finally, take z^{2} = −1 and q^{2} ≠ −1. In this case the restriction M to T L_{n}^{a}(q) of ϕ_{α}^{∗}L_{t,z}(n) is (cf. the proof of (10.10) the direct sum L_{0,z}^{+}(n) ⊕ L_{0,z}^{−}(n). The two (irreducible) direct summands are interchanged by τ, and V therefore acts on ϕ_{α}^{∗}L_{t,z}(n) via the matrix

[ \begin{matrix}0 & ατ_{n} \\ ατ_{n} & 0 \end{matrix} ] .

But it is shown in [GL3, (2.8.1)] that the isomorphism class of the T̂ L_{n}^{a}(q)-module M on which T L_{n}^{a}(q) acts according to the given structure and on which V acts via the matrix

[ \begin{matrix}0 & α_{1}τ_{n} \\ α_{2}τ_{n} & 0 \end{matrix} ]

is determined by the product α_{1}α_{2}. Hence if z^{2} = −1, ϕ_{α}^{∗}L_{0,z}(n) ≅ ϕ_{α′}^{∗}L_{0,±z}(n) if and only if α′ = ±α, which is the required statement for this case. This completes the proof of Theorem (10.12). □

(10.13) Corollary. In the notation of (9.12), let (s, N) be a pair in P (recall that this means that N = N_{k} is a 2-step nilpotent matrix and s = s(a, b) is a diagonal matrix such that sNs^{−1} = q^{2}N). Then there is a well-defined Ĥ_{n}^{a}(q)-module L_{s,N}, which is either 0 or a composition factor of M_{s,N}, with the following properties. The non-zero modules L_{s,N} are pairwise non-isomorphic, and the composition factors of all the modules M_{s′,N′} ((s′, N′) ∈ P) are among the L_{s,N}. The condition that L_{s(a,b),N_{k}} be zero is that

(10.13.1)  q^{2} = −1, n is even, k = \frac{n}{2}, a = α, b = −α, for some α ∈ ℂ^{×}.

Proof. If (s, N) corresponds to the triple (t, α, z) ∈ Ω, take L_{s,N} to be the top quotient L_{t,z}(α) of W_{t,z}(α). The fact that the non-zero modules L_{s,N} are distinct follows from (10.12). Moreover, by (9.12), M_{s,N} and W_{t,z}(α) have the same composition factors. But the composition factors of W_{t,z}(α) are clearly the inflations via θ_{α} of those of the T_{n}^{a}(q)-module W_{t,z}(n). The result is now clear from (10.5), which shows that the composition factors of W_{t,z}(n) are all top quotients of some W t′, z′(n). The last assertion concerning the cases when L_{s,N} = 0 is obtained simply by translating the condition in Theorem (10.2) (viz. (t, z) = (0, ±q) when q^{2} = −1) into the language of pairs using (9.9.1). □

Our final task is to translate the precise results (10.3) et seq. concerning the composition factors of the modules W_{t,z}(n) into the language of pairs (s, N). From the argument in the proof of (10.13), it is clear that in the Grothendieck group Γ(Ĥ_{n}^{a}(q)), for any (t, z) ∈ Λ^{a}(n)^{+} and α ∈ ℂ^{×},

(10.14)  W_{t,z}(α) = ∑_{(t,z)⪯(s,y)^{(s,y)∈Λ^{a}(n)^{0}}} L_{s,y}(α).

In order to describe the composition multiplicities of the standard modules M_{s,N} for (s, N) ∈ P, we therefore need to interpret the order relation ≺ in terms of pairs.

(10.15) Proposition. Suppose that under the correspondence of (9.12), the triples (t_{1}, α, z_{1}), (t_{2}, α, z_{2}) correspond to the pairs (s(a_{1}, b_{1}), N_{k_{1}}), (s(a_{2}, b_{2}), N_{k_{2}}) respectively. Then (t_{1}, z_{1}) ≺^{∘} (t_{2}, z_{2}) if and only if there exists ℓ > 0 and ϵ = ±1 such that if we write 2k_{i} = n − t_{i} for i = 1, 2, then

