THE SYMMETRIC GROUP REPRESENTATION ON COHOMOLOGY OF THE REGULAR ELEMENTS OF A

MAXIMAL TORUS OF THE SPECIAL LINEAR GROUP

ANTHONY HENDERSON

Abstract. We give a formula for the character of the representation of the Weyl group S_{n} on each isotypic component of the cohomology of the set of regular elements of a maximal torus of SL_{n}, with respect to the action of the centre. As a consequence, we find that this representation is induced from a cohomology representation of a wreath product subgroup.

1. Introduction

For any positive integers r and n, define

T (r, n) := {(z_{1}, z_{2}, ⋯, z_{n}) ∈ (ℂ^{×})^{n} | z_{1}^{r}, ⋯, z_{n}^{r} distinct}.

This is an open subvariety of (ℂ^{×})^{n}, and in fact a hyperplane complement in ℂ^{n}. Let S_{n} be the nth symmetric group, μ_{r} the cyclic group of rth complex roots of 1, and W (r, n) the wreath product μ_{r} ≀ S_{n} = S_{n} ⋉ μ_{r}^{n}. We will write the elements of W (r, n) in the form w(ζ_{1}, ⋯, ζ_{n}) where w ∈ S_{n} and ζ_{i} ∈ μ_{r}, so that the group law is

w(ζ_{1}, ⋯, ζ_{n})w′(ζ′_{1}, ⋯, ζ′_{n}) = ww′(ζ_{w′(1)}ζ′_{1}, ⋯, ζ_{w′(n)}ζ′_{n}).

Clearly W (r, n) acts on T (r, n) via the rules

\begin{aligned}w.(z_{1}, ⋯, z_{n}) & = (z_{w^{−1}(1)}, ⋯, z_{w^{−1}(n)}), \\ (ζ_{1}, ⋯, ζ_{n}).(z_{1}, ⋯, z_{n}) & = (ζ_{1}z_{1}, ⋯, ζ_{n}z_{n}). \end{aligned}

This action arises naturally: for r = 1 it is the action of the Weyl group S_{n} on the set of regular elements of a maximal torus of GL_{n}(ℂ), and for r ≥ 2 it is the action of the reflection group G(r, 1, n) ≅ W (r, n) on the complement of its reflecting hyperplanes. W (r, n) also acts on ℙT (r, n) ⊂ ℙ^{n−1}(ℂ), the set of lines lying in T (r, n). Thus the cohomology groups H^{i}(T (r, n)) and H^{i}(ℙT (r, n)) – always taken with complex coefficients – are representations of W (r, n). Lehrer has given quite explicit formulas for the characters of these representations – see (3.1) and (3.2) below. (Their dimensions, i.e. the Betti numbers of T (r, n) and ℙT (r, n), were already known by work of Orlik and Solomon.)

This work was supported by Australian Research Council grant DP0344185.

Now consider

ST (1, n) := {(z_{i}) ∈ (ℂ^{×})^{n} | z_{1}, ⋯, z_{n} distinct, z_{1}z_{2} ⋯ z_{n} = 1}.

This can be identified with the set of regular elements of a maximal torus of SL_{n}(ℂ). Of course S_{n} still acts, and there is a commuting action of μ_{n} (the centre of SL_{n}(ℂ)) by scaling. Thus we have a direct sum decomposition of S_{n}-modules:

(1.1)  H^{i}(ST (1, n)) ≅ ⨁_{χ∈μ̂_{n}} H^{i}(ST (1, n))_{χ},

where H^{i}(ST (1, n))_{χ} is the χ-isotypic component of H^{i}(ST (1, n)). In this paper we address the following problem, suggested by Lehrer:

Problem 1.1. Give a formula for the character of the representation of S_{n} on each H^{i}(ST (1, n))_{χ}.

Our solution is given in §3 (see especially (3.6)). The resulting formula for the total representation on H^{i}(ST (1, n)) is essentially equivalent to a result of Fleischmann and Janiszczak (see Remark 3.4).

As a consequence, we will derive the following curious statement. Let α_{r,m} : W (r, m) → ℂ^{×} be the one-dimensional character taking the following values on generators:

α_{r,m}(s_{i}) = (−1)^{r−1}, α_{r,m}((ζ, 1, ⋯, 1)) = (−1)^{r−1}ζ,

where ζ is a primitive rth root of 1. Note that W (r, m) can be embedded in S_{rm} as the centralizer of the product of m disjoint rcycles. The character α_{r,m} is the product of the determinant character det : W (r, m) → ℂ^{×} (viewing W (r, m) as G(r, 1, m) ⊂ GL_{m}(ℂ)) and the sign character on S_{rm}.

