NILPOTENT ORBITS OF LINEAR AND CYCLIC QUIVERS AND KAZHDAN-LUSZTIG POYNOMIALS

OF TYPE A

ANTHONY HENDERSON

Abstract. The intersection cohomologies of closures of nilpotent orbits of linear (respectively, cyclic) quivers are known to be described by Kazhdan-Lusztig polynomials for the symmetric group (respectively, the affine symmetric group). We explain how to simplify this description using a combinatorial cancellation procedure, and derive some consequences for representation theory.

Contents

1. Introduction 1 2. Cancellation for the symmetric group 10 3. Nilpotent orbits of the linear quiver 18 4. Cancellation for the affine symmetric group 22 5. Nilpotent orbits of the cyclic quiver 28 References 33

1. Introduction

This paper is concerned with formulas for the intersection cohomologies of closures of nilpotent orbits of linear and cyclic quivers. By fundamental results in geometric representation theory, these intersection cohomologies control certain features of the representations of affine Hecke algebras and quantum affine algebras. There is a well-known formula in the linear case due to Zelevinsky, using Kazhdan-Lusztig polynomials of the symmetric group; there is an analogous formula in the cyclic case due to Lusztig, using Kazhdan-Lusztig polynomials of the affine symmetric group. The main point of this paper is that both formulas can be rewritten in terms of Kazhdan-Lusztig polynomials for different (potentially smaller) symmetric or affine symmetric groups, by applying a combinatorial ‘cancellation’ procedure due to Billey and Warrington. The rewritten formulas in the linear quiver case have already appeared, in a representation-theoretic guise, in the work of Suzuki and others; in the cyclic quiver case they are new.

This work was supported by Australian Research Council grant DP0344185.

In the remainder of the introduction we will survey the main results and their representation-theoretic consequences; the other sections give the proofs, concentrating on the combinatorial side of the story. Sections 2 and 4 are purely combinatorial, explaining the concept of ‘cancellation’ for the symmetric group and affine symmetric group respectively. Most (perhaps all) of the results in Section 2 are known, but we will go over them in detail to provide a reference for the generalizations to the affine case in Section 4. Sections 3 and 5 connect these combinatorial results to the problem of computing intersection cohomology.

Throughout the paper, all vector spaces, algebras and varieties are over ℂ.

Consider the linear quiver of type A_{∞}, with vertex set ℤ and arrows i → i + 1 for all i ∈ ℤ. Finite-dimensional representations of this quiver are parametrized by multisegments: a segment is a nonempty finite interval [i, j] in ℤ, and a multisegment is a finite formal sum of segments. Now fix a ℤ-graded finite-dimensional vector space V = ⨁_{i∈ℤ} V_{i}. Let d_{i} = dim V_{i}, d = dim V. A representation of the quiver on V is simply an element of

N_{V} = {φ ∈ End(V) | φ(V_{i}) ⊆ V_{i+1}, ∀i ∈ ℤ}.

Two such representations are isomorphic if they are in the same orbit for G_{V} = {g ∈ GL(V) | g(V_{i}) = V_{i}, ∀i}, acting on N_{V} by conjugation. Since all elements of N_{V} are nilpotent as endomorphisms of V, we call these nilpotent orbits. They are clearly in bijection with M_{(d_{i})}, the set of multisegments such that each i occurs d_{i} times as an element of a segment. For m ∈ M_{(d_{i})}, let O_{m} denote the corresponding orbit in N_{V}. We put a partial order ⪯ on M_{(d_{i})} by setting m ⪯ m′ if and only if O_{m} is contained in the closure \overline{O_{m′}} of O_{m′}.

The extent to which \overline{O_{m′}} is singular at the points of O_{m} is measured by an intersection cohomology polynomial IC_{m,m′} ∈ ℕ[q], defined by

IC_{m,m′} = ∑_{i} dim H_{m}^{2i}IC(\overline{O_{m′}}) q^{i},

where IC(\overline{O_{m′}}) is the intersection cohomology complex of \overline{O_{m′}}, and H_{m}^{2i} denotes the stalk at a point of O_{m} of the (2i)th cohomology sheaf (it turns out that all odd-degree cohomology sheaves of IC(\overline{O_{m′}}) vanish). Note that IC_{m,m′} is nonzero if and only if m ⪯ m′, and is 1 if m = m′. Hence the inverse matrix (IC_{m,m′}^{⟨−1⟩})_{m,m′∈M_{(d_{i})}} of (IC_{m,m′})_{m,m′∈M_{(d_{i})}} has entries in ℤ[q]; moreover IC_{m,m′}^{⟨−1⟩} is zero unless m ⪯ m′, and is 1 if m = m′.

From the viewpoint of geometric representation theory, the poset M_{(d_{i})}, together with these IC polynomials, is a model for certain “blocks” of representations of Lie-theoretic algebras of type A. More concretely, the algebras listed below each have a collection of finite-dimensional standard modules {M_{m} | m ∈ M_{(d_{i})}} and a collection of finite-dimensional simple modules {L_{m} | m ∈ M_{(d_{i})}}, which are related by the following (equivalent) equations in the Grothendieck group of modules:

[M_{m}] = ∑_{m′∈M_{(d_{i})}} IC_{m,m′}(1) [L_{m′}], ∀m ∈ M_{(d_{i})},

[L_{m}] = ∑_{m′∈M_{(d_{i})}} IC_{m,m′}^{⟨−1⟩}(1) [M_{m′}], ∀m ∈ M_{(d_{i})}.

(1.1)

So the sum of the coefficients of IC_{m,m′} is a composition multiplicity of a standard module; the individual coefficients record the composition multiplicities in a certain Jantzen-like filtration. For each algebra, the general definition of standard modules allows segments of arbitrary complex numbers, not just integers; but the problem of computing composition multiplicities can be reduced to the integer case. The algebras in question, and references to the definitions and results, are as follows.

(1) The affine Hecke algebra Ĥ_{d} attached to GL_{d}, specialized at a parameter which is not a root of unity (as in [5, Definition 12.3.1]). The standard and simple modules were defined by Zelevinksy in [24], and (1.1) was conjectured in [25] (see also [19]). Ginzburg proved (1.1) for standard modules defined in a geometric way (see [6, Theorem 8.6.23]). The fact that Ginzburg’s standard modules coincide with Zelevinsky’s in the Grothendieck group is usually deduced from the Induction Theorem of Kazhdan and Lusztig (see [2] – nowadays the best version of the Induction Theorem to use is [15, Theorem 7.11]), though it should be regarded as a comparatively easy case of that result.

(2) The corresponding degenerate affine Hecke algebra (as in [5, Definition 12.3.2]). The definitions of standard modules (as in [21, Section 2.2]) are analogous to case (1), and indeed (1.1) in this case can be deduced from case (1) by the results of Lusztig in [11] – he also gave a proof specific to this case in [13].

(3) The quantum affine algebra U_{ϵ}(𝔰̂𝔩_{r}), specialized at a parameter ϵ which is not a root of unity (as in [5, Section 12.2A]). Here the standard modules are tensor products of fundamental evaluation modules corresponding to the segments, so we need r to be greater than or equal to the length of the longest segment involved (otherwise, we could just stipulate that any module indexed by a multisegment containing a segment of length > r is zero). Equation (1.1) can be deduced from case (1) by Frobenius-Schur duality (see [5, Section 12.3D]). Alternatively, with a geometric definition of standard modules, (1.1) was proved by Ginzburg and Vasserot (see [23, Theorem 3]). It then follows that the two kinds of standard modules are the same in the Grothendieck group (see [23, Proposition 18]), which can presumably also be proved directly.

(4) The Yangian Y (𝔰𝔩_{r}) (as in [5, Section 12.1A]). The standard modules as defined by Drinfeld in [7] are analogous to those in case (3), and (1.1) in this case can be deduced either from case (3) or case (2) using the results in [7].

This profusion of representation-theoretic meanings of the polynomials IC_{m,m′} and IC_{m,m′}^{⟨−1⟩} is the main reason to be interested in computing them; but it is also why, in this paper, the clear-cut geometric definition is given greater prominence.

A classic result of Zelevinsky ([26, Corollary 1], see Theorem 3.2 below) identifies the polynomials IC_{m,m′} with Kazhdan-Lusztig polynomials of the symmetric group S_{d}. More precisely, it provides an isomorphism of posets between M_{(d_{i})} and a lower ideal M′_{(d_{i})} of the poset of maximal-length representatives of S_{(d_{i})}–S_{(d_{i})} double cosets (under Bruhat order), where S_{(d_{i})} is the parabolic (i.e. Young) subgroup of S_{d} determined by the composition d = ∑_{i} d_{i}; and the polynomials attached to these posets (IC polynomials for M_{(d_{i})}, and Kazhdan-Lusztig polynomials for M′_{(d_{i})}) coincide under this isomorphism. Zelevinsky’s proof is geometric, embedding the nilpotent orbits O_{m} as open subvarieties of certain Schubert varieties, and using the fact that the intersection cohomologies of the latter are described by Kazhdan-Lusztig polynomials; but the point is that the poset M′_{(d_{i})} and its Kazhdan-Lusztig polynomials can be defined (and, in principle, computed) purely combinatorially.

More recent work of Suzuki ([21]) implicitly generalizes this result, providing a family of poset isomorphisms between various upper ideals of M_{(d_{i})} and combinatorially-defined posets. To explain this we adopt the notation of [17], which views multisegments as “generalized skewshapes”. For λ, μ ∈ ℤ^{k}, write λ ⊇ μ if λ_{i} ≥ μ_{i} for all 1 ≤ i ≤ k, and if this holds define a multisegment

(1.2)  λ/μ = ∑_{i=1}^{k} [μ_{i} − i + 1, λ_{i} − i],

where any “empty segments” of the form [s + 1, s] are ignored. The reason for the notation is that if λ and μ are partitions, i.e. λ_{1} ≥ ⋯ ≥ λ_{k} ≥ 0 and similarly for μ, then the segments are exactly the rows of the skew-shape diagram usually called λ/μ, where each box is replaced by its content. Since the order of terms in (1.2) is unimportant, λ/μ = (w · λ)/(w · μ) for all w ∈ S_{k}, where the “dot action” of S_{k} on ℤ^{k} is defined as usual by

(1.3)  (w · λ)_{i} − i = λ_{w^{−1}(i)} − w^{−1}(i).

A fundamental domain for this dot action is

D_{k} := {λ ∈ ℤ^{k} | λ_{1} − 1 ≥ λ_{2} − 2 ≥ ⋯ ≥ λ_{k} − k},

so we can write any multisegment in the standard form

(1.4)  λ/(w · μ) where \{ \begin{matrix}λ, μ ∈ D_{k}, w ∈ S_{k}, λ ⊇ w · μ, and \\ w has maximal length in W_{λ}wW_{μ}. \end{matrix}

Here W_{λ} and W_{μ} are the stabilizers of λ and μ for the dot action, which are clearly parabolic subgroups of S_{k}. Note that this expression in standard form is not uniquely determined by the multisegment, but rather by the multisegment together with a chosen multiset of empty segments.

Example. Let m be the multisegment [1, 2] + [2, 2] + [3, 3]. The most economical way to express this in the form (1.4) is to take k = 3, λ = (4, 4, 5), μ = (3, 3, 3), and w to be the transposition (2, 3). Another way is to take k = 4, λ = (4, 4, 5, 5), μ = (3, 3, 4, 4), and w to be the transposition (2, 4); this effectively adds the empty segment [2, 1].

If λ, μ ∈ D_{k}, we define

S_{k}[λ, μ] = {w ∈ S_{k} | λ ⊇ w · μ} and
S_{k}[λ, μ]^{∘} = {w ∈ S_{k}[λ, μ] | w has maximal length in W_{λ}wW_{μ}}.

These are posets under Bruhat order; in fact we will see that S_{k}[λ, μ] is a lower ideal of S_{k}, so in particular S_{k}[λ, μ] ≠ ∅ ⇔ λ ⊇ μ. With this notation, the generalized form of Zelevinsky’s result can be stated as follows.

Theorem 1.1. Let λ, μ ∈ D_{k} be such that λ ⊇ μ, λ/μ ∈ M_{(d_{i})}.