(10.15.1)  \begin{matrix}k_{2} = & k_{1} − ℓ ≥ 0 \\ a_{1}b_{1}^{−1} = & q^{t_{1}+ϵt_{2}} \\ (a_{2}, b_{2}) = & (a_{1}, b_{1}) if ϵ = 1 \\ (b_{1}, a_{1}) if ϵ = −1 \end{matrix}

Proof. From equations (9.9.1), we see that z_{1}^{2} = a_{1}b_{1}^{−1}. Hence from the equations (10.3), t_{2} = t_{1}+2ℓ for some ℓ > 0, and a_{1}b_{1}^{−1}q^{−t_{1}} = q^{ϵt_{2}} for some ϵ = ±1. Since z_{2} = z_{1}q^{−ϵℓ}, the equations (9.10.1) show that a_{2} = a_{1}q^{ℓ(1−ϵ)} and b_{2} = b_{1}q^{−ℓ(1−ϵ)}. The relations (10.15.1) follow after a little rearrangement. Conversely, if the relations (10.15.1) hold, the equations (9.9.1) and (9.10.1) may be used as above to deduce the relations (10.3a) and (10.3b). □

The relation ≺^{∘} defined by the equations (10.15.1) generates a partial order ⪯ on the set P of pairs (s, N) described in (9.12). This partial order may be interpreted as ordering the G-conjugacy classes of pairs in the obvious way; viz. two classes are related if they contain related elements. Let P_{0} be the set of (conjugacy classes of) pairs (s, N) ∈ P, where s = s(a, b), N = N_{k} and (s, N) is not of the form in (10.13.1). By Theorem (10.12) P_{0} parametrises the composition factors of the standard modules M_{s,N}, (s, N) ∈ P. Theorems (10.5) and (10.6) may now be applied as follows.

(10.16) Theorem. Let P and P_{0} be the sets of semisimple- nilpotent pairs defined in the previous paragraph, and let ⪯ be the partial order on P generated by the relation (10.15.1). Then in the Grothendieck group of finite-dimensional Ĥ_{n}^{a}(q)- modules,

(10.16.1)  [M_{s,N}] = ∑_{(s,N)⪯(s′,N′)^{(s′,N′)∈P_{0}}} [L_{s′,N′}] for any pair (s, N) ∈ P,

and

(10.16.2)  [L_{s,N}] = ∑_{(s,N)⪯(s′,N′)^{(s′,N′)∈P}} n_{s,N}^{s′,N′} [M_{s′,N′}]

where n_{s,N}^{s′,N′} = 0 or ±1.

We conclude by giving some applications to the structure of the standard modules.

(10.17) Corollary.

(i) The standard modules M_{s,N} ((s, N) ∈ P) are multiplicity free.

(ii) If q is not a root of unity, the standard modules have at most 2 composition factors.

(iii) In all cases, M_{s,N} has composition length bounded by [\frac{n}{2}].

(iv) The maximum composition length of M_{s,N} is [\frac{n}{2}], and therefore is unbounded as n → ∞.

These statements are easy consequences of Theorem (10.16).

11. Concluding remarks

We first wish to explain how our results generalise certain aspects of the theory in [Z], which involves the notion of “multisegments”. Without discussing the general notion in [op. cit.], let us define these in our context as follows.

(11.1) Definition. Let R be an integral domain and q ∈ R^{×}. A segment in R is a sequence

(11.1.1)  I(a, m) = (a, aq^{−2}, aq^{−4}, …, aq^{−2(m−1)})

(a ∈ R^{×}, m ∈ ℤ_{>0}) of elements of R^{×}. In addition, there is a unique segment of length 0, which is the empty sequence. In what follows, some remarks apply only to segments of length ≠ 0, and we shall rely on the context to distinguish these. Note that the elements of I(a, m) are distinct if q is not a root of unity. Otherwise, there may be repetitions in I(a, m). Denote by |I(a, m)| the multiset (i.e. set with multiplicities) underlying I(a, m). Then I(a, m) is determined by |I(a, m)|, together with its initial element.

The length of I(a, m) is m, and we may speak of a subsegment of I(a, m), with the obvious meaning, and similarly for initial and final subsegments of I(a, m).