Theorem 1.2. Let e be the order of χ ∈ μ̂_{n}. For every i, we have an isomorphism of S_{n}-modules:

H^{i}(ST (1, n))_{χ} ≅ Ind_{W (e,n/e)}^{S_{n}}(α_{e,n/e}^{−1} ⊗ H^{i−n+n/e}(ℙT (e, n/e))).

The proof of this Theorem given below merely equates the characters of both sides; I lack a more conceptual understanding of the isomorphism. A small part of Theorem 1.2 is the fact that

(1.2) dim H^{i}(ST (1, n))_{χ} = \frac{n!}{e^{n/e}(n/e)!} dim H^{i−n+n/e}(ℙT (e, n/e)),

both sides being nonzero iff n − n/e ≤ i ≤ n − 1.

Clearly the quotient of ST (1, n) by μ_{n} can be identified with ℙT (1, n). Thus H^{i}(ST (1, n))_{χ} ≅ H^{i}(ℙT (1, n), L_{χ}) where L_{χ} is the local system of rank 1 on ℙT (1, n) corresponding to χ. In particular, H^{i}(ST (1, n))^{μ_{n}} ≅ H^{i}(ℙT (1, n)), which is the e = 1 case of Theorem 1.2.

The other extreme is the e = n case. Since W (n, 1) ≅ μ_{n} and ℙT (n, 1) is a point, Theorem 1.2 says that for faithful χ ∈ μ̂_{n},

(1.3)  H^{i}(ST (1, n))_{χ} ≅ \{ \begin{matrix}Ind_{μ_{n}}^{S_{n}}(α_{n,1}), & if i = n − 1, \\ 0, & otherwise. \end{matrix}

This result for prime n was proved in [1, §4.4]. Note that the character α_{n,1} ∈ μ̂_{n}, which is defined by α_{n,1}(ζ) = (−1)^{n−1}ζ for ζ a primitive nth root of 1, is itself faithful precisely when n ≢ 2 (mod 4).

2. Equivariant weight polynomials

Suppose X is an irreducible complex variety which is minimally pure in the sense that H_{c}^{i}(X) is a pure Hodge structure of weight 2i−2 dim X for all i (see [1]). Let Γ be a finite group acting on X. We define the equivariant weight polynomials of this action by

P (γ, X, q) := ∑_{i}(−1)^{i} tr(γ, H_{c}^{i}(X)) q^{i−dim X},

for all γ ∈ Γ, where q is an indeterminate (= t^{2} in the notation of [1]). We also define

P^{Γ}(X, q) := ∑_{i}(−1)^{i} [H_{c}^{i}(X)] q^{i−dim X} ∈ R(Γ)[q],

where R(Γ) is the complexified representation ring of Γ. If Δ is an abelian finite group acting on X whose action commutes with that of Γ, and χ is a character of Δ, we define

P (γ, χ, X, q) := ∑_{i}(−1)^{i} tr(γ, H_{c}^{i}(X)_{χ}) q^{i−dim X}

= ∑_{i}(−1)^{i} \frac{1}{|Δ|} ∑_{δ∈Δ} χ(δ)^{−1}tr((γ, δ), H_{c}^{i}(X)) q^{i−dim X}

= \frac{1}{|Δ|} ∑_{δ∈Δ} χ(δ)^{−1}P ((γ, δ), X, q),

and similarly

P^{Γ}(χ, X, q) := ∑_{i}(−1)^{i} [H_{c}^{i}(X)_{χ}] q^{i−dim X} ∈ R(Γ)[q].

If X is nonsingular, we can translate knowledge of H_{c}^{i}(X) and H_{c}^{i}(X)_{χ} into knowledge of H^{i}(X) and H^{i}(X)_{χ} by Poincaré duality.

Now T (r, n) (respectively, ℙT (r, n)) is a nonsingular irreducible minimally pure variety of dimension n (respectively, n − 1); minimal purity is a standard property of hyperplane complements ([1, Example 3.3]). Also, ST (1, n) is clearly a nonsingular irreducible variety of dimension n − 1. To show that it is minimally pure, one can use [1, Corollary 4.2], or else observe that it is the quotient of ℙT (n, n) by the free action of a finite group, as follows. Define a surjective map φ : ℙT (n, n) → ST (1, n) by

φ([x_{1} : x_{2} : ⋯: x_{n}]) = \frac{1}{x_{1}x_{2} ⋯ x_{n}} (x_{1}^{n}, x_{2}^{n}, ⋯, x_{n}^{n}).