(1) The map w ↦ λ/(w · μ) is an isomorphism of posets between S_{k}[λ, μ]^{∘} and {m′ ∈ M_{(d_{i})} | λ/μ ⪯ m′}.

(2) For w, w′ ∈ S_{k}[λ, μ]^{∘}, IC_{λ/(w·μ),λ/(w′·μ)} = P_{w,w′}, a Kazhdan-Lusztig polynomial of S_{k}.

Zelevinsky’s original result is the special case where k = d, λ is such that each integer i occurs d_{i} times in (λ_{1} − 1, ⋯, λ_{d} − d), and μ = λ − (1, 1, ⋯, 1). In this case λ/μ is the trivial multisegment ∑_{i} d_{i}[i, i] (corresponding to the zero orbit), so the image of the isomorphism in part (1) is all of M_{(d_{i})}; the parabolic subgroups W_{λ} and W_{μ} both equal S_{(d_{i})}, and S_{d}[λ, μ]^{∘} is the poset M′_{(d_{i})} mentioned above.

The fact that Theorem 1.1 is true in general means that in expressing m and m′ as λ/(w · μ) and λ/(w′ · μ), any of the empty segments which occur in the Zelevinsky case can be “cancelled” without changing the Kazhdan-Lusztig polynomial (or, indeed, new ones can be added). For example, at the extreme, Theorem 1.1 shows that IC_{m,m′} can be identified with a Kazhdan-Lusztig polynomial of S_{k(m)}, where k(m) is the number of segments of m (smaller than d, unless m is trivial). In Section 3 we will use a result of Billey and Warrington, which provides for just such cancellations in Kazhdan-Lusztig polynomials of symmetric groups, to deduce Theorem 1.1 from Zelevinsky’s theorem. The essence of the result is stated in Theorem 3.3, using some matrix notation which will be introduced in §2.

As already mentioned, Theorem 1.1 cannot be considered new, since in the context of the representation theory of the degenerate affine Hecke algebra (case (2) above) it follows from Suzuki’s results in [21]. With notation as in the Theorem, he defines an exact functor F_{λ} from the category O of representations of 𝔤𝔩_{k} to the category of finitedimensional modules for the degenerate affine Hecke algebra associated to GL_{d}, and shows that it takes the Verma module M(w · μ) to the standard module M_{λ/(w·μ)} and the simple module L(w · μ) to the simple module L_{λ/(w·μ)} for all w ∈ S_{k}[λ, μ]^{∘}. Part (1) and the q = 1 specialization of part (2) of Theorem 1.1 then follow from the known Kazhdan-Lusztig conjecture for 𝔤𝔩_{k} (combine [21, (5.2.1) and (5.2.2)]

– the historical remarks following [21, (5.2.3)] properly apply only to the Zelevinsky case). Moreover, by [21, Theorem 5.3.5] the Kazhdan-Lusztig polynomials P_{w,w′} for w′ ∈ S_{k}[λ, μ]^{∘} record multiplicities in a Jantzen-type filtration of M_{λ/(w·μ)}, whose definition clearly depends only on the multisegment (i.e. not on the empty segments); since (2) of Theorem 1.1 is true in the Zelevinsky case, it must be true in general. (As well as [21], see [17] and [1] for the analogous results in the case of the affine Hecke algebra and Yangian respectively).

One corollary concerns those multisegments λ/μ where λ, μ ∈ D_{k} satisfy W_{λ} = W_{μ} = {1}, i.e. λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{k}, μ_{1} ≥ μ_{2} ≥ ⋯ ≥ μ_{k} (these are the “placed skew-shapes” of [18]): for such λ/μ, it follows from Theorem 1.1 that

(1.5)  IC_{λ/μ,λ/(w·μ)}^{⟨−1⟩} = ε(w), for all w ∈ S_{k}[λ, μ],

where ε denotes the sign character. Thus the corresponding simple modules (called calibrated for the affine Hecke algebra in [18] and tame for the Yangian in [16]) can be written as an alternating sum of standard modules in the Grothendieck group. Representation-theoretically, this reflects the existence of a BGG-like resolution of these simple modules (transferred by the appropriate functor from the BGG resolution of the 𝔤𝔩_{k}-module L(μ)) – see [21, Theorem 5.1.1] and [17, (4.13)].

The justification for re-proving Theorem 1.1 in §3 below is that the combinatorics involved generalizes immediately to the case of cyclic quivers, as we will now explain.

Fix a positive integer n, and consider the cyclic quiver of type { \atop A_{n−1}}, with vertex set ℤ/nℤ and arrows ī → \overline{i + 1} for all ī ∈ ℤ/nℤ. Finitedimensional nilpotent representations of this quiver are parametrized by multisegments as before, except that there is no difference between segments [i, j] and [i′, j′] when i′ − i = j′ − j is a multiple of n. Fix a (ℤ/nℤ)-graded finite-dimensional vector space V = ⨁_{ī∈ℤ/nℤ} Vī, and set d_{i} = dim Vī, d = dim V. We define

N_{V} = \{φ ∈ End(V) | φ(Vī) ⊆ V\overline{i+1}, ∀ī ∈ ℤ/nℤ, φ nilpotent\},

and consider G_{V} -orbits in N_{V}. These are in bijection with M_{(d_{i}),n}, the set of multisegments (in this modulo n sense) such that each congruence class ī occurs d_{i} times among the elements of the segments. For m ∈ M_{(d_{i}),n}, let O_{m} denote the corresponding nilpotent orbit, and define a partial order ⪯ and polynomials IC_{m,m′} ∈ ℕ[q] and IC_{m,m′}^{⟨−1⟩} ∈ ℤ[q] in the same way as before.

These polynomials too have representation-theoretic significance. The specialized quantum affine algebra U_{ζ}(𝔰̂𝔩_{r}), where ζ^{2} is a primitive nth root of 1, has a collection of standard modules {M_{m} | m ∈ M_{(d_{i}),n}} and a collection of simple modules {L_{m} | m ∈ M_{(d_{i}),n}}, satisfying the equivalent equations

[M_{m}] = ∑_{m′∈M_{(d_{i}),n}} IC_{m,m′}(1) [L_{m′}], ∀m ∈ M_{(d_{i}),n},

[L_{m}] = ∑_{m′∈M_{(d_{i}),n}} IC_{m,m′}^{⟨−1⟩}(1) [M_{m′}], ∀m ∈ M_{(d_{i}),n}.

(1.6)

(See [23, Theorem 3] – again, for small r we have to disregard multisegments containing a segment of length > r.) The same is true for the affine Hecke algebra Ĥ_{d} specialized at a primitive nth root of unity, except that there the simple modules are parametrized by the smaller set of aperiodic multisegments (see [10, Section 2]), so we have to set [L_{m}] = 0 if m is not aperiodic.

The analogue of Zelevinsky’s result for cyclic quivers was proved by Lusztig in [12, §11] (it is stated below as Theorem 5.2). This identifies IC_{m,m′} with a Kazhdan-Lusztig polynomial of the affine symmetric group S̃_{d} (the Coxeter group of type { \atop A_{d−1}}). In Section 4, we will show that a version of Billey and Warrington’s cancellation works for the affine symmetric group. As a consequence, we get an analogue of Theorem 1.1 in this setting, for which a representation-theoretic proof does not yet exist.

To state it requires extending the dot action of S_{k} on ℤ^{k} to S̃_{k}, so that the extra Coxeter generator s_{0} acts by

(s_{0} · λ)_{i} = \{ \begin{matrix}λ_{k} − k + 1 + n, & if i = 1, \\ λ_{i}, & if 2 ≤ i ≤ k − 1, \\ λ_{1} − 1 + k − n, & if i = k. \end{matrix}

It is then clear that (w ·λ)/(w ·μ) = λ/μ for all w ∈ S̃_{k}, where the multisegments are now interpreted in the modulo n sense. A fundamental domain for the action of S̃_{k} on ℤ^{k} is

D̃_{k} := {λ ∈ ℤ^{k} | λ_{1} − 1 ≥ λ_{2} − 2 ≥ ⋯ ≥ λ_{k} − k ≥ λ_{1} − n − 1},

and the corresponding standard form for multisegments is

(1.7)  λ/(w · μ) where \{ \begin{matrix}λ, μ ∈ D̃_{k}, w ∈ S̃_{k}, λ ⊇ w · μ, and \\ w has maximal length in W̃_{λ}wW̃_{μ}, \end{matrix}

where W̃_{λ} and W̃_{μ} denote the stabilizers of λ and μ in S̃_{k} (proper parabolic subgroups, hence finite). For λ, μ ∈ D̃_{k}, we define

S̃_{k}[λ, μ] = {w ∈ S̃_{k} | λ ⊇ w · μ} and
S̃_{k}[λ, μ]^{∘} = {w ∈ S̃_{k}[λ, μ] | w has maximal length in W̃_{λ}wW̃_{μ}}.

We will see in §5 that, as in the symmetric group case, S̃_{k}[λ, μ] is a (finite) lower ideal of S̃_{k} for Bruhat order. We can now state a generalization of Lusztig’s result.

Theorem 1.2. Let λ, μ ∈ D̃_{k} be such that λ ⊇ μ, λ/μ ∈ M_{(d_{i}),n}.

(1) The map w ↦ λ/(w · μ) is an isomorphism of posets between S̃_{k}[λ, μ]^{∘} and {m′ ∈ M_{(d_{i}),n} | λ/μ ⪯ m′}.

(2) For w, w′ ∈ S̃_{k}[λ, μ]^{∘}, IC_{λ/(w·μ),λ/(w′·μ)} = P_{w,w′}, a Kazhdan-Lusztig polynomial of S̃_{k}.

(An alternative statement using matrix notation is given in Theorem 5.3.) It is natural to wonder whether there is a representation-theoretic functor which “explains” this Theorem too.

As in the linear quiver case, Theorem 1.2 implies that IC_{m,m′} can be identified with a Kazhdan-Lusztig polynomial of { \atop S_{k(m)}}, where k(m) is the number of segments of m; this immediately implies the main result of [8], that IC_{m,m′} = 1 when m ⪯ m′, k(m) = 2.

Another consequence of Theorem 1.2 is an analogue of (1.5), concerning those multisegments λ/μ where λ, μ ∈ D̃_{k} satisfy W̃_{λ} = W̃_{μ} = {1}; this means that λ_{1} ≥ λ_{2} ≥ ⋯ λ_{k} ≥ λ_{1} − n + k, μ_{1} ≥ μ_{2} ≥ ⋯ ≥ μ_{k} ≥ μ_{1} − n + k. For such λ/μ, it follows from Theorem 1.2 that

(1.8)  IC_{λ/μ,λ/(w.μ)}^{⟨−1⟩} = ε(w), for all w ∈ S̃_{k}[λ, μ].

So once more the corresponding simple modules can be written as an alternating sum of standard modules in the Grothendieck group; probably this indicates a BGG-like resolution.

Theorems 1.1 and 1.2 combine well with the method used by Varagnolo and Vasserot in [22] to determine the decomposition numbers of U_{ζ}(𝔤𝔩_{r}) where ζ^{2} is a primitive nth root of 1. Suppose we want to compute the multiplicity of the simple module L_{ζ}(μ′) in the Weyl module V_{ζ}(λ′), where λ and μ are partitions with at most k parts all of size ≤ r, and λ′ and μ′ are the transpose partitions (regarded as dominant integral weights for 𝔤𝔩_{r}). By definition, V_{ζ}(λ′) is the specialization at ζ of the simple module V_{q}(λ′) for the generic U_{q}(𝔤𝔩_{r}). Now using a suitable normalization of the evaluation map U_{q}(𝔰̂𝔩_{r}) → U_{q}(𝔤𝔩_{r}), we can regard V_{q}(λ′) as the simple U_{q}(𝔰̂𝔩_{r})-module L_{λ/0} (see [22, Section 12.2]). By (1.5), we have the equation

(1.9)  [L_{λ/0}] = ∑_{w∈S_{k}[λ,0]} ε(w) [M_{λ/(w·0)}].