Suppose I(a_{1}, m_{1}) and I(a_{2}, m_{2}) are two segments. We say that I(a_{1}, m_{1}) precedes I(a_{2}, m_{2}) if a_{2} = a_{1}q^{−2i} for some i ∈ {m_{1} −m_{2} +1, m_{1} −m_{2} +2, ⋯_{1} −1, m_{1}}, where only the non-negative among these integers are included; i.e. if the initial element of I(a_{2}, m_{2}) coincides with one of the final m_{2} elements of I(a_{1}, m_{1} + 1). If I(a_{1}, m_{1}) precedes I(a_{2}, m_{2}), we may define a concatenation I(a_{1}, m_{1})#I(a_{2}, m_{2}) as any segment which starts with I(a_{1}, m_{1} + 1) and finishes with I(a_{2}, m_{2}). Such a segment exists by definition, and any concatenation clearly has length strictly greater than the maximum of m_{1} and m_{2}.

Say that two segments I(a_{1}, m_{1}) and I(a_{2}, m_{2}) are linked if either one precedes the other. Observe that when q is a root of unity, each may precede the other, and there may be many different concatenations I(a_{1}, m_{1})#I(a_{2}, m_{2}) and I(a_{2}, m_{2})#I(a_{1}, m_{1})

Note that in the “generic” case, when q is not a root of unity, this definition (of linked segments) coincides with the definition [Z, 4.1, p. 184].

We may now state our results concerning composition factors as follows. First note that the pairs (s(a, b), N_{k}) ∈ P (cf. (9.12)) correspond bijectively to pairs I_{1} = I(a, n−k) and I_{2} = I(b, k) of segments of length n−k = t+k and k respectively, where n − k = t + k ≥ k. In the statement below, we write M_{I_{1},I_{2}} and L_{I_{1},I_{2}} respectively for the corresponding standard modules M_{s(a,b),N_{k}} and irreducibles L_{s(a,b),N_{k}} (see §10 above)).

(11.2) Theorem.

(i) The standard module M_{s,N} is irreducible if and only if the segments I_{1}, I_{2} are not linked.

(ii) If I_{1} precedes I_{2}, let I′_{1} = I_{1}#I_{2} be any concatenation of I_{1} and I_{2}. This is a segment of length n − k′, where k′ < k (see above). Define I′_{2} = I(b, k′). Then the irreducible Ĥ_{n}^{a}(q)-module L_{I′_{1},I′_{2}} is a composition factor of M_{I_{1},I_{2}}.

(iii) If I_{2} precedes I_{1}, take I′_{1} to be any concatenation I_{2}#I_{1}. This is a segment of length n − k′, where again k′ < k. Define I′_{2} = I(a, k′). Then the irreducible Ĥ_{n}^{a}(q)-module L_{I′_{1},I′_{2}} is a composition factor of M_{I_{1},I_{2}}.

(iv) All composition factors of M_{I_{1},I_{2}} have multiplicity one, and are of the form L_{I′_{1},I′_{2}}, where (I′_{1}, I′_{2}) arises from (I_{1}, I_{2}) by a sequence of transformations of the form described in (ii) and (iii) above.

(v) We have L_{I_{1},I_{2}} = 0 if and only if q^{2} = −1 and I_{2} = −I_{1}, so that n = 2k. (vi) Say that (I_{1}, I_{2}) ⪯ (I′_{1}, I′_{2}) if (I′_{1}, I′_{2}) is obtained from (I_{1}, I_{2}) by a sequence

of transformations of the form (ii) and (iii) above. Then in the Grothendieck group of finite dimensional Ĥ_{n}^{a}(q) modules, we have

[L_{I_{1},I_{2}}] = ∑_{(I_{1},I_{2})⪯(I′_{1},I′_{2})} n_{I_{1},I_{2}}^{I′_{1},I′_{2}}[M_{I′_{1},I′_{2}}]

where n_{I_{1},I_{2}}^{I′_{1},I′_{2}} = 0 or ±1.

These statements are simply restatements of the facts treated in (10.15) and (10.16) above.

Note the similarity between our (11.2)(i) and [Z, Theorem 4.2]. The latter is of course false when q is a root of unity without our modification of the meaning of linked segments. Note also that our statement holds for any q ∈ ℂ, even q = 1.