The fibres of φ are clearly the orbits of the normal subgroup Sμ_{n}^{n} ⊂ W (n, n) defined by

Sμ_{n}^{n} := {(ζ_{1}, ⋯, ζ_{n}) ∈ μ_{n}^{n} | ζ_{1}ζ_{2} ⋯ ζ_{n} = 1}.

The action of Sμ_{n}^{n} on ℙT (n, n) becomes free once one factors out the subgroup {(ζ, ζ, ⋯, ζ)}, which acts trivially. Thus ST (1, n) is minimally pure, and solving Problem 1.1 amounts to computing the polynomials P (w, χ, ST (1, n), q), for all w ∈ S_{n} and χ ∈ μ̂_{n}.

Consider the quotient of T (n, m) by Sμ_{n}^{m} for arbitrary m ≥ 1. This can be identified with

T^{(n)}(1, m) := {((z_{i}), z) ∈ T (1, m) × ℂ^{×} | z^{n} = z_{1} ⋯ z_{m}}.

The quotient map ψ : T (n, m) → T^{(n)}(1, m) is given by

ψ(x_{1}, x_{2}, ⋯, x_{m}) = ((x_{1}^{n}, x_{2}^{n}, ⋯, x_{m}^{n}), x_{1}x_{2} ⋯ x_{m}).

The group S_{m} × μ_{n} ≅ W (n, m)/Sμ_{n}^{m} acts on the quotient T^{(n)}(1, m) in the obvious way: S_{m} acts on the T (1, m) component, and μ_{n} acts by scaling z.

When m = n, we have an isomorphism

\begin{matrix}T^{(n)}(1, n) & →^{∼} ST (1, n) × ℂ^{×} \\ ((z_{1}, ⋯, z_{n}), z) & ↦ ((z_{1}z^{−1}, ⋯, z_{n}z^{−1}), z), \end{matrix}

which respects the S_{n}-action, and transforms the μ_{n}-action on T^{(n)}(1, n) into the inverse of the μ_{n}-action on ST (1, n), and a scaling action on ℂ^{×}. Since the latter has no effect on cohomology,

(2.1)  P (w, χ, ST (1, n), q) = \frac{1}{q − 1}P (w, χ^{−1}, T^{(n)}(1, n), q).

So we aim to compute P (w, χ^{−1}, T^{(n)}(1, n), q); it turns out to be convenient to compute the polynomials P (w, χ^{−1}, T^{(n)}(1, m), q) for all m ≥ 1 and w ∈ S_{m} together.

Remark 2.1. One can see a priori that allowing m ≠ n incurs no extra work, thanks to the following neat argument, pointed out to me by Lehrer. If d = gcd(m, n), the action of μ_{n/d} ⊂ μ_{n} on T^{(n)}(1, m) is part of the action of the connected group ℂ^{×} defined by

t.((z_{i}), z) = ((t^{n/d}z_{i}), t^{m/d}z).

Hence μ_{n/d} acts trivially on cohomology, so P (w, χ^{−1}, T^{(n)}(1, m), q) = 0 unless χ|_{μ_{n/d}} = 1, i.e. χ^{d} = 1, i.e. e|m, where e is the order of χ. Moreover, if e|m, then writing χ^{∘} for the character of μ_{e} such that χ(ζ) = χ^{∘}(ζ^{n/e}) for all ζ ∈ μ_{n}, and χ′ for the character of μ_{m} defined by χ′(ζ) = χ^{∘}(ζ^{m/e}) for all ζ ∈ μ_{m}, we have

\begin{aligned}P (w, χ^{−1}, T^{(n)}(1, m), q) & = P (w, (χ^{∘})^{−1}, T^{(e)}(1, m), q) \\ & = P (w, (χ′)^{−1}, T^{(m)}(1, m), q). \end{aligned}

We will not actually use this observation.

