To appear: Journal London Math. Soc.

On the left cell representations of Iwahori–Hecke algebras of finite Coxeter groups

Andrew Mathas

Abstract

In this paper we investigate the left cell representations of the Iwahori–Hecke algebras associated to a finite Coxeter group W. Our main result shows that T_{w_{0}}, where w_{0} is the element of longest length in W, acts essentially as an involution upon the canonical bases of a cell representation. We describe some properties of this involution, use it to further describe the left cells, and finally show how to realize each cell representation as a submodule of ℋ . Our results rely upon certain positivity properties of the structure constants of the Kazhdan–Lusztig bases of the Hecke algebra and so have not yet been shown to apply to all finite Coxeter groups.

Introduction

Let (W, S) be a finite Coxeter system with length function ℓ and let ℋ = ℋ (W, S) be the associated Iwahori–Hecke algebra defined over the Laurent polynomial ring 𝒜 = ℤ[q^{\frac{1}{2}} , q^{− \frac{1}{2}} ]. Thus ℋ is the free associative 𝒜 –algebra with unit T_{1} with 𝒜 –free basis { T_{w} : w ∈ W } and multiplication determined by the formulae:

T_{s}T_{w} = \{ \begin{matrix}T_{sw} & ℓ(sw) > ℓ(w) \\ (q − 1)T_{w} + qT_{sw} & ℓ(sw) < ℓ(w), \end{matrix}

where s ∈ S and w ∈ W. Note in particular that if w = s_{1} … s_{k} is a reduced expression for w then T_{w} = T_{s_{1}} … T_{s_{k}} (s_{1}, …, s_{k} ∈ S), and consequently T_{w} is invertible for all w since T_{s}^{−1} = q^{−1}(T_{s} − (q − 1)T_{1}) for any s ∈ S. Further details can be found, for example, in Humphreys’ book [4].

In this paper we will be concerned with the left cells of W and the associated representations. We emphasis that almost all of our results apply only to those finite Coxeter groups where the positivity results (1.1) and (1.6) are known to hold. In particular they apply to finite Weyl groups and to finite dihedral groups. For brevity we assume throughout that W is a Weyl group. It should be noted that both (1.1) and (1.6) are almost certainly true for all Coxeter groups.

This paper is organized as follows: in section 1 we recall results from the literature concerning the Kazhdan–Lusztig bases of ℋ and Lusztig’s 𝔞–function and set our notation.

In section 2 we give bounds upon the degrees of various structure constants that arise from the action of the T –basis upon the Kazhdan–Lusztig bases. In section 3 we apply these results to prove that the element of longest length T_{w_{0}} acts essentially as a involution upon the cell representations. We also describe the fix points of this involution, and use it to give a characterization of when a cell representation is irreducible. Finally, in section 4 we give a canonical embedding of each cell representation as a submodule of ℋ .

1 Left cells and Lusztig’s 𝔞–function

In this section we set our notation and recall the results from the literature which we shall need. Most of these preliminary results are due to either Kazhdan and Lusztig [5, 6] or Lusztig [10, 13]; a good reference is Curtis’ survey article [1]. For brevity we assume throughout that W is a finite Weyl group.

The ring 𝒜 has two semi–linear involutions and ε given by \overline{q^{\frac{1}{2}}} = q^{− \frac{1}{2}} and ε(q^{\frac{1}{2}} ) = −q^{− \frac{1}{2}} . Extend these to involutions on ℋ by setting \overline{T_{x}} = T_{x^{−1}}^{−1} and ε(T_{x}) = ϵ_{x}q_{x}^{− \frac{1}{2}} T_{x} for all x ∈ W

where ϵ_{x} = (−1)^{ℓ(x)} and qx^{\frac{1}{2}} = q^{\frac{1}{2} ℓ(x)}. Note that and ε commute and that \overline{h_{1}h_{2}} = \overline{h_{1}} \overline{h_{2}} and ϵ(h_{1}h_{2}) = ϵ(h_{1})ϵ(h_{2}) for all h_{1}, h_{2} ∈ ℋ .

Let τ : ℋ ⟶ 𝒜 be the canonical trace function on ℋ ; thus if h = ∑_{x∈W} a_{x}T_{x} ∈ ℋ then τ(h) = a_{1}. It is easy to see that τ(h_{1}h_{2}) = τ(h_{2}h_{1}) for any h_{1}, h_{2} ∈ ℋ .