Now let w_{λ}, w_{μ}, w_{0} ∈ S̃_{k} be such that w_{λ} · λ, w_{μ} · μ, w_{0} · 0 ∈ D̃_{k}. As noted in [22, Section 12.3], the specialization at ζ of the standard module M_{λ/(w·0)} is merely the U_{ζ}(𝔰̂𝔩_{r})-standard module of the same name, which in standard form is M_{(w_{λ}·λ)/(w_{λ}ww_{0}^{−1})^{∘}·(w_{0}·0)}, where (w_{λ}ww_{0}^{−1})^{∘} is

the longest element of { \atop W_{w_{λ}·λ}}w_{λ}ww_{0}^{−1}{ \atop W_{w_{0}·0}}. So in the Grothendieck group of U_{ζ}(𝔰̂𝔩_{r})-modules,

(1.10)  [V_{ζ}(λ′)] = ∑_{w∈S_{k}[λ,0]} ε(w) [M_{(w_{λ}·λ)/(w_{λ}ww_{0}^{−1})^{∘}·(w_{0}·0)}].

Now as noted in [22, Section 12.2], L_{ζ}(μ′) regarded as a simple U_{ζ}(𝔰̂𝔩_{r})- module is L_{μ/0} = L_{(w_{μ}·μ)/(w_{μ}w_{0}^{−1})^{∘}·(w_{0}·0)}, where (w_{μ}w_{0}^{−1})^{∘} is the longest

element of { \atop W_{w_{μ}·μ}}w_{μ}w_{0}^{−1}{ \atop W_{w_{0}·0}}. Using Theorem 1.2, we obtain

[V_{ζ}(λ′) : L_{ζ}(μ′)] = \{ \begin{matrix}∑_{w∈S_{k}} ε(w) P_{(w_{λ}ww_{0}^{−1})^{∘},(w_{μ}w_{0}^{−1})^{∘}}(1), & if μ ∈ S̃_{k} · λ, \\ 0, & otherwise. \end{matrix} (1.11)

In the first case, summing over all of S_{k} rather than just S_{k}[λ, 0] introduces no new terms, since (w_{μ}w_{0}^{−1})^{∘} ∈ S̃_{k}[w_{μ} · μ, w_{0} · 0] = S̃_{k}[w_{λ} · λ, w_{0}·0], so the Kazhdan-Lusztig polynomial can only be nonzero when w_{λ}ww_{0}^{−1} ∈ S̃_{k}[w_{λ} · λ, w_{0} · 0], i.e. λ ⊇ w · 0.

In the special case that λ and μ have trivial stabilizers in S̃_{k} (i.e. λ_{1} − 1, ⋯, λ_{k} − k have different residues modulo n, and similarly for μ – this requires k ≤ n, which automatically implies w_{0} = 1), (1.11) becomes

(1.12) [V_{ζ}(λ′) : L_{ζ}(μ′)] = \{ \begin{matrix}∑_{w∈S_{k}} ε(w) P_{w_{λ}w,w_{μ}}(1), & if μ ∈ S̃_{k} · λ, \\ 0, & otherwise. \end{matrix}

This is the form of the answer given by Soergel in [20, Conjecture 7.1] for the equivalent problem of computing tilting module multiplicities for U_{ζ}(𝔤𝔩_{k}). Acknowledgements. This work grew out of stimulating conversations with my colleagues A. Mathas and A. Molev, and I would like to thank them for their help. I am also grateful to G. Lusztig, K. McGerty, A. Parker, A. Ram, J. Ramagge, and G. Warrington for their valuable comments on an earlier version of this paper.

2. Cancellation for the symmetric group

In this section we explain the combinatorial result of Billey and Warrington on which our approach depends. Fix a positive integer d, and let S_{d} be the group of permutations of [1, d] = {1, ⋯, d}. For i ∈ [1, d − 1], we define s_{i} ∈ S_{d} to be the transposition interchanging i and i + 1; as everyone knows, s_{1}, ⋯, s_{d−1} form a set of Coxeter generators for S_{d} of type A_{d−1}. We thus have a length function ℓ : S_{d} → ℕ, a Bruhat order ≤, and Kazhdan-Lusztig polynomials P_{y,w} ∈ ℕ[q] for y, w ∈ S_{d} (which are nonzero iff y ≤ w). Good references for Kazhdan-Lusztig polynomials are [9, Chapter 7] and [20] (where the notation is somewhat different).

The length function and the Bruhat order have well-known combinatorial descriptions. Define the inversion statistics

inv_{i}(w) = |{i′ < i | w(i′) > w(i)}|, Inv_{i}(w) = |{i′ > i | w(i′) < w(i)}|,

for any w ∈ S_{d} and i ∈ [1, d]. These are related by Inv_{i}(w) = inv_{i}(w) + w(i) − i. Then

(2.1)  ℓ(w) = ∑_{i∈[1,d]} inv_{i}(w) = ∑_{i∈[1,d]} Inv_{i}(w).

A special case of Bruhat order is that for all i ∈ [1, d − 1],

(2.2)  ws_{i} < w if and only if w(i) > w(i + 1).

The general description, due to Deodhar, is as follows:

Proposition 2.1. If y, w ∈ S_{d}, y ≤ w if and only if for all i, j ∈ [1, d],

|{i′ ≤ i | y(i′) ≥ j}| ≤ |{i′ ≤ i | w(i′) ≥ j}|.

In other words, for all i ∈ [1, d] and m ∈ [1, i], the mth largest element in y[1, i] is less than or equal to the mth largest element in w[1, i]. If y ≤ w, we write [y, w] for the Bruhat interval {x ∈ S_{d} | y ≤ x ≤ w}.

We now come to the key definition.

Definition. If y ≤ w in S_{d}, we say that i ∈ [1, d] is cancellable for the interval [y, w] if y(i) = w(i), inv_{i}(y) = inv_{i}(w), and Inv_{i}(y) = Inv_{i}(w). (Clearly any two of these conditions imply the third.)

The reason for the name ‘cancellable’ is that Bruhat order and Kazhdan-Lusztig polynomials are preserved under the operation of ‘cancelling the common action on i’ from the permutations in question, in the following sense. For all i ∈ [1, d], let σ_{i} : [1, d] \ {i} → [1, d − 1] be the unique order-preserving bijection. For w ∈ S_{d}, we define wî ∈ S_{d−1} by

wî = σ_{w(i)} ∘ w ∘ σ_{i}^{−1}.

It is clear from either formula in (2.1) that

(2.3)  ℓ(wî) = ℓ(w) − inv_{i}(w) − Inv_{i}(w).

The following result combines Lemmas 17 and 39 of [3], but we will spell out the proof for later reference.

Proposition 2.2. Suppose that i is cancellable for [y, w].

(1) For any x ∈ [y, w], x(i) = y(i) and inv_{i}(x) = inv_{i}(y). Hence i is cancellable for any sub-interval of [y, w].

(2) x ↦ xî is an isomorphism of posets between [y, w] and [yî, wî], which reduces all lengths by the same amount.

(3) For any u, v ∈ [y, w], P_{u,v} = P_{uî,vî}.

Proof. Set j = y(i) = w(i), m = inv_{i}(y) + 1 = inv_{i}(w) + 1, and suppose y ≤ x ≤ w. Now y[1, i] and w[1, i] each have exactly m elements ≥ j and m − 1 elements > j. By Proposition 2.1, the same is true of x[1, i]. Similarly, y[1, i − 1] and w[1, i − 1] each have exactly m − 1 elements ≥ j and m − 1 elements > j, so the same is true of x[1, i − 1]. Thus x(i) = j and inv_{i}(x) = m − 1, proving (1). Moreover, it is clear from Proposition 2.1 that yî ≤ xî ≤ wî. The construction of the map [yî, wî] → [y, w] : x ↦ x̃ inverse to x ↦ xî is easy:

x̃(i′) = \{ \begin{matrix}j, & if i′ = i, \\ σ_{j}^{−1}(x(σ_{i}(i′))), & if i′ ≠ i. \end{matrix}

This proves the isomorphism part of (2), and the statement about lengths follows from (1). In light of parts (1) and (2), it clearly suffices to prove (3) in the case u = y, v = w. We prove this by induction on ℓ(w), it being trivial if w = 1. Choose one of the Coxeter generators, say s, such that ws < w. We now have three cases. Case 1: s = s_{i−1}. This means that w(i − 1) > j, so w[1, i − 2] has only m − 2 elements > j. Therefore the same is true of y[1, i − 2], so y(i − 1) > j, i.e. ys < y. Moreover, ws[1, i − 1] has only m − 2 elements > j, so y ≰ ws. Under these circumstances we have (see (2.5) below)

(2.4)  P_{y,w} = P_{ys,ws}.

Obviously i − 1 is cancellable for [ys, ws], so by the induction hypothesis, P_{ys,ws} = P_{(ys){ \atop i−1},(ws){ \atop i−1}}. But (ys){ \atop i−1} = yî and (ws_{i−1}){ \atop i−1} = wî, so we have the result. Case 2: s = s_{i}. This means that w(i + 1) < j, so w[1, i + 1] has only m − 1 elements > j. Therefore the same is true of y[1, i + 1], so y(i + 1) < j, i.e. ys < y. Moreover, ws[1, i] has only m − 1 elements ≥ j, so y ≰ ws. The proof proceeds as in Case 1, with i + 1 in place of i − 1. Case 3: s ≠ s_{i−1}, s_{i}. The fundamental recursive property of Kazhdan-Lusztig polynomials ([9, Section 7.11, (23)]) tells us that

(2.5) P_{y,w} = P_{y′,ws} + qP_{y′s,ws} − ∑_{\begin{matrix}y≤z<ws \\ zs<z \end{matrix}} μ(z, ws) q^{(ℓ(w)−ℓ(z))/2}P_{y,z},

where μ(z, ws) is the coefficient of q^{(ℓ(ws)−ℓ(z)−1)/2} in P_{z,ws}, and y′ is the minimum of y and ys in Bruhat order. All the nonzero Kazhdan-Lusztig polynomials involved in the right-hand side are indexed by elements of the interval [y′, ws], for which i is cancellable. By the induction hypothesis, they can all be replaced by the analogous polynomials for the interval [(y′)î, (ws)î], and the result follows. □

We now recall (and extend slightly) the matrix notation used in [26]. Let (b_{i})_{i∈[1,n]} be an n-tuple of nonnegative integers whose sum is d, and let (c_{j})_{j∈[1,n′]} be an n′-tuple of nonnegative integers whose sum is also d. To avoid notational clutter, we make the convention for the rest of this section that the range of the variables i and i′ will be [1, n] unless otherwise specified, and that of the variables j and j′ will be [1, n′]. We will use boldface letters such as m and m′ for the (n × n′)-matrices whose entries are written with the corresponding ordinary letters m_{i,j} and m′_{i,j}. Let M_{(b_{i});(c_{j})} be the set of all (n × n′)-matrices m satisfying:

(1) m_{i,j} ∈ ℕ, for all i, j,

(2) ∑_{j} m_{i,j} = b_{i}, for all i, and

(3) ∑_{i} m_{i,j} = c_{j}, for all j. If any b_{i} or c_{j} is 0, the corresponding row or column must always be zero and is therefore irrelevant, but it will be convenient to allow this possibility. We will use an obvious notation for the sums of various sectors of a matrix:

m_{≤i,≥j} = ∑_{\begin{matrix}i′≤i \\ j′≥j \end{matrix}} m_{i′,j′}, m_{≤i,j} = ∑_{i′≤i} m_{i′,j}, m_{i,≥j} = ∑_{j′≥j} m_{i,j′},

and similarly m_{≥i,≤j}, etc. Note that for m ∈ M_{(b_{i});(c_{j})},

\begin{aligned}m_{≥i,≤j} & = c_{1} + c_{2} + ⋯ + c_{j} − m_{≤i−1,≤j} \\ & = c_{1} + ⋯ + c_{j} − b_{1} − ⋯ − b_{i−1} + m_{≤i−1,≥j+1}. \end{aligned} ; (2.6)

The matrices in M_{(b_{i});(c_{j})} parametrize double cosets of S_{d} with respect to certain parabolic subgroups. Namely, write [1, d] as the disjoint union of blocks B_{1}, ⋯, B_{n} such that all elements of B_{i} are less than all elements of B_{i+1}, and |B_{i}| = b_{i}. (Because we are allowing some b_{i} to be zero, some of these blocks could be empty.) Let S_{(b_{i})} be the subgroup of S_{d} which preserves each B_{i} separately; this is a parabolic subgroup isomorphic to S_{b_{1}} × ⋯ × S_{b_{n}}. Similarly define blocks C_{j} of sizes c_{j}, and the parabolic subgroup S_{(c_{j})}. We define a surjective map ψ : S_{d} → M_{(b_{i});(c_{j})} by

ψ(w)_{i,j} = |w(B_{i}) ∩ C_{j}|.