The identification of T^{(n)}(1, m) with the quotient of T (n, m) by Sμ_{n}^{m} has the following consequence for equivariant weight polynomials:

Proposition 2.2. For any w ∈ S_{m} and χ ∈ μ̂_{n},

\begin{matrix}P (w, χ^{−1}, & T^{(n)}(1, m), q) \\ = \frac{1}{n^{m}} ∑_{(ζ_{i})∈μ_{n}^{m}} χ(ζ_{1} ⋯ ζ_{m})P (w(ζ_{1}, ⋯, ζ_{m}), T (n, m), q). \end{matrix}

Proof. It is well known that if V is a representation of the finite group G and V^{H} is the subspace invariant under the normal subgroup H, the character of G/H on V^{H} is given by

tr(gH, V^{H}) = \frac{1}{|H|} ∑_{h∈H} tr(gh, V).

Now apply this with V = H_{c}^{i}(T (n, m)), G = W (n, m), and H = Sμ_{n}^{m}, so that V^{H} ≅ H_{c}^{i}(T^{(n)}(1, m)) and G/H ≅ S_{m} × μ_{n}. We find that for all ζ ∈ μ_{n},

P ((w, ζ), T^{(n)}(1, m), q) = \frac{1}{n^{m−1}} ∑_{\begin{matrix}(ζ_{i})∈μ_{n}^{m} \\ ζ_{1}···ζ_{m}=ζ \end{matrix}} P (w(ζ_{1}, ⋯, ζ_{m}), T (n, m), q).

Combining this with the fact that

P (w, χ^{−1}, T^{(n)}(1, m), q) = \frac{1}{n} ∑_{ζ∈μ_{n}} χ(ζ)P ((w, ζ), T^{(n)}(1, m), q)

gives the desired result. □

3. Generating functions

In this section we will compute the sum in Proposition 2.2 using Lehrer’s formulas for the equivariant weight polynomials of T (n, m). As is usual in dealing with characters of symmetric groups and wreath products, the computations become easier if one uses suitable “equivariant generating functions”.

For any r ≥ 1, let Λ(r) denote the polynomial ring ℂ[p_{i}(ζ)] in countably many independent variables p_{i}(ζ) where i is a positive integer and ζ ∈ μ_{r}. Define an ℕ-grading on Λ(r) by deg(p_{i}(ζ)) = i. Also let Λ(r)[q] := Λ(r) ⊗_{ℂ} ℂ[q], with the ℕ-grading given by the first factor (so deg(q) = 0). Let 𝔸(r) = ℂ[[p_{i}(ζ)]] be the completion of Λ(r), and set 𝔸(r)[q] = 𝔸(r) ⊗ ℂ[q].

As in [7, Chapter I, Appendix B], we define an isomorphism ch_{W (r,m)} :

R(W (r, m)) →^{∼} Λ(r)_{m} by

ch_{W (r,m)}([M]) = \frac{1}{r^{m}m!} ∑_{y∈W (r,m)} tr(y, M) p_{y},

where p_{y} = ∏_{i,ζ} p_{i}(ζ)^{a_{i}(ζ)} if y has a_{i}(ζ) cycles of length i and type ζ. Note that to recover tr(y, M) from ch_{W (r,m)}([M]) one must multiply the coefficient of ∏_{i,ζ} p_{i}(ζ)^{a_{i}(ζ)} by the order of the centralizer of y, which is ∏_{i,ζ} a_{i}(ζ)!(ri)^{a_{i}(ζ)}. Write ch_{W (r,m)} also for the induced isomorphism

R(W (r, m))[q] →^{∼} Λ(r)[q]_{m}.

Lehrer’s results on the equivariant weight polynomials for T (r, m) can be conveniently stated in terms of the equivariant generating function P (r, q) ∈ 𝔸(r)[q] defined by

\begin{aligned}P (r, q) & := 1 + ∑_{m≥1} ch_{W (r,m)}(P^{W (r,m)}(T (r, m), q)) \\ & = 1 + ∑_{m≥1} \frac{1}{r^{m}m!} ∑_{y∈W (r,m)} P (y, T (r, m), q) p_{y}. \end{aligned}

In the following result μ denotes the Möbius function.

Theorem 3.1. If R_{r,i,θ} := ∑_{d|i} |{ζ ∈ μ_{r} | ζ^{d} = θ}|μ(d)(q^{i/d}−1) ∈ ℂ[q],

P (r, q) = ∏_{\begin{matrix}i≥1 \\ θ∈μ_{r} \end{matrix}}(1 + p_{i}(θ))^{R_{r,i,θ}/ri}.