For convenience we renormalise the T –basis of ℋ and write T̃_{x} = q_{x}^{− \frac{1}{2}} T_{x} for all x ∈ W. Then ε(T̃_{x}) = T̃_{x} and τ(T̃_{x}T̃_{y}) = δ_{x,y^{−1}} for all x, y ∈ W (Kronecker delta).

Let ≤ denote the Bruhat ordering on W and recall the Kazhdan–Lusztig bases {C_{x}} and {C′_{x}} of ℋ from [5] (x ∈ W). These elements are characterized as the unique elements of ℋ such that (i) \overline{C_{x}} = C_{x} and \overline{C′_{x}} = C′_{x} and

(ii) C′_{x} = q_{x}^{− \frac{1}{2}} ∑_{y≤x} qy^{\frac{1}{2}} P_{y,x}T̃_{y} and C_{x} = ϵ_{x}qx^{\frac{1}{2}} ∑_{y≤x} ϵ_{y}q_{y}^{− \frac{1}{2}} \overline{P_{y,x}} T̃_{y}

where P_{y,x} ∈ ℤ[q] is a polynomial such that P_{x,x} = 1 for all x and if y < x then P_{y,x} has degree (in q) of at most \frac{1}{2}(ℓ(x) − ℓ(y) − 1). The polynomials P_{y,x} are the celebrated Kazhdan–Lusztig polynomials. From (ii) it is immediate that ε(C_{x}) = C′_{x} for all x ∈ W.

Let 𝒜^{+} = ℕ[q^{\frac{1}{2}} , q^{− \frac{1}{2}} ]. By interpreting the Kazhdan–Lusztig polynomials as Poincarè series of intersection cohomology complexes Kazhdan and Lusztig [6] proved the following deep fact:

(1.1)  P_{y,x} ∈ 𝒜^{+} for all x, y ∈ W.

We shall need the preorders ≤_{L}, ≤_{R}, and ≤_{LR} from [5] and the associated equivalence relations ∼_{L}, ∼_{R} and ∼_{LR} for the left, right and two–sided cells of W respectively. Thus, x ≤_{L} y if and only if there exists some h ∈ ℋ such that C′_{x} appears as a non–zero summand of hC′_{y} when this product is expressed in the C′–basis; x ≤_{R} y if and only if x^{−1} ≤_{L} y^{−1}; and ≤_{LR} is the preorder generated by ≤_{L} and ≤_{R}.

Given a left cell Γ and x ∈ W we shall write x ≤_{L} Γ if x ≤_{L} y for some (equivalently all) y ∈ Γ, and x <_{L} Γ if x ≤_{L} Γ and x ∉ Γ. We use similar notations for the right and two–sided cells.

Associated to each cell Γ of W are the (left) cell representations ℋ_{Γ} and its “ε–dual” ℋ′_{Γ} defined by

ℋ_{Γ} = ∑_{x≤_{L}Γ} 𝒜 C_{x}/ ∑_{x<_{L}Γ} 𝒜 C_{x} and ℋ′_{Γ} = ∑_{x≤_{L}Γ} 𝒜 C′_{x}/ ∑_{x<_{L}Γ} 𝒜 C′_{x}.

Thus ℋ_{Γ} and ℋ′_{Γ} differ by the sign representation ϵ when specialized to the group algebra of W; equivalently, ℋ′_{Γ} = ε(ℋ_{Γ}). From the definition of the preorder ≤_{L} it follows that these are well defined ℋ –modules. Below we show how to realize, canonically, each cell representation as a submodule of ℋ (see Theorem 4.1).

Following Lusztig [10], define bases {D_{x}} and {D′_{x}} dual to the Kazhdan–Lusztig bases by

τ(C_{x}D_{y}) = δ_{x,y^{−1}} and τ(C′_{x}D′_{y}) = δ_{x,y^{−1}},

for all x, y ∈ W. Then

(1.2)  D_{x} = q_{x}^{− \frac{1}{2}} ∑_{y≥x} \overline{Q_{x,y}} qy^{\frac{1}{2}} T̃_{y}  and D′_{x} = ϵ_{x}qx^{\frac{1}{2}} ∑_{y≥x} ϵ_{y}q_{y}^{− \frac{1}{2}} Q_{x,y}T̃_{y}.

where the polynomials Q_{x,y} are the inverse Kazhdan–Lusztig polynomials. Consequently ε(D_{x}) = D′_{x}. By (1.2) the Q–polynomials are determined by the inversion formulae:

(1.3)  ∑_{x≤y≤z} ϵ_{x}ϵ_{y}Q_{x,y}P_{y,z} = δ_{x,z}.