The fibres of ψ are exactly the double cosets S_{(c_{j})}wS_{(b_{i})}, so ψ induces a bijection S_{(c_{j})}\S_{d} / S_{(b_{i})} ↔ M_{(b_{i});(c_{j})}. For m ∈ M_{(b_{i});(c_{j})}, let w_{m} ∈ S_{d} be the longest element in the corresponding double coset.

Note that in the case when n = n′ = d and all b_{i} = c_{j} = 1, the parabolic subgroups are trivial, and we have merely passed from elements of S_{d} to the corresponding permutation matrices (or their transposes, depending on your convention). In general, the permutation w_{m} can be constructed from the matrix m as follows: assuming that the images of B_{i′} for i′ < i have been determined, we send successive various-sized sub-blocks of B_{i} to the various C_{j}s, according to the entries of the ith row of m read from right to left. Within each sub-block, we successively take the largest element of C_{j} still unused. More formally, if a is the sth element of B_{i}, then w_{m}(a) ∈ C_{j} where j is maximal such that m_{i,≥j} ≥ s. Specifically, w_{m}(a) is the tth largest element of C_{j} where

(2.7)  t = m_{≤i−1,j} + s − m_{i,≥j+1}.

Example. Take d = 9, n = n′ = 4, and define b_{i}, c_{j} so that

\begin{matrix}B_{1} = \{1\}, B_{2} = \{2, 3, 4, 5\}, B_{3} = \{6, 7, 8\}, B_{4} = \{9\}, \\ C_{1} = \{1, 2\}, C_{2} = \{3, 4, 5\}, C_{3} = \{6, 7, 8\}, C_{4} = \{9\}. \end{matrix}

Let us construct w_{m} where

m = (\begin{matrix}1 & 0 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) .

The first row tells us that w_{m}(1) is an element of C_{1}; we take the largest element, namely 2. The second row tells us that w_{m}(B_{2}) consists of one element of C_{3}, two elements of C_{2}, and one of C_{1}, in that order. Taking the largest elements not yet used, we set w_{m}(2) = 8, w_{m}(3) = 5, w_{m}(4) = 4, and w_{m}(5) = 1. Continuing in this way, we see that w_{m} is the permutation 285417639 (in ‘one-line’ notation).

We define a length function ℓ : M_{(b_{i});(c_{j})} → ℕ by ℓ(m) = ℓ(w_{m}), and a partial order on M_{(b_{i});(c_{j})} by

m ≤ m′ ⇔ w_{m} ≤ w_{m′}.

These can be described as follows.

Proposition 2.3. Let m, m′ ∈ M_{(b_{i});(c_{j})}.

(1) ℓ(m) = ∑_{i,j} m_{i,j}m_{≤i,≥j} − ∑_{i,j} \binom{m_{i,j}+1}{2}.

(2) m ≤ m′ if and only if, for all i, j,

m_{≤i,≥j} ≤ m′_{≤i,≥j}.

(3) m ≤ m′ if and only if, for all i, j,

m_{≥i,≤j} ≤ m′_{≥i,≤j}.

Proof. Let a be the largest element of B_{i} ∩ w_{m}^{−1}(C_{j}). Then for 1 ≤ k ≤ m_{i,j}, a − k + 1 is the kth largest element of B_{i} ∩ w_{m}^{−1}(C_{j}). Clearly

(2.8)  inv_{a−k+1}(w_{m}) = m_{≤i,≥j} − k.

Summing this over all i, j, and 1 ≤ k ≤ m_{i,j} gives (1). To prove (2), fix i and j, and let b be the largest element of ∪_{i′≤i}B_{i′} and c the smallest element of ∪_{j′≥j}C_{j′}. If m ≤ m′, then by Proposition 2.1, we have

|{a ≤ b | w_{m}(a) ≥ c}| ≤ |{a ≤ b | w_{m′}(a) ≥ c}|,

which exactly says that m_{≤i,≥j} ≤ m′_{≤i,≥j}. Conversely, suppose we know that m_{≤i,≥j} ≤ m′_{≤i,≥j} and m_{≤i−1,≥j} ≤ m′_{≤i−1,≥j}. For all 1 ≤ k ≤ b_{i}, we have

(2.9) |{a ≤ b − k + 1 | w_{m}(a) ≥ c}| = max{m_{≤i,≥j} − k + 1, m_{≤i−1,≥j}}

and similarly for m′, so our assumption implies

|{a ≤ b − k + 1 | w_{m}(a) ≥ c}| ≤ |{a ≤ b − k + 1 | w_{m′}(a) ≥ c}|.

Combining these statements for all j tells us that for all m, the mth largest element of w_{m}[1, b−k+1] lies in a block C_{j} prior or equal to that containing the mth largest element of w_{m′}[1, b − k + 1]. Remembering how w_{m} and w_{m′} are constructed from m and m′, this implies that for all m, the mth largest element of w_{m}[1, b − k + 1] is less than or equal to the mth largest element of w_{m′}[1, b − k + 1]. Letting i and k vary, we get w_{m} ≤ w_{m′} by Proposition 2.1, so (2) is proved. One way to deduce (3) from (2) is to use (2.6). Another way is to recall that w_{m} ≤ w_{m′} if and only if w_{m}^{−1} ≤ w_{m′}^{−1}; clearly the inverse of w_{m} is the permutation w_{m^{t}} associated to the transpose matrix m^{t} ∈ M_{(c_{j});(b_{i})}, and the condition in (3) is the transpose of the condition in (2). □

We can also define Kazhdan-Lusztig polynomials indexed by pairs of elements of M_{(b_{i});(c_{j})}: P_{m,m′} = P_{w_{m},w_{m′}}. By definition we have

(2.10)  P_{m,m′} ≠ 0 ⇒ m ≤ m′, and P_{m,m} = 1.

So the inverse matrix (P_{m,m′}^{⟨−1⟩})_{m,m′∈M_{(b_{i});(c_{j})}} of (P_{m,m′})_{m,m′∈M_{(b_{i});(c_{j})}} has entries in ℤ[q] which also satisfy (2.10). In fact, we can express these entries in terms of those of the original matrix, as follows. Recall the Kazhdan-Lusztig inversion formula ([9, Section 7.14, (24)]):

(2.11)  ∑_{x∈S_{d}} ε(xy) P_{xw_{0}^{(d)},yw_{0}^{(d)}} P_{x,w} = δ_{y,w},

where ε(z) = (−1)^{ℓ(z)} and w_{0}^{(d)} is the longest element of S_{d}. Using the fact that P_{x,w_{m′′}} = P_{x′,w_{m′′}} for all x′ ∈ S_{(c_{j})}xS_{(b_{i})} ([9, Section 7.14, Corollary]), we get

(2.12)  P_{m,m′}^{⟨−1⟩} = ∑_{x∈S_{(c_{j})}w_{m′}S_{(b_{i})}}ε(xw_{m}) P_{xw_{0}^{(d)},w_{m}w_{0}^{(d)}}.

A general Kazhdan-Lusztig polynomial P_{y,w}, y, w ∈ S_{d}, can be expressed in the form P_{m,m′} in various ways. The most trivial takes n = n′ = d and all b_{i} = c_{j} = 1, so that there is no difference between permutations and matrices. At the other extreme of usefulness, we can take (B_{i}) to be the collection consisting of the maximal intervals on which w is decreasing, and (C_{j}) the same thing for w^{−1}. With these choices, w is clearly the longest element in its double coset S_{(c_{j})}wS_{(b_{i})}, so P_{y,w} depends only on the double coset of y; in other words, P_{y,w} = P_{ψ(y),ψ(w)} where ψ : S_{d} → M_{(b_{i});(c_{j})} is as above.

Example. Let y = 128456379 and w = 587429316 in S_{9}. The blocks B_{i} and C_{j} determined by w are exactly those used in the previous example. Indeed, w = w_{m′} where

m′ = (\begin{matrix}0 & 1 & 0 & 0 \\ 1 & 1 & 2 & 0 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}) .

Now ψ(y) is the matrix m from the previous example, so the permutation w_{m} = 285417639 found there is the longest element in the double coset S_{(c_{j})}yS_{(b_{i})}. Using the criteria in Proposition 2.3, it is easy to check that m ≤ m′. The above principle means in this case that

P_{y,w} = P_{m,m′} = P_{w_{m},w}.

The advantage of the latter form is that 2 is cancellable for [w_{m}, w]. Since wˆ2 m = 25417638 and wˆ2 = 57428316, we get P_{y,w} = P_{25417638,57428316}.

In order to be able to perform such a cancellation directly on matrices, we note the following.

Proposition 2.4. Let m ∈ M_{(b_{i});(c_{j})}, a ∈ B_{i} ∩ w_{m}^{−1}(C_{j}). Let e be the matrix with e_{i,j} = 1, all other entries zero.

(1) wâ ; m = w_{m−e}.

(2) ℓ(m) − ℓ(m − e) equals each of the following:

\begin{aligned}m_{≤i,≥j} + m_{≥i,≤j} − m_{i,j} − 1 & = m_{≤i−1,≥j} + m_{≥i+1,≤j} + b_{i} − 1 \\ & = m_{≤i,≥j+1} + m_{≥i,≤j−1} + c_{j} − 1. \end{aligned}

Proof. Part (1) is clear from the explicit construction of w_{m} given above, and (2) follows easily from (1) of Proposition 2.3. □

Definition. If m ≤ m′ in M_{(b_{i});(c_{j})}, we say that (i, j) ∈ [1, n] × [1, n′] is cancellable for the interval [m, m′] if

(1) m_{i,j} ≥ 1.

(2) m_{≤i−1,≥j} = m′_{≤i−1,≥j}, or equivalently m_{≥i,≤j−1} = m′_{≥i,≤j−1}.

(3) m_{≤i,≥j+1} = m′_{≤i,≥j+1}, or equivalently m_{≥i+1,≤j} = m′_{≥i+1,≤j}. These equivalences follow from (2.6).

Proposition 2.5. Suppose that m ≤ m′ in M_{(b_{i});(c_{j})} and (i, j) is cancellable for [m, m′]. Let e be the matrix with e_{i,j} = 1, all other entries zero.

(1) For any m^{1} ∈ [m, m′], (a) m_{i,j}^{1} ≥ m_{i,j},

(b) m_{≤i−1,≥j}^{1} = m_{≤i−1,≥j}, and

(c) m_{≤i,≥j+1}^{1} = m_{≤i,≥j+1}.

Hence (i, j) is cancellable for any sub-interval of [m, m′].

(2) The map m^{1} ↦ m^{1} − e is an isomorphism of posets between [m, m′] and [m − e, m′ − e], which reduces all lengths by the same amount.

(3) For any m^{1}, m^{2} ∈ [m, m′], P_{m^{1},m^{2}} = P_{m^{1}−e,m^{2}−e}.

(4) For any m^{1}, m^{2} ∈ [m, m′], P_{m^{1},m^{2}}^{⟨−1⟩} = P_{m^{1}−e,m^{2}−e}^{⟨−1⟩}.

Proof. Let m^{1} ∈ [m, m′]. By (2) of Proposition 2.3, we have

m_{≤i−1,≥j} ≤ m_{≤i−1,≥j}^{1} ≤ m′_{≤i−1,≥j} = m_{≤i−1,≥j},

which proves (1b), and (1c) is similar. It follows that

m_{i,j}^{1} − m_{i,j} = (m_{i,j}^{1} + m_{≤i−1,≥j}^{1} + m_{≤i,≥j+1}^{1}) − (m_{i,j} + m_{≤i−1,≥j} + m_{≤i,≥j+1})
= (m_{≤i,≥j}^{1} + m_{≤i−1,≥j+1}^{1}) − (m_{≤i,≥j} + m_{≤i−1,≥j+1})
= (m_{≤i,≥j}^{1} − m_{≤i,≥j}) + (m_{≤i−1,≥j+1}^{1} − m_{≤i−1,≥j+1})
≥ 0,

by Proposition 2.3 again. So (1a) is proved. Thus m^{1} − e ∈ M_{(b̃_{i});(c̃_{j})}, where

b̃_{i′} = b_{i′} − δ_{i,i′}, c̃_{j′} = c_{j′} − δ_{j,j′}.