Proof. Mostly this is a reformulation of results in [4] (r = 2) and [6] (r ≥ 2). See [3, Theorem 8.4] for a proof for all r ≥ 2 (T (r, m) is the same as what is there called M(r, m)). The r = 1 case can be proved by the same method (note that T (1, m) is different from the variety M(1, m) considered in [3, Theorem 8.2], since it has the extra condition of nonzero coordinates). □

Recovering the traces of individual elements by the above rule, we get an equivalent statement, closer to Lehrer’s in [6, §6]: if y ∈ W (r, m) has a_{i}(ζ) cycles of length i and type ζ,

(3.1) P (y, T (r, m), q) = ∏_{\begin{matrix}i≥1 \\ ζ∈μ_{r} \end{matrix}} R_{r,i,ζ}(R_{r,i,ζ} −ri) ⋯ (R_{r,i,ζ} −(a_{i}(ζ)−1)ri).

This also gives a character formula for ℙT (r, m), since it is easy to see that

(3.2)  P (y, ℙT (r, m), q) = \frac{1}{q − 1}P (y, T (r, m), q).

There is an alternative description of the polynomials R_{r,i,θ}. Define

R_{i}^{(r)} := ∑_{\begin{matrix}d|i \\ gcd(d,r)=1 \end{matrix}} μ(d)(q^{i/d} − 1) ∈ ℂ[q].

Lemma 3.2. If t(θ) denotes the order of θ,

R_{r,i,θ} = ∑_{s| gcd(r/t(θ),i)}sμ(s)R_{i/s}^{(r)}.

Proof. Since μ_{r} is cyclic of order r, we have

|\{ζ ∈ μ_{r} | ζ^{d} = θ\}| = \{ \begin{matrix}gcd(d, r), & if gcd(d, r) | (r/t(θ)), \\ 0, & otherwise. \end{matrix}

Hence

R_{r,i,θ} = ∑_{s|(r/t(θ))} s ∑_{\begin{matrix}d|i \\ gcd(d,r)=s \end{matrix}}μ(d)(q^{i/d} − 1).

The sum over d has no terms unless s|i, in which case it equals

∑_{\begin{matrix}d|(i/s) \\ gcd(d,r)=1 \end{matrix}}μ(ds)(q^{i/ds} − 1).

Since gcd(d, r) = 1 implies μ(ds) = μ(d)μ(s), this is μ(s)R_{i/s}^{(r)}. □

This Lemma makes it clear that the r = 2 case of (3.1) is indeed equivalent to [4, Theorem 5.6].

Now for any χ ∈ μ̂_{n}, define the generating function

\begin{aligned}P (χ, q) & := 1 + ∑_{m≥1} ch_{S_{m}}(P^{S_{m}}(χ^{−1}, T^{(n)}(1, m), q)) \\ & = 1 + ∑_{m≥1} \frac{1}{m!} ∑_{w∈S_{m}} P (w, χ^{−1}, T^{(n)}(1, m), q) p_{w} ∈ 𝔸(1)[q]. \end{aligned}

We want a formula for this similar to Theorem 3.1. Define

P_{i}^{(r)} := ∏_{s| gcd(r,i)}(1 − (−p_{i})^{r/s})^{sμ(s)R_{i/s}^{(r)}/ri} ∈ 𝔸(1)[q].

Theorem 3.3. If χ ∈ μ̂_{n} has order e, P (χ, q) = ∏_{i≥1} P_{i}^{(e)}. Proof. Using Proposition 2.2, we see that P (χ, q) equals

1 + ∑_{m≥1} \frac{1}{n^{m}m!} ∑_{\begin{matrix}w∈S_{m} \\ (ζ_{i})∈μ_{n}^{m} \end{matrix}} χ(ζ_{1} ⋯ ζ_{m})P (w(ζ_{1}, ⋯, ζ_{m}), T (n, m), q) p_{w},

which is precisely the result of applying to P (n, q) the specialization p_{i}(θ) → χ(θ)p_{i}. So by Theorem 3.1,

P (χ, q) = exp ∑_{\begin{matrix}i≥1 \\ θ∈μ_{n} \end{matrix}} \frac{R_{n,i,θ}}{ni} log(1 + χ(θ)p_{i})

= exp ∑_{\begin{matrix}i≥1 \\ θ∈μ_{n} \\ m≥1 \end{matrix}} \frac{−R_{n,i,θ}}{nmi} χ(θ)^{m}(−p_{i})^{m}