(11.3). We shall now give a more explicit description of the poset P of bisegments P = (I_{1}, I_{2}), ordered by the relation ⪯, with a view to giving a more explicit version of (11.2)(vi). It is convenient to change notation as follows. We have defined a segment above as I(a, m) = (a, aq^{−2}, …, f) where f = aq^{−2(m−1)}. Write [a, f]_{m} for this segment. This notation applies when a, f ∈ R^{×} and m is a positive integer such that f = aq^{−2(m−1)}. We call a and f the endpoints of the segment. The set P of bisegments is parametrised by unordered pairs P = {[a, f]_{l}, [b, q]_{k}} of segments such that k + l = n. We say P has has length n = k + l and write k(P) =min(k, l). The corresponding standard module and irreducible modules will be denoted M_{P} , L_{P} respectively.

Suppose P and Q are bisegments of the same length. We say that P and Q are transverse if they are of the form {[a, f]_{l}, [b, g]_{k}} and {[a, g]_{i}, [b, f]_{j}} respectively; we say P and Q are parallel if they are of the form {[a, f]_{l}, [b, g]_{k}} and {[a, f]_{i}, [b, g]_{j}} respectively. (Note that the subscripts may be zero.)

Given a pair Q, Q′ of parallel bisegments of the same length, we say that Q and Q′ have the same orientation if they are of the form {[a, f]_{l}, [b, g]_{k}} and {[a, f]_{i}, [b, g]_{j}} respectively with k ≤ l and j ≤ i; otherwise Q and Q′ have opposite orientations.

The following assertions concerning the combinatorics of the poset P may be easily checked.

(i) If P, P′ ∈ P and P ⪯ P′, then P′ is either transverse or parallel to P (possibly both).

(ii) If P′ is transverse to P and k(P′) < k(P), then P ≺^{∘} P′.

(iii) If P is transverse to itself, then {P′ ∈ P | P ⪯ P′} is totally ordered. If P_{1} is the minimal element of {P′ ∈ P | P ≺ P′} (i.e. P_{1} ≠ P) then

[L_{P}] = [M_{P}] − [M_{P_{1}}].

(iv) If P, P′ are parallel, have the same orientation, k(P′) < k(P), and there exists Q with P ≺ Q, then there is a bisegment P_{1} ∈ P such that P ≺^{∘} P_{1} ≺^{∘} P′.

(v) Say that P ≠ P′ ∈ P are opposed if they are parallel and not comparable. Then given Q ∈ P such that there exists P ∈ P with Q ≺ P, there exists at most one element Q′ ∈ P such that Q, Q′ are opposed.

The proof of (v) proceeds by showing that if Q is opposed to both Q′ and Q′′ then Q′, Q′′ are opposed to each other, and hence by (iv) they must have opposite orientations. But they both have orientation opposite to that of Q, whence two such elements Q′, Q′′ cannot exist.

(vi) Suppose that P_{1} is not transverse to itself. If P_{1} ≺ Q ≺ P_{2}, then there exists a unique bisegment Q′ which is opposed to Q.

(vii) If P ≺ Q, then the number of bisegments R in the interval [P, Q] := {R | P ⪯ R ⪯ Q} which are parallel to P is equal to the number of R which are transverse to P.

The structure of the poset P is completely determined by the properties (i) to (vii) above, which are all straightforward to prove. Moreover it is an easy consequence of (vii) that

(11.4) Proposition. Suppose P ∈ P is not transverse to itself. Then

[L_{P}] = ∑_{P ⪯Q} n_{P}^{Q}[M_{Q}],

where

n_{P}^{Q} = \{ \begin{matrix}1 if Q is parallel to P \\ −1 if Q is transverse to P . \end{matrix}

We remark also that given the results of [LLT], [A] and [G], our statement (11.2)(vi) implies that the Kazhdan-Lusztig polynomials which arise from intersection complexes on the closure of the Z_{G}(s) orbit of the nilpotent element N are equal to 1.

Finally, we observe that the results of this paper may be used to discuss aspects of the representation theory of the affine Hecke algebra Ĥ_{n}^{a}(q) over any algebraically closed field of positive characteristic, i.e. the “modular case”. This is carried out for the algebras T LB_{n}(q, Q) in [CGM].

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