Given this, the first part of (2) is obvious from either description of the partial order given in Proposition 2.3, and the second part from (2) of Proposition 2.4. To prove (3), let a be the largest element of B_{i} ∩ w_{m}^{−1}(C_{j}), i.e. the (m_{i,≥j})th element of B_{i}. We want to show that a is cancellable for [w_{m}, w_{m′}]. Let

δ = m′_{i,≥j} − m_{i,≥j} = m′_{≤i,j} − m_{≤i,j} = m′_{≤i,≥j} − m_{≤i,≥j} ≥ 0.

(These are equal because (i, j) is cancellable, and nonnegative because m ≤ m′.) By the above chain of equalities applied to m^{1} = m′, we have

m′_{i,j} − m_{i,j} = δ + (m′_{≤i−1,≥j+1} − m_{≤i−1,≥j+1}) ≥ δ,

so

m′_{i,≥j+1} = m′_{i,≥j} − m′_{i,j} = m_{i,≥j} + δ − m′_{i,j} < m_{i,≥j} ≤ m′_{i,≥j},

which means that w_{m′}(a) ∈ C_{j}. From (2.7) we see that w_{m′}(a) = w_{m}(a). Moreover,

inv_{a}(w_{m}) = m_{≤i,≥j} − 1 = m′_{≤i,≥j} − δ − 1 = inv_{a}(w_{m′}),

so a is cancellable for [w_{m}, w_{m′}]. In particular, for any m^{1} ∈ [m, m′], w_{m^{1}}(a) ∈ C_{j}, which by (1) of Proposition 2.4 implies wâ ; m^{1} = w_{m^{1}−e}. Then part (3) follows from (3) of Proposition 2.2, and part (4) follows formally from parts (2) and (3). □

Example. With matrices m, m′ defined as in previous examples, (2, 3) is cancellable for [m, m′], corresponding to the fact that 2 is cancellable for [w_{m}, w_{m′}]. Performing the cancellation directly on the matrices, we get

m − e = (\begin{matrix}1 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) , m′ − e = (\begin{matrix}0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}) .

The reader can check that these matrices correspond to the permutations 25417638 and 57428316 found earlier.

3. Nilpotent orbits of the linear quiver

We now return to the set-up of the first part of the introduction, so V is a d-dimensional ℤ-graded vector space, with d_{i} = dim V_{i}. For convenience, we adjust the grading so that d_{i} ≠ 0 ⇒ i ∈ [1, n], for some positive integer n (so we are effectively considering the linear quiver of type A_{n}). Throughout this section, the variables i, j range over [1, n] unless otherwise specified.

We saw in §1 that the G_{V} -orbits in N_{V} are in bijection with the set M_{(d_{i})} of multisegments in which i occurs d_{i} times as an element of a segment. Following [26], we change this parametrization by multisegments to a parametrization by matrices. We identify each m ∈ M_{(d_{i})} with the (n × n)-matrix (m_{i,j}), where (3.1)

m_{i,j} = \{ \begin{matrix}multiplicity of the segment [i, j], & if i ≤ j, \\ number of segments [k, l] where k ≤ j, l ≥ i, & if j = i − 1, \\ 0, & if j < i − 1. \end{matrix}

It is clear that this matrix lies in the set M_{(d_{i});(d_{j})}, as defined in the previous section. So we have identified M_{(d_{i})} with a subset of M_{(d_{i});(d_{j})}, which can be described as follows.

Proposition 3.1. Let M′_{(d_{i})} = {m ∈ M_{(d_{i});(d_{j})} | m_{i,j} = 0, ∀j < i − 1}.

(1) M′_{(d_{i})} is a lower ideal of the poset M_{(d_{i});(d_{j})}.

(2) If m ∈ M′_{(d_{i})}, then for all i ≥ j, m_{≤i,≥j} = d_{j} + d_{j+1} + ⋯ + d_{i}.

(3) For m, m′ ∈ M′_{(d_{i})}, m ≤ m′ if and only if m_{≤i,≥j} ≤ m′_{≤i,≥j} for all i < j.

(4) If m ∈ M′_{(d_{i})}, then for all i ∈ [2, n], m_{i,i−1} = m_{≤i−1,≥i}.

(5) M′_{(d_{i})} = M_{(d_{i})}.

Proof. An element m ∈ M_{(d_{i});(d_{j})} lies in M′_{(d_{i})} if and only if m_{≥i,≤j} = 0 for all i, j such that j < i − 1, so (1) follows from (3) of Proposition 2.3. For (2), since m_{≥i+1,≤j−1} = 0, (2.6) gives

m_{≤i,≥j} = d_{1} + ⋯ + d_{i} − d_{1} − ⋯ − d_{j−1} = d_{j} + ⋯ + d_{i},

as required. Part (3) then follows from (2) of Proposition 2.3. For (4), we have

m_{i,i−1} = d_{i} − m_{i,≥i} = m_{≤i,≥i} − m_{i,≥i} = m_{≤i−1,≥i}.

From (4) and the i = j case of (2) it follows that every matrix in M′_{(d_{i})} arises from a multisegment in M_{(d_{i})} by the rule (3.1), whence (5). □

As mentioned in the introduction, the identification of M_{(d_{i})} with M′_{(d_{i})} is a poset isomorphism: the geometrically-defined partial order ⪯ on M_{(d_{i})} is the restriction of the partial order ≤ on M_{(d_{i});(d_{j})}. This is part of Zelevinsky’s result [26, Corollary 1], which we can state (with some supplementary detail) as follows.

Theorem 3.2. Let m, m′ ∈ M_{(d_{i})}.

(1) dim O_{m} = ℓ(m) − ∑_{i} \binom{d_{i}}{2}.

(2) O_{m} ⊆ \overline{O_{m′}} ⇔ m ≤ m′.

(3) H^{i}IC(\overline{O_{m′}}) = 0 for i odd.

(4) IC_{m,m′} = P_{m,m′}.

(5) IC_{m,m′}^{⟨−1⟩} = P_{m,m′}^{⟨−1⟩}.

Proof. For reference in §5, we recall Zelevinsky’s proof. Define the partial flag variety B_{(d_{i})} to be the set of collections of subspaces (W_{i})_{i∈[0,n]} of V such that W_{0} = 0, and for all i ∈ [1, n], W_{i−1} ⊂ W_{i} and dim W_{i}/W_{i−1} = d_{i}; this is naturally a nonsingular projective variety of dimension \binom{d}{2} − ∑_{i} \binom{d_{i}}{2}. We define a ‘base-point’ (U_{i}) in B_{(d_{i})} by U_{i} = V_{1} ⊕ ⋯ ⊕ V_{i}. Relative to this base-point, B_{(d_{i})} decomposes into Schubert cells B_{m} for m ∈ M_{(d_{i});(d_{j})}. Explicitly, B_{m} consists of those (W_{i}) such that for all i, j ∈ [1, n],

dim \frac{W_{i} ∩ U_{j}}{W_{i} ∩ U_{j−1} + W_{i−1} ∩ U_{j}} = m_{i,j}.

The analogues of (1)–(4) for these Schubert cells (for all of M_{(d_{i});(d_{j})}) are well known. Let B′_{(d_{i})} be the closed subvariety of B_{(d_{i})} defined by requiring W_{i} ⊃ U_{i−1}; from the description of M_{(d_{i})} as M′_{(d_{i})}, it is easy to see that B′_{(d_{i})} = ⋃_{m∈M_{(d_{i})}} B_{m}.

Now we define a morphism N_{V} → B′_{(d_{i})} : φ ↦ (W_{i}(φ)) by the rule

W_{i}(φ) = U_{i−1} ⊕ {v + φ(v) + φ^{2}(v) + ⋯ + φ^{n−i}(v) | v ∈ V_{i}}.

An easy check shows that this morphism maps O_{m} into B_{m} for all m ∈ M_{(d_{i})}. Moreover, it gives an isomorphism between N_{V} and the open subvariety of B′_{(d_{i})} defined by requiring

W_{i} ∩ ⨁_{i′>i} V_{i′} = 0, ∀i ∈ [1, n].

Hence each O_{m} is embedded as an open subvariety of the Schubert cell B_{m}, and (1)–(4) follow. Since M_{(d_{i})} is a lower ideal of M_{(d_{i});(d_{j})}, (5) is an automatic consequence of (4). □

In view of (3) of Proposition 3.1, part (2) of Theorem 3.2 says that O_{m} ⊆ \overline{O_{m′}} if and only if for all i < j, m_{≤i,≥j} ≤ m′_{≤i,≥j}. Now if φ ∈ O_{m}, then for i ≤ j, m_{≤i,≥j} = rk φ^{j−i}|_{V_{i}}. So we recover the well-known fact that O_{m} ⊆ \overline{O_{m′}} if and only if for all i < j, rk φ^{j−i}|_{V_{i}} ≤ rk (φ′)^{j−i}|_{V_{i}} for any φ ∈ O_{m} and φ′ ∈ O_{m′}. (Of course the “only if” direction is obvious.) We can define an element m^{max} ∈ M_{(d_{i})} uniquely by the requirement that for i ≤ j, m_{≤i,≥j}^{max} equals the maximum possible rank, namely min{d_{i}, d_{i+1}, ⋯, d_{j}}. It follows that m ≤ m^{max} for all m ∈ M_{(d_{i})}, and the orbit O_{m^{max}} is dense in N_{V}.

As foreshadowed in the introduction, Theorem 3.2 is only one of many possible ways to express a particular IC_{m,m′} as a Kazhdan-Lusztig polynomial: the below-diagonal entries prescribed by (3.1) correspond to one particular choice of “empty segments”. A more general statement is the following.

Theorem 3.3. Let b_{1}, ⋯, b_{n}, c_{1}, ⋯, c_{n} ∈ ℕ be such that

b_{1} = d_{1}, c_{n} = d_{n}, and d_{i} − b_{i} = d_{i−1} − c_{i−1}, ∀i ∈ [2, n].

Define an (n × n)-matrix a by

a_{i,j} = \{ \begin{matrix}d_{i} − b_{i}, & if j = i − 1, \\ 0, & otherwise. \end{matrix}

Let M_{(d_{i})}^{(b_{i});(c_{j})} = {m ∈ M_{(d_{i})} | m_{i,i−1} ≥ d_{i} − b_{i}, ∀i ∈ [2, n]}.

(1) M_{(d_{i})}^{(b_{i});(c_{j})} is an upper ideal of M_{(d_{i})}.

(2) The map m ↦ m − a is an isomorphism of posets between M_{(d_{i})}^{(b_{i});(c_{j})} and {m̃ ∈ M_{(b_{i});(c_{j})} | m̃_{i,j} = 0, ∀j < i − 1}.

(3) For any m′, m′′ ∈ M_{(d_{i})}^{(b_{i});(c_{j})}, IC_{m′,m′′} = P_{m′−a,m′′−a}.

(4) For any m′, m′′ ∈ M_{(d_{i})}^{(b_{i});(c_{j})}, IC_{m′,m′′}^{⟨−1⟩} = P_{m′−a,m′′−a}^{⟨−1⟩}.