= exp ∑_{\begin{matrix}i≥1 \\ d|i \\ ζ∈μ_{n} \\ m≥1 \end{matrix}} \frac{−μ(d)}{nmi}χ(ζ)^{dm}(q^{i/d} − 1)(−p_{i})^{m}

= exp ∑_{\begin{matrix}i≥1 \\ d|i \\ m≥1, e|dm \end{matrix}}\frac{−μ(d)}{mi}(q^{i/d} − 1)(−p_{i})^{m},

since ∑_{ζ∈μ_{n}} χ(ζ)^{dm} = n if χ^{dm} is trivial, and vanishes otherwise. Thus we need only show that

(3.3)  P_{i}^{(e)} = exp ∑_{\begin{matrix}d|i \\ m≥1, e|dm \end{matrix}}\frac{−μ(d)}{mi}(q^{i/d} − 1)(−p_{i})^{m}.

The condition e|dm is equivalent to (e/ gcd(d, e))|m. Writing f for gcd(d, e), the right-hand side becomes

exp ∑_{\begin{matrix}f|e \\ d|i, gcd(d,e)=f \\ m≥1, (e/f)|m \end{matrix}}\frac{−μ(d)}{mi}(q^{i/d} − 1)(−p_{i})^{m}

= exp ∑_{f| gcd(e,i)} \frac{f}{ei} log(1 − (−p_{i})^{e/f}) ∑_{\begin{matrix}d|i \\ gcd(d,e)=f \end{matrix}}μ(d)(q^{i/d} − 1).

By the same argument as in the proof of Lemma 3.2, the sum over d equals μ(f)R_{i/f}^{(e)} as required. □

Note that P (χ, q) depends only on e, not on n or χ, and that, as predicted in Remark 2.1, its nonzero homogeneous components all have degree divisible by e.

There is no formula as neat as (3.1) for the individual polynomials P (w, χ^{−1}, T^{(n)}(1, m), q). But if w ∈ S_{m} has a_{i} cycles of length i, we know that

(3.4) P (w, χ^{−1}, T^{(n)}(1, m), q) = ∏_{i≥1} a_{i}! i^{a_{i}}(coefficient of p_{i}^{a_{i}} in P_{i}^{(e)}).

Note that for the right-hand side to be nonzero, a_{i} must be divisible by (e/ gcd(e, i)) for all i. In the special case that gcd(e, i) = 1, we have

P_{i}^{(e)} = (1 − (−p_{i})^{e})^{R_{i}^{(e)}/ei},

and the coefficient of p_{i}^{a_{i}}, where a_{i} is divisible by e, is

(3.5) (−1)^{a_{i}−a_{i}/e} \frac{R_{i}^{(e)}(R_{i}^{(e)} − ei)(R_{i}^{(e)} − 2ei) ⋯ (R_{i}^{(e)} − (a_{i} − e)i)}{(ei)^{a_{i}/e}(a_{i}/e)!}.

Some further special cases of note: the e = 1 case of Theorem 3.3 says that

P (triv, q) = ∏_{i≥1}(1 + p_{i})^{R_{i}^{(1)}/i} = P (1, q),

reflecting the fact that the quotient of T^{(n)}(1, m) by μ_{n} is T (1, m). Slightly more interesting is the e = 2 case. We have

P_{i}^{(2)} = \{ \begin{matrix}(1 − p_{i}^{2})^{R_{i}^{(2)}/2i}, & if i is odd, \\ (1 − p_{i}^{2})^{R_{i}^{(2)}/2i}(1 + p_{i})^{−R_{i/2}^{(2)}/i}, & if i is even. \end{matrix}

Hence if i is even, the coefficient of p_{i}^{a_{i}} is

∑_{j=0}^{⌊a_{i}/2⌋}(−1)^{j}\binom{R_{i}^{(2)}/2i}{j}\binom{−R_{i/2}^{(2)}/i}{a_{i} − 2j}.

Returning to Problem 1.1, equations (2.1) and (3.4) tell us that if w ∈ S_{n} has a_{i} cycles of length i,

(3.6) P (w, χ, ST (1, n), q) = \frac{1}{q − 1} ∏_{i≥1} a_{i}! i^{a_{i}}(coefficient of p_{i}^{a_{i}} in P_{i}^{(e)}).

Since there are ϕ(e) characters χ ∈ μ̂_{n} of order e, we deduce that (3.7)

P (w, ST (1, n), q) = \frac{1}{q − 1} ∑_{e|n} ϕ(e) ∏_{i≥1} a_{i}! i^{a_{i}}(coefficient of p_{i}^{a_{i}} in P_{i}^{(e)}).