Proof. For m ∈ M_{(d_{i})}, m_{i,i−1} equals m_{≥i,≤i−1}, so (1) follows from (3) of Proposition 2.3. For (2), the fact that the given map is a bijection is obvious, and by Proposition 2.3 it preserves partial orders. Now this map is the composition of maps of the form m ↦ m − e as in Proposition 2.5 and their inverses m ↦ m + e, where the positions which are being altered are all of the form (i, i − 1). Since all matrices involved have zero entries in positions (i, j) where j < i − 1, conditions (2) and (3) of the definition of cancellability always hold. So Proposition 2.5 implies (3) and (4) with P_{m′,m′′} and P_{m′,m′′}^{⟨−1⟩} in place of IC_{m′,m′′} and IC_{m′,m′′}^{⟨−1⟩}, and Theorem 3.2 gives the result. □

Corollary 3.4. For m ∈ M_{(d_{i})}, define an (n × n)-matrix m^{−} by

m_{i,j}^{−} = \{ \begin{matrix}m_{i,j}, & if j = i − 1, \\ 0, & otherwise. \end{matrix}

Let ⟨m⟩ = {m′ ∈ M_{(d_{i})} | m ≤ m′} = [m, m^{max}].

(1) For all m′ ∈ ⟨m⟩, m′ − m^{−} ∈ M_{(d_{i}−m_{i,i−1});(d_{j}−m_{j+1,j})}.

(2) The map m′ ↦ m′ − m^{−} is an isomorphism of posets between ⟨m⟩ and [m − m^{−}, m^{max} − m^{−}].

(3) For any m′, m′′ ∈ ⟨m⟩, IC_{m′,m′′} = P_{m′−m^{−},m′′−m^{−}}.

(4) For any m′, m′′ ∈ ⟨m⟩, IC_{m′,m′′}^{⟨−1⟩} = P_{m′−m^{−},m′′−m^{−}}^{⟨−1⟩}.

Proof. Apply Theorem 3.3 with

b_{i} = d_{i} − m_{i,i−1}, c_{i−1} = d_{i−1} − m_{i,i−1}

for all i ∈ [2, n], and restrict to the upper ideal ⟨m⟩ of M_{(d_{i})}^{(b_{i});(c_{j})}. □

Note that the polynomials P_{m′−m^{−},m′′−m^{−}} in (3) are Kazhdan-Lusztig polynomials for S_{k(m)}, where k(m) is the number of segments of m, which is also the sum of the entries of m − m^{−}.

Finally, we have to connect Theorem 3.3 to the notation of the introduction, in order to prove Theorem 1.1. We have elements λ, μ ∈ D_{k}; we can clearly assume that all λ_{s} − s, μ_{s} − s + 1 for 1 ≤ s ≤ k lie in [1, n]. Define

b_{i} = |{s | μ_{s} − s + 1 = i}|, c_{j} = |{s | λ_{s} − s = j}|.

Then the subgroups S_{(b_{i})} and S_{(c_{j})} of S_{k} are exactly the conjugates under w_{0}^{(k)} of the dot stabilizers W_{μ} and W_{λ} (this reversal comes about because the sequences (λ_{s} − s) and (μ_{s} − s) are decreasing). The map ψ : S_{k} → M_{(b_{i});(c_{j})} as defined in the previous section satisfies

(3.2) ψ(w_{0}^{(k)}ww_{0}^{(k)})_{i,j} = |{s | μ_{s} − s + 1 = i, λ_{w(s)} − w(s) = j}|.

So S_{k}[λ, μ] = {w ∈ S_{k} | ψ(w_{0}^{(k)}ww_{0}^{(k)})_{i,j} = 0, ∀j < i − 1}, which shows that it is indeed a lower ideal of S_{k}. Moreover, w ↦ ψ(w_{0}^{(k)}ww_{0}^{(k)}) gives an isomorphism of posets between S_{k}[λ, μ]^{∘} and {m̃ ∈ M_{(b_{i});(c_{j})} | m̃_{i,j} = 0, ∀j < i−1}, and the polynomials attached to these posets correspond, since

(3.3)  P_{w,w′} = P_{w_{0}^{(k)}ww_{0}^{(k)},w_{0}^{(k)}w′w_{0}^{(k)}} = P_{ψ(w_{0}^{(k)}ww_{0}^{(k)}),ψ(w_{0}^{(k)}w′w_{0}^{(k)})}

for all w, w′ ∈ S_{k}[λ, μ]^{∘}. Now as in Theorem 1.1, assume that λ/μ ∈ M_{(d_{i})}; it follows immediately that (b_{i}) and (c_{j}) satisfy the conditions of Theorem 3.3. By (3.2), for all w ∈ S_{k}[λ, μ]^{∘}, the multisegment λ/(w · μ) when viewed as a matrix has the same diagonal and abovediagonal entries as ψ(w_{0}^{(k)}ww_{0}^{(k)}); hence λ/(w · μ) = ψ(w_{0}^{(k)}ww_{0}^{(k)}) + a where a is as in Theorem 3.3. Thus Theorem 1.1 follows from Theorem 3.3.

4. Cancellation for the affine symmetric group

We now want to extend the results of §2 to the affine symmetric group. Again fix a positive integer d. Let Ŝ_{d} be the group of permutations w of the set ℤ such that w(i + d) = w(i) + d, for all i ∈ ℤ. An element w ∈ Ŝ_{d} is determined by its window (w(1), w(2), ⋯, w(d)), which can be any collection of representatives of the congruence classes mod d, in any order. The subgroup of Ŝ_{d} which preserves [1, d] is clearly isomorphic to S_{d}.

The group Ŝ_{d} is the ‘extended’ affine symmetric group: it can be written as a semi-direct product ⟨τ⟩ ⋉ S̃_{d}, where

S̃_{d} = {w ∈ Ŝ_{d} | ∑_{i=1}^{d} w(i) = ∑_{i=1}^{d} i}

is the actual affine symmetric group, and τ is the element of infinite order sending i to i + 1 for all i ∈ ℤ. In general, w ∈ τ^{a(w)}S̃_{d} where

a(w) = \frac{1}{d}(∑_{i=1}^{d} w(i) − ∑_{i=1}^{d} i).

Note that for any i ∈ ℤ, the set w(−∞, i] can be obtained from (−∞, i + a(w)] by changing finitely many elements (keeping distinctness). In other words, for m sufficiently large, the mth largest element in w(−∞, i] is i + a(w) − m + 1.

If d = 1, S̃_{d} is the trivial group and Ŝ_{d} = ⟨τ⟩. If d ≥ 2, we define s_{i} ∈ S̃_{d} for all i ∈ ℤ by

s_{i}(j) = \{ \begin{matrix}j + 1, & if j ≡ i mod d, \\ j − 1, & if j ≡ i + 1 mod d, \\ j, & otherwise. \end{matrix}

Thus s_{i} = s_{i′} iff i ≡ i′ mod d. It is well known that s_{0}, s_{1}, ⋯, s_{d−1} form a set of Coxeter generators for S̃_{d} of type { \atop A_{d−1}}. They thus determine a length function ℓ : S̃_{d} → ℕ, a Bruhat order ≤ on S̃_{d}, and Kazhdan-Lusztig polynomials P_{y,w} ∈ ℕ[q] for y, w ∈ S̃_{d}, all of which are invariant under conjugation by τ. We extend these to Ŝ_{d} in the standard way:

ℓ(w) = ℓ(τ^{−a(w)}w),
y ≤ w ⇔ a(y) = a(w), τ^{−a(y)}y ≤ τ^{−a(w)}w,
P_{y,w} = \{ \begin{matrix}P_{τ^{−a(y)}y,τ^{−a(w)}w}, & if a(y) = a(w), \\ 0, & otherwise. \end{matrix}

We define inversion statistics as in the finite case:

inv_{i}(w) = |{i′ < i | w(i′) > w(i)}|, Inv_{i}(w) = |{i′ > i | w(i′) < w(i)}|, for any w ∈ Ŝ_{d} and i ∈ ℤ (these sets are finite, even though i′ runs over ℤ.) Clearly inv_{i+d}(w) = inv_{i}(w), Inv_{i+d}(w) = Inv_{i}(w), and

Inv_{i}(w) − inv_{i}(w)

= |(−∞, w(i)] \ w(−∞, i]| − |w(−∞, i] \ (−∞, w(i)]|

= |(−∞, w(i)] \ (−∞, i + a(w)]| − |(−∞, i + a(w)] \ (−∞, w(i)]| = w(i) − i − a(w).

The formula for ℓ on Ŝ_{d} is analogous to that for S_{d} (see [4, Proposition 4.1(ii)]):

(4.1)  ℓ(w) = ∑_{i∈[1,d]} inv_{i}(w) = ∑_{i∈[1,d]} Inv_{i}(w).

We also have

(4.2)  ws_{i} < w if and only if w(i) > w(i + 1),

and a general description of Bruhat order along the lines of Proposition 2.1 (this is a rephrasing of [4, Theorem 6.5], trivially extended from S̃_{d} to Ŝ_{d}):

Proposition 4.1. If y, w ∈ Ŝ_{d}, y ≤ w if and only if for all i ∈ ℤ,

|{i′ ≤ i | y(i′) ≥ j}| ≤ |{i′ ≤ i | w(i′) ≥ j}|, ∀j, with equality for j ≪ 0.

In other words, for all positive integers m, the mth largest element in y(−∞, i] is less than or equal to the mth largest element in w(−∞, i], with equality for m ≫ 0. It suffices to check this for i ∈ [1, d].

The definition of cancellability is identical to the finite case:

Definition. If y ≤ w in Ŝ_{d}, we say that i ∈ ℤ is cancellable for the interval [y, w] if y(i) = w(i), inv_{i}(y) = inv_{i}(w), and Inv_{i}(y) = Inv_{i}(w). (Clearly any two of these conditions imply the third, and i is cancellable for [y, w] iff i + d is.)

However, the process of cancellation is not as uniquely defined as in the finite case: we need to choose order-preserving bijections σī : ℤ \ ī → ℤ for all congruence classes ī mod d. Then for any w ∈ Ŝ_{d}, we define wî ∈ Ŝ_{d−1} by

wî = σ\overline{w(i)} ∘ w ∘ σ−1 ; ī.

Note that using different σ’s would have the effect of multiplying wî on left and right by powers of τ. Independently of the choice, we have

(4.3)  ℓ(wî) = ℓ(w) − inv_{i}(w) − Inv_{i}(w).

Example. Take d = 3, y = τs_{1}s_{2} and w = τs_{2}s_{1}s_{0}s_{2}. Then y has window (3, 4, 2) and w has window (0, 7, 2). Since inv_{3}(y) = inv_{3}(w) = 2, 3 is cancellable for [y, w]. If we normalize σ¯2 and σ¯3 by requiring that they preserve 1, then yˆ3 and wˆ3 are the elements of Ŝ_{2} with windows (2, 3) and (0, 5), namely τ and τs_{1}s_{0}.

We can now extend Proposition 2.2 to the affine case.

Proposition 4.2. Suppose that i is cancellable for [y, w].

(1) For any x ∈ [y, w], x(i) = y(i) and inv_{i}(x) = inv_{i}(y). Hence i is cancellable for any sub-interval of [y, w].

(2) x ↦ xî is an isomorphism of posets between [y, w] and [yî, wî], which reduces all lengths by the same amount.

(3) For any u, v ∈ [y, w], P_{u,v} = P_{uî,vî}.

Proof. The proof of part (1) is identical to that of part (1) of Proposition 2.2, with [1, i] replaced by (−∞, i] and so on, and of course using Proposition 4.1 instead of Proposition 2.1. Similarly with part (2), where the inverse map [yî, wî] → [y, w] : x ↦ x̃ is now defined by

x̃(i′) = \{ \begin{matrix}y(i) + kd, & if i′ = i + kd, \\ σ−1 ; \overline{y(i)}(x(σī(i′))), & if i′ ∉ ī. \end{matrix}

The proof of (3) is also mostly unchanged. Apart from replacing [1, i] by (−∞, i] and so on, the only change is that in Case 1, we need not have (ys){ \atop i−1} = yî and (ws){ \atop i−1} = wî, but rather have

(ys){ \atop i−1} = τ^{a}yîτ^{b}, (ws){ \atop i−1} = τ^{a}wîτ^{b} for some a, b ∈ ℤ,

which still implies P_{(ys){ \atop i−1},(ws){ \atop i−1}} = P_{yî,wî} as required. □

We now introduce some affine matrix notation very similar to that in [14]. Let (b_{i})_{i∈ℤ} be a ℤ-tuple of nonnegative integers, periodic with period n ≥ 1, such that ∑_{i=1}^{n} b_{i} = d; and let (c_{j})_{j∈ℤ} be another such, with period n′ ≥ 1, such that ∑_{j=1}^{n′} c_{j} = d. Our notational convention now is that the range of the variables i, i′, j, j′ is all of ℤ unless otherwise specified. Let M_{(b_{i}),n;(c_{j}),n′} be the set of all (ℤ × ℤ)-matrices m satisfying:

(1) m_{i,j} ∈ ℕ, for all i, j,

(2) m_{i+n,j+n′} = m_{i,j}, for all i, j,

(3) ∑_{j} m_{i,j} = b_{i}, for all i, and

(4) ∑_{i} m_{i,j} = c_{j}, for all j. It is easy to see that for m ∈ M_{(b_{i}),n;(c_{j}),n′}, m_{i,j} = 0 for |j − i| ≫ 0; so sums of the form m_{i,≥j}, m_{≤i,j}, m_{≤i,≥j} and m_{≥i,≤j} are finite. We have the following substitute for (2.6). For fixed i, m_{≥i,≤j_{0}} = 0 for all j_{0} sufficiently negative, and for j greater than such j_{0},

(4.4)  m_{≥i,≤j} = c_{j_{0}+1} + ⋯ + c_{j} − m_{≤i−1,≥j_{0}+1} + m_{≤i−1,≥j+1}.