Remark 3.4. If q is specialized to a prime power congruent to 1 mod n, the right-hand side of (3.7) equals the formula given in [2, Theorem 5.8] for the number of 𝔽_{q}-points of the regular set of a maximal torus of SL_{n}(𝔽_{q}) obtained from a maximally split one by twisting with w. (To see this, use the expression (3.3) for P_{i}^{(e)}; the coefficient of p_{i}^{a_{i}} is called R_{a_{i},n}^{i}(q) in [2].) We have effectively reproved that result, since

P (w, ST (1, n), q) = |ST (1, n)(\overline{𝔽_{q}})^{wF} |

by Grothendieck’s Frobenius trace formula and a suitable comparison theorem of complex and ℓ-adic cohomology. Actually, everything we have done with equivariant weight polynomials P (γ, X, q) would have worked equally well with the numbers of fixed points |X(\overline{𝔽_{q}})^{γF} | of twisted Frobenius maps.

4. Induction

We now aim to interpret the generating function P (χ, q) in terms of induced characters. Recall that W (r, m) is embedded in S_{rm} as the centralizer of the product of m disjoint r-cycles. For any θ ∈ μ_{r}, let t(θ) denote the order of θ.

Lemma 4.1. For any W (r, m)-module M,

ch_{S_{rm}}([Ind_{W (r,m)}^{S_{rm}}(M)]) = ch_{W (r,m)}([M])|_{p_{i}(θ)→p_{it(θ)}^{r/t(θ)}}.

Proof. This is a direct consequence of Frobenius’ formula for induced characters, once one observes that a cycle of length i and type θ in W (r, m) beomes the product of r/t(θ) disjoint it(θ)-cycles when regarded as an element of S_{rm}. □

Lemma 4.2. For any W (r, m)-module M,

\begin{aligned}ch_{S_{rm}}([Ind_{W (r,m)}^{S_{rm}} & (α_{r,m} ⊗ M)]) \\ & = (−1)^{rm−m}ch_{W (r,m)}([M])|_{p_{i}(θ)→−θ(−p_{it(θ)})^{r/t(θ)}}. \end{aligned}

Proof. Let y ∈ W (r, i) be a cycle of length i and type θ. Since α_{r,i} is the product of det : W (r, i) → ℂ^{×} and the sign character on S_{ri},

α_{r,i}(y) = (−1)^{i−1+(it(θ)−1)r/t(θ)}θ = (−1)^{i−1+ri−r/t(θ)}θ.

Also, if y ∈ W (r, m),

p_{y}|_{p_{i}(θ)→(−1)^{ri+i}p_{i}(θ)} = (−1)^{rm−m}p_{y}.

Hence

ch_{W (r,m)}([α_{r,m} ⊗ M]) = (−1)^{rm−m}ch_{W (r,m)}([M])|_{p_{i}(θ)→−θ(−1)^{r/t(θ)}p_{i}(θ)},

and the result follows by applying the previous Lemma. □

Now define an element P′(r, q) ∈ 𝔸(1)[q] by

P′(r, q) := P (r, q)|_{p_{i}(θ)→−θ(−p_{it(θ)})^{r/t(θ)}}

= 1 + ∑_{m≥1} ch_{W (r,m)}(P^{W (r,m)}(T (r, m), q))|_{p_{i}(θ)→−θ(−p_{it(θ)})^{r/t(θ)}}

= 1 + ∑_{m≥1}(−1)^{rm−m}ch_{S_{rm}}(Ind_{W (r,m)}^{S_{rm}}(α_{r,m} ⊗ P^{W (r,m)}(T (r, m), q))).

Proposition 4.3. P′(r, q) = ∏_{i≥1} P_{i}^{(r)}.

Proof. By Theorem 3.1 and Lemma 3.2, we have

\begin{aligned}P′(r, q) & = ∏_{\begin{matrix}i≥1 \\ θ∈μ_{r} \end{matrix}}(1 − θ(−p_{it(θ)})^{r/t(θ)})^{R_{r,i,θ}/ri} \\ & = ∏_{\begin{matrix}i≥1 \\ θ∈μ_{r} \\ s| gcd(r/t(θ),i) \end{matrix}} (1 − θ(−p_{it(θ)})^{r/t(θ)})^{sμ(s)R_{i/s}^{(r)}/ri}. \end{aligned}

Applying to this the Möbius inversion formula for cyclotomic polynomials, namely that

∏_{\begin{matrix}θ∈μ_{r} \\ t(θ)=t \end{matrix}}(1 − θX) = ∏_{u|t}(1 − X^{t/u})^{μ(u)},

we obtain

P′(r, q) = ∏_{\begin{matrix}i≥1 \\ t|r \\ s| gcd(r/t,i) \\ u|t \end{matrix}} (1 − (−p_{it})^{r/u})^{sμ(s)μ(u)R_{i/s}^{(r)}/ri}.