The matrices in M_{(b_{i}),n;(c_{j}),n′} parametrize double cosets of Ŝ_{d} with respect to proper parabolic subgroups of S̃_{d}. Namely, write ℤ as the disjoint union of (possibly empty) blocks B_{i} such that all elements of B_{i} are less than all elements of B_{i+1}, and |B_{i}| = b_{i}. It follows that B_{i+n} = B_{i} + d. Note that the collection (B_{i}) is determined by (b_{i}) up to translation (i.e. a power of τ). Let S_{(B_{i})} be the subgroup of Ŝ_{d} which preserves each B_{i} separately; this is a parabolic subgroup of S̃_{d} isomorphic to S_{b_{1}} ×⋯×S_{b_{n}}. (It is determined by (b_{i}) up to conjugation by a power of τ.) Similarly define blocks C_{j} of sizes c_{j}, and the parabolic subgroup S_{(C_{j})}. We define a surjective map ψ : Ŝ_{d} → M_{(b_{i}),n;(c_{j}),n′} by

ψ(w)_{i,j} = |w(B_{i}) ∩ C_{j}|.

The fibres of ψ are exactly the double cosets S_{(C_{j})}wS_{(B_{i})}, so ψ induces a bijection S_{(C_{j})} \Ŝ_{d} / S_{(B_{i})} ↔ M_{(b_{i}),n;(c_{j}),n′}. For m ∈ M_{(b_{i}),n;(c_{j}),n′}, let w_{m} ∈ Ŝ_{d} be the longest element in the corresponding double coset.

The permutation w_{m} can be read off the matrix m by exactly the same prescription as in the finite case (remembering that i, i′, j, j′ now range over all of ℤ).

Example. Let d = 7, n = 2, n′ = 3, and define

b_{i} = \{ \begin{matrix}3, & if i ≡ 1 mod 2, \\ 4, & if i ≡ 0 mod 2, \end{matrix} c_{j} = \{ \begin{matrix}3, & if j ≡ 1 mod 3, \\ 2, & if j ≡ 2 mod 3, \\ 2, & if j ≡ 0 mod 3. \end{matrix}

Let m ∈ M_{(b_{i}),2;(c_{j}),3} be the following matrix:

...  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮
2 0 0 0 0 1 0 0 0 0 0 0
⋯ 0 2 0 0 0 1 0 0 0 0 0 0 ⋯
1 0 0 2 0 0 0 0 1 0 0 0
⋯ 0 0 0 0 2 0 0 0 1 0 0 0 ⋯
0 0 0 1 0 0 2 0 0 0 0 1
⋯ 0 0 0 0 0 0 0 2 0 0 0 1 ⋯
0 0 0 0 0 0 1 0 0 2 0 0
⋮  ⋮  ⋮  ⋮  ⋮  ⋮ ...

where the 0 is the (1, 1) entry. We choose (B_{i}) and (C_{j}) so that B_{1} = C_{1} = {1, 2, 3}. The row containing 0 tells us that w_{m}(B_{1}) consists of one element of C_{6} = {13, 14} and two of C_{2} = {4, 5}, in that order. Since the (0, 6) entry is 1, the largest element of C_{6} ‘has already been used’ in w_{m}(B_{0}), so we set w_{m}(1) = 13, w_{m}(2) = 5, w_{m}(3) = 4. Treating the next row similarly, we find that w_{m} is the element of Ŝ_{7} with window (13, 5, 4, 21, 10, 9, 1).

We define a length function ℓ : M_{(b_{i}),n;(c_{j}),n′} → ℕ by ℓ(m) = ℓ(w_{m}), and a partial order on M_{(b_{i}),n;(c_{j}),n′} by

m ≤ m′ ⇔ w_{m} ≤ w_{m′}.

Since the map m ↦ w_{m} depends on the choice of (B_{i}) and (C_{j}) only modulo left and right multiplication by fixed powers of τ, these definitions are independent of this choice. Indeed, they can be described in an analogous way to Proposition 2.3:

Proposition 4.3. Let m, m′ ∈ M_{(b_{i}),n;(c_{j}),n′}.

(1) ℓ(m) = ∑_{i∈[1,n],j} m_{i,j}m_{≤i,≥j} − ∑_{i∈[1,n],j} \binom{m_{i,j}+1}{2}.

(2) m ≤ m′ if and only if, for all i ∈ ℤ,

m_{≤i,≥j} ≤ m′_{≤i,≥j}, ∀j, with equality for j ≪ 0.

(3) m ≤ m′ if and only if, for all j ∈ ℤ,

m_{≥i,≤j} ≤ m′_{≥i,≤j}, ∀i, with equality for i ≪ 0.

Proof. The proof is mostly identical to that of Proposition 2.3, using (4.1) and Proposition 4.1 instead of (2.1) and Proposition 2.1. In the proof of (3), the argument using (2.6) no longer makes sense, but the argument using transposes does. □

As in §2, we define P_{m,m′} = P_{w_{m},w_{m′}} for m, m′ ∈ M_{(b_{i}),n;(c_{j}),n′}. From (2) of Proposition 4.3 it is clear that each interval [m, m′] in the poset M_{(b_{i}),n;(c_{j}),n′} is finite, so the inverse matrix (P_{m,m′}^{⟨−1⟩})_{m,m′∈M_{(b_{i}),n;(c_{j}),n′}} of (P_{m,m′})_{m,m′∈M_{(b_{i}),n;(c_{j}),n′}} is well defined.

The matrix definition of cancellability is identical to the finite case.

Definition. If m ≤ m′ in M_{(b_{i}),n;(c_{j}),n′}, we say that (i, j) ∈ ℤ × ℤ is cancellable for [m, m′] if

(1) m_{i,j} ≥ 1.

(2) m_{≤i−1,≥j} = m′_{≤i−1,≥j}, or equivalently m_{≥i,≤j+1} = m′_{≥i,≤j+1}.

(3) m_{≤i,≥j+1} = m′_{≤i,≥j+1}, or equivalently m_{≥i+1,≤j} = m′_{≥i+1,≤j}. These equivalences follow from (4.4), bearing in mind (2) of Proposition 4.3. Clearly (i, j) is cancellable iff (i + n, j + n′) is.

Proposition 4.4. Suppose that m ≤ m′ in M_{(b_{i}),n;(c_{j}),n′} and (i, j) is cancellable for [m, m′]. Let e be the matrix with e_{i+kn,j+kn′} = 1 for all k, all other entries zero.

(1) For any m^{1} ∈ [m, m′], (a) m_{i,j}^{1} ≥ m_{i,j},

(b) m_{≤i−1,≥j}^{1} = m_{≤i−1,≥j}, and

(c) m_{≤i,≥j+1}^{1} = m_{≤i,≥j+1}.

Hence (i, j) is cancellable for any sub-interval of [m, m′].

(2) The map m^{1} ↦ m^{1} − e is an isomorphism of posets between [m, m′] and [m − e, m′ − e], which reduces all lengths by the same amount.

(3) For any m^{1}, m^{2} ∈ [m, m′], P_{m^{1},m^{2}} = P_{m^{1}−e,m^{2}−e}.

(4) For any m^{1}, m^{2} ∈ [m, m′], P_{m^{1},m^{2}}^{⟨−1⟩} = P_{m^{1}−e,m^{2}−e}^{⟨−1⟩}.

Proof. Completely analogous to the proof of Proposition 2.5, using the analogue of Proposition 2.4. □

5. Nilpotent orbits of the cyclic quiver

We now return to the set-up of the latter part of §1, so V is a ddimensional ℤ/nℤ-graded vector space, and d_{i} = dim Vī for all i ∈ ℤ. We saw in §1 that the G_{V} -orbits in N_{V} are in bijection with the set M_{(d_{i}),n} of multisegments (in the modulo n sense) such that each congruence class ī occurs d_{i} times among the elements of the segments. As in §3, we will identify each m ∈ M_{(d_{i}),n} with a matrix (m_{i,j}), this time in M_{(d_{i}),n;(d_{j}),n}; the definition of m_{i,j} is exactly the same as (3.1). The resulting subset of M_{(d_{i}),n;(d_{j}),n} is described as follows.

Proposition 5.1. Let M′′_{(d_{i}),n} = {m ∈ M_{(d_{i}),n;(d_{j}),n} | m_{i,j} = 0, ∀j < i − 1}.

(1) M′′_{(d_{i}),n} is a lower ideal of the poset M_{(d_{i}),n;(d_{j}),n}.

(2) For all m ∈ M′′_{(d_{i}),n}, there is some f(m) ∈ ℤ such that

m_{≤i,≥j} + f(m) = d_{j} + ⋯ + d_{i}, ∀i ≥ j.

(3) For m, m′ in M′′_{(d_{i}),n}, m ≤ m′ if and only if f(m) = f(m′) and m_{≤i,≥j} ≤ m′_{≤i,≥j} for all i < j.

(4) M′_{(d_{i}),n} = {m ∈ M′′_{(d_{i}),n} | f(m) = 0} is a lower ideal of the poset M_{(d_{i}),n;(d_{j}),n}.

(5) If m ∈ M′_{(d_{i}),n}, then m_{i,i−1} = m_{≤i−1,≥i} for all i.

(6) M′_{(d_{i}),n} = M_{(d_{i}),n}.

Proof. As in the finite case, (1) is immediate from (3) of Proposition 4.3. (2) comes from the fact that for i ≥ j,

m_{≤i,≥j} = d_{j} + ⋯ + d_{i−1} + m_{≤i,≥i} = d_{j+1} + ⋯ + d_{i} + m_{≤j,≥j}.

Using this, (3) comes from (2) of Proposition 4.3, and (4) is an immediate consequence of (1) and (3). (5) is proved in the same way as (4) of Proposition 3.1. From (5) and the i = j case of (2) it follows that every matrix in M′_{(d_{i}),n} arises from a multisegment in M_{(d_{i}),n}, whence (6). □

We can now state Lusztig’s affine analogue of Theorem 3.2:

Theorem 5.2. Let m, m′ ∈ M_{(d_{i}),n}.

(1) dim O_{m} = ℓ(m) − ∑_{i∈[1,n]} \binom{d_{i}}{2}.

(2) O_{m} ⊆ \overline{O_{m′}} ⇔ m ≤ m′.

(3) H^{i}IC(\overline{O_{m′}}) = 0 for i odd.

(4) IC_{m,m′} = P_{m,m′}.

(5) IC_{m,m′}^{⟨−1⟩} = P_{m,m′}^{⟨−1⟩}.

Proof. As with Theorem 3.2, (5) follows from (4) because M_{(d_{i}),n} is a lower ideal of M_{(d_{i}),n;(d_{j}),n}. Parts (1)–(4) were proved by Lusztig in [12, §11], but since the conventions there are slightly different, a sketch of a proof along the lines of the above proof of Theorem 3.2 may be helpful.

Form V = ℂ((t)) ⊗_{ℂ} V, and consider lattices (free ℂ[[t]]-submodules of rank d) in V. Define B̂_{(d_{i}),n} to be the set of collections of lattices (M_{i})_{i∈ℤ} such that for all i ∈ ℤ:

(1) M_{i−1} ⊂ M_{i},

(2) dim_{ℂ} M_{i}/M_{i−1} = d_{i}, and

(3) M_{i−n} = tM_{i}. It is well known that B̂_{(d_{i}),n} has the structure of an increasing union of projective varieties. We define a base-point (L_{i}) in B̂_{(d_{i}),n} as follows. For any i ∈ ℤ, let V_{i} denote t^{k}Vī where k is defined by i + kn ∈ {1, ⋯, n}. Define

L_{i} = ; ⨁_{j≤i}V_{j}, ∀i ∈ ℤ,

where ˆ⨁ denotes completed direct sum. Relative to this base-point, B̂_{(d_{i}),n} decomposes into affine Schubert cells B̂_{m} for m ∈ M_{(d_{i}),n;(d_{j}),n}. Explicitly, B̂_{m} consists of those (M_{i}) such that for all i, j ∈ ℤ,

dim_{ℂ} \frac{M_{i} ∩ L_{j}}{M_{i} ∩ L_{j−1} + M_{i−1} ∩ L_{j}} = m_{i,j}.

The analogues of (1)–(4) for these affine Schubert cells (for all of M_{(d_{i}),n;(d_{j}),n}) are well known. Let B̂′_{(d_{i}),n} be the closed subvariety of B̂_{(d_{i}),n} defined by requiring M_{i} ⊇ L_{i−1}, dim_{ℂ} M_{i}/L_{i−1} = d_{i}; from the description of M_{(d_{i}),n} as M′_{(d_{i}),n}, it is easy to see that B̂′_{(d_{i}),n} = ⋃_{m∈M_{(d_{i}),n}} B̂_{m}.

Now we define a morphism N_{V} → B̂′_{(d_{i}),n} : φ ↦ (M_{i}(φ)) by the rule

M_{i}(φ) = L_{i−1} ⊕ {v + φ(v) + φ^{2}(v) + ⋯ | v ∈ V_{i}}.

(Since φ is nilpotent, this sum is actually finite.) An easy check shows that this morphism maps O_{m} into B̂_{m} for all m ∈ M_{(d_{i}),n}. All that remains is to verify that it gives an isomorphism between N_{V} and the open subvariety of B̂′_{(d_{i}),n} defined by requiring

M_{i} ∩ ⨁_{i′>i} V_{i′} = 0, ∀i ∈ ℤ.

The “dual” statement to this is what is proved in [12, §11]. □

Note that in contrast to the situation in §3, the poset M_{(d_{i}),n} may have more than one maximal element.

We now come to the affine analogue of Theorem 3.3, a generalization of Theorem 5.2.

Theorem 5.3. Let b_{i}, c_{j} ∈ ℕ be such that

b_{i+n} = b_{i}, c_{j+n} = c_{j}, and d_{i} − b_{i} = d_{i−1} − c_{i−1}, ∀i, j ∈ ℤ.

Define a (ℤ × ℤ)-matrix a by

a_{i,j} = \{ \begin{matrix}d_{i} − b_{i}, & if j = i − 1, \\ 0, & otherwise. \end{matrix}

Let M_{(d_{i}),n}^{(b_{i});(c_{j})} = {m ∈ M_{(d_{i}),n} | m_{i,i−1} ≥ d_{i} − b_{i}, ∀i ∈ ℤ}.

(1) M_{(d_{i}),n}^{(b_{i});(c_{j})} is an upper ideal of M_{(d_{i}),n}.

(2) The map m ↦ m − a is an isomorphism of posets between M_{(d_{i}),n}^{(b_{i});(c_{j})} and {m̃ ∈ M_{(b_{i}),n;(c_{j}),n} | m̃_{i,j} = 0, ∀j < i − 1}.

(3) For any m′, m′′ ∈ M_{(d_{i}),n}^{(b_{i});(c_{j})}, IC_{m′,m′′} = P_{m′−a,m′′−a}.

(4) For any m′, m′′ ∈ M_{(d_{i}),n}^{(b_{i});(c_{j})}, IC_{m′,m′′}^{⟨−1⟩} = P_{m′−a,m′′−a}^{⟨−1⟩}.

Proof. Completely analogous to the proof of Theorem 3.3, using Proposition 4.3, Proposition 4.4, and Theorem 5.2 in place of Proposition 2.3, Proposition 2.5, and Theorem 3.2. □

Corollary 5.4. For m ∈ M_{(d_{i}),n}, define a (ℤ × ℤ)-matrix m^{−} by

m_{i,j}^{−} = \{ \begin{matrix}m_{i,j}, & if j = i − 1, \\ 0, & otherwise. \end{matrix}

Let ⟨m⟩ = {m′ ∈ M′_{(d_{i}),n} | m ≤ m′}, and let m_{1}^{max}, ⋯, m_{t}^{max} be the maximal elements of ⟨m⟩.

(1) For all m′ ∈ ⟨m⟩, m′ − m^{−} ∈ M_{(d_{i}−m_{i,i−1}),n;(d_{j}−m_{j+1,j}),n}.

(2) The map m′ ↦ m′ − m^{−} is an isomorphism of posets between ⟨m⟩ and ⋃_{s=1}^{t}[m − m^{−}, m_{s}^{max} − m^{−}].

(3) For any m′, m′′ ∈ ⟨m⟩, IC_{m′,m′′} = P_{m′−m^{−},m′′−m^{−}}.

(4) For any m′, m′′ ∈ ⟨m⟩, IC_{m′,m′′}^{⟨−1⟩} = P_{m′−m^{−},m′′−m^{−}}^{⟨−1⟩}.

Proof. Apply Theorem 5.3 with

b_{i} = d_{i} − m_{i,i−1}, c_{i−1} = d_{i−1} − m_{i,i−1}

for all i ∈ ℤ, and restrict to the upper ideal ⟨m⟩ of M_{(d_{i}),n}^{(b_{i});(c_{j})}. □

Note that the polynomials P_{m′−m^{−},m′′−m^{−}} in (3) are Kazhdan-Lusztig polynomials for { \atop S_{k(m)}}, where k(m) is the number of segments in m, which is also the sum of the entries in rows 1 to n of m − m^{−}. As a corollary, we recover the main result of [8]:

Corollary 5.5. If m ≤ m′ in M_{(d_{i}),n}, and k(m) = 2, then IC_{m,m′} = 1.

Proof. In S̃_{2} all nonzero Kazhdan-Lusztig polynomials are 1. □

Example. Let d = 6, n = 3, d_{1} = d_{2} = d_{3} = 2. Let m ∈ M_{(d_{i}),3} be the multisegment [1, 2] + [2, 3] + [3, 4]. Then ⟨m⟩ has three maximal elements,

m_{1}^{max} = [1, 6], m_{2}^{max} = [2, 7], and m_{3}^{max} = [3, 8].

Displaying only the rows indexed by 1, 2, 3, we have

m = (\begin{matrix}⋯ & 1 & 0 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ ⋯ & 0 & 1 & 0 & 1 & 0 & 0 & 0 & ⋯ \\ ⋯ & 0 & 0 & 1 & 0 & 1 & 0 & 0 & ⋯ \end{matrix}) ,

m_{1}^{max} = (\begin{matrix}⋯ & 1 & 0 & 0 & 0 & 0 & 0 & 1 & ⋯ \\ ⋯ & 0 & 2 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ ⋯ & 0 & 0 & 2 & 0 & 0 & 0 & 0 & ⋯ \end{matrix}) ,

m − m^{−} = (\begin{matrix}⋯ & 0 & 0 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ ⋯ & 0 & 0 & 0 & 1 & 0 & 0 & 0 & ⋯ \\ ⋯ & 0 & 0 & 0 & 0 & 1 & 0 & 0 & ⋯ \end{matrix}) ,

m_{1}^{max} − m^{−} = (\begin{matrix}⋯ & 0 & 0 & 0 & 0 & 0 & 0 & 1 & ⋯ \\ ⋯ & 0 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ ⋯ & 0 & 0 & 1 & 0 & 0 & 0 & 0 & ⋯ \end{matrix}) ,

where the 0 is the (1, 1)-entry. Setting B_{i} = {i} and C_{j} = {j − 1}, so that w_{m−m^{−}} is the identity of S̃_{3}, we find that w_{m_{1}^{max}−m^{−}} has window (5, 0, 1), and is therefore s_{1}s_{0}s_{2}s_{1}. Similarly w_{m_{2}^{max}−m^{−}} = s_{2}s_{1}s_{0}s_{2} and w_{m_{3}^{max}−m^{−}} = s_{0}s_{2}s_{1}s_{0}. So m′ ↦ w_{m′−m^{−}} is an isomorphism between ⟨m⟩ and [1, s_{1}s_{0}s_{2}s_{1}] ∪ [1, s_{2}s_{1}s_{0}s_{2}] ∪ [1, s_{0}s_{2}s_{1}s_{0}]. Moreover,

IC_{m,m_{1}^{max}} = P_{1,s_{1}s_{0}s_{2}s_{1}} = q + 1,

and similarly IC_{m,m_{2}^{max}} = IC_{m,m_{3}^{max}} = q + 1, while

IC_{m,m′}^{⟨−1⟩} = P_{1,w_{m′−m^{−}}}^{⟨−1⟩} = ε(w_{m′−m^{−}})

for all m′ ∈ ⟨m⟩ (an example of (1.8)).

Finally, we must prove Theorem 1.2. We have elements λ, μ ∈ D̃_{k}; define λ_{s} and μ_{s} for all s ∈ ℤ by the rule

λ_{s+k} = λ_{s} + k − n, μ_{s+k} = μ_{s} + k − n.

Then λ_{s} − s ≥ λ_{s+1} − (s + 1) for all s ∈ ℤ, and similarly for μ; also,

(w · μ)_{s} − s = μ_{w^{−1}(s)} − w^{−1}(s), ∀w ∈ S̃_{k}, s ∈ [1, k].

Define

B_{i} = {−s | μ_{s} − s + 1 = i}, C_{j} = {−s | λ_{s} − s = j}.

Then the subgroups S_{(B_{i})} and S_{(C_{j})} of S̃_{k} are exactly the images of W̃_{μ} and W̃_{λ} under the automorphism τ : Ŝ_{k} → Ŝ_{k} defined by τ(w)(i) = −w(−i). The map ψ : Ŝ_{k} → M_{(b_{i}),n;(c_{j}),n} as defined in §4 satisfies

(5.1)  ψ(τ(w))_{i,j} = |{s ∈ ℤ | μ_{s} − s + 1 = i, λ_{w(s)} − w(s) = j}|.

So S̃_{k}[λ, μ] = {w ∈ S̃_{k} | ψ(τ(w))_{i,j} = 0, ∀j < i − 1}, which shows that it is indeed a lower ideal of S̃_{k}. Moreover, w ↦ ψ(τ(w)) gives an isomorphism of posets between S̃_{k}[λ, μ]^{∘} and

{m̃ ∈ M_{(b_{i}),n;(c_{j}),n} | m̃ ≥ ψ(1) and m̃_{i,j} = 0, ∀j < i − 1},

and the polynomials attached to these posets correspond, since

(5.2)  P_{w,w′} = P_{τ(w),τ(w′)} = P_{ψ(τ(w)),ψ(τ(w′))}

for all w, w′ ∈ S̃_{k}[λ, μ]^{∘}. Now make the assumption of Theorem 1.2, that λ/μ ∈ M_{(d_{i}),n}; it follows immediately that (b_{i}) and (c_{j}) satisfy the conditions of Theorem 5.3. By (5.1), for all w ∈ S̃_{k}[λ, μ]^{∘}, the multisegment λ/(w ·μ) when viewed as a matrix has the same diagonal and above-diagonal entries as ψ(τ(w)); hence λ/(w · μ) = ψ(τ(w)) + a where a is as in Theorem 5.3. Thus Theorem 1.2 follows from Theorem 5.3.

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School of Mathematics and Statistics, University of Sydney, NSW 2006, AUSTRALIA

  E-mail address: anthonyh@maths.usyd.edu.au