Write this as ∏_{i≥1} Q_{i}^{(r)}, where Q_{i}^{(r)} is the product of all factors involving the variable p_{i}. Thus

Q_{i}^{(r)} = exp ∑_{\begin{matrix}t| gcd(r,i) \\ s| gcd(r/t,i/t) \\ u|t \end{matrix}}\frac{stμ(s)μ(u)}{ri}R_{i/st}^{(r)} log(1 − (−p_{i})^{r/u})

= exp ∑_{\begin{matrix}v| gcd(r,i) \\ u|v \\ s|(v/u) \end{matrix}}\frac{vμ(s)μ(u)}{ri}R_{i/v}^{(r)} log(1 − (−p_{i})^{r/u}),

where we have set v = st. Since ∑_{s|(v/u)} μ(s) is nonzero if and only if u = v, we find that Q_{i}^{(r)} = P_{i}^{(r)} as required. □

Corollary 4.4. If χ ∈ μ̂_{n} has order e, P (χ, q) = P′(e, q).

Proof. Combine Theorem 3.3 and Proposition 4.3. □

Corollary 4.5. If χ ∈ μ̂_{n} has order e, and e|m, we have the following equality in R(S_{m})[q]:

\begin{aligned}P^{S_{m}} & (χ^{−1}, T^{(n)}(1, m), q) \\ & = (−1)^{m−m/e}Ind_{W (e,m/e)}^{S_{m}}(α_{e,m/e} ⊗ P^{W (e,m/e)}(T (e, m/e), q)). \end{aligned}

Proof. Under the isomorphism ch_{S_{m}}, the left-hand side corresponds to the degree-m term of P (χ, q), and the right-hand side corresponds to the degree-m term of P′(e, q). □

Taking coefficients of q^{i−m} on both sides and multiplying by (−1)^{i}, we get an isomorphism of S_{m}-modules:

(4.1) H_{c}^{i}(T^{(n)}(1, m))_{χ^{−1}} ≅ Ind_{W (e,m/e)}^{S_{m}}(α_{e,m/e} ⊗ H_{c}^{i−m+m/e}(T (e, m/e)).

By Poincaré duality, this is equivalent to

(4.2) H^{i}(T^{(n)}(1, m))_{χ} ≅ Ind_{W (e,m/e)}^{S_{m}}(α_{e,m/e}^{−1} ⊗ H^{i−m+m/e}(T (e, m/e)).

An equivalent way to state this isomorphism is:

(4.3) H^{i}(T (1, m), L) ≅ Ind_{W (e,m/e)}^{S_{m}}(α_{e,m/e}^{−1} ⊗ H^{i−m+m/e}(T (e, m/e)),

where L is any of the rank-one local systems on T (1, m) obtained from the covers T^{(n)}(1, m), and e is its order in the group of rank-one local systems.

Finally, we prove Theorem 1.2. Equation (2.1) and Corollary 4.5 together imply

P^{S_{n}}(χ, ST (1, n), q)

= \frac{(−1)^{n−n/e}}{q − 1} Ind_{W (e,n/e)}^{S_{n}}(α_{e,n/e} ⊗ P^{W (e,n/e)}(T (e, n/e), q))

= (−1)^{n−n/e}Ind_{W (e,n/e)}^{S_{n}}(α_{e,n/e} ⊗ P^{W (e,n/e)}(ℙT (e, n/e), q)).

Taking coefficients of q^{i−n+1} on both sides and multiplying by (−1)^{i}, we get an isomorphism of S_{n}-modules:

(4.4) H_{c}^{i}(ST (1, n))_{χ} ≅ Ind_{W (e,n/e)}^{S_{n}}(α_{e,n/e} ⊗ H_{c}^{i−n+n/e}(ℙT (e, n/e))).

Since the right-hand side depends only on n and the order of χ, this remains true if χ is replaced by χ^{−1}. Then Theorem 1.2 follows by Poincaré duality.

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

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