One can check that if w_{0} is the element of longest length in W then

D_{x} = T̃_{w_{0}}C′_{w_{0}x} = C′_{xw_{0}} T̃_{w_{0}} and D′_{x} = T̃_{w_{0}}C_{w_{0}x} = C_{xw_{0}} T̃_{w_{0}}.

Using (1.1) it is straightforward to show that

(1.4)  D′_{x}C′_{y} ≠ 0 if and only if D_{x}C_{y} ≠ 0 if and only if x ≤_{L} y^{−1}.

Multiplying on the left by T̃_{w_{0}}^{−1} we obtain

(1.5)  C_{x}C′_{y} ≠ 0 if and only if C′_{x}C_{y} ≠ 0 if and only if w_{0}x ≤_{L} y^{−1}.

Following Lusztig [13], define polynomials h_{x,y,z} ∈ 𝒜 by

C′_{x}C′_{y} = ∑_{z∈W} h_{x,y,z}C′_{z}.

For all x, y, z ∈ W it is clear that h_{x,y,z} = h_{y^{−1},x^{−1},z^{−1}}, \overline{h_{x,y,z}} = h_{x,y,z} and that if h_{x,y,z} ≠ 0 then z ≤_{L} y and z ≤_{R} x. We shall apply these facts below without further comment. Lusztig’s h–polynomials are actually defined using the C–basis of ℋ ; however, these polynomials are essentially the same (just apply the involution ε). We use the C′–basis because of the following deep positivity result (see [16]), analogous to (1.1):

(1.6).  h_{x,y,z} ∈ 𝒜^{+} for all x, y, z ∈ W

Lusztig’s 𝔞–function 𝔞: W ⟶ℕ is defined by

𝔞(z) = max\{ deg_{q^{\frac{1}{2}}} (h_{x,y,z}) : x, y ∈ W \}.

Note that q^{− \frac{1}{2} 𝔞(z)} is the smallest power of q^{\frac{1}{2}} which appears in h_{x,y,z} since \overline{h_{x,y,z}} = h_{x,y,z}. Also, 𝔞(z) = 𝔞(z^{−1}) for all z ∈ W since h_{x,y,z} = h_{y^{−1},x^{−1},z^{−1}}. Again Lusztig defines the 𝔞–function using the structure constants of multiplication with respect to the C–basis; however by the above remarks these definitions are equivalent.

The two positivity results, (1.1) and (1.6), together with an otherwise elementary argument [13] show that

(1.7)  𝔞(x) ≤ ℓ(x) − 2δ(x) for all x ∈ W,

where δ(x) is the degree of P_{1,x} as a polynomial in q. Let 𝒟 = { d ∈ W : 𝔞(d) = ℓ(d)−2δ(d) }. Then every element of 𝒟 is an involution and 𝒟 is called the set of Duflo involutions of W.

Let γ_{x,y,z} ∈ ℤ be the coefficient of q^{\frac{1}{2} 𝔞(z)} in h_{x,y,z}; this is also the coefficient of q^{− \frac{1}{2} 𝔞(z)} in h_{x,y,z} (note: Lusztig defines γ_{x,y,z} to be the coefficient of q^{\frac{1}{2} 𝔞(z)} in h_{x,y,z^{−1}}; we follow Curtis’[1] convention). By (1.6), γ_{x,y,z} ∈ ℕ. If d ∈ 𝒟 and x, y ∈ W then

(1.8)  γ_{x,y,d} ≠ 0 if and only if x = y^{−1} and y ∼_{L} d; moreover γ_{y^{−1},y,d} = 1.

Moreover, for all d ∈ 𝒟

(1.9)  P_{1,d} = q^{δ(d)} + terms of lower degree in q.

We also have that

(1.10)  if x ≤_{LR} y then 𝔞(x) ≥ 𝔞(y),

and in addition

(1.11)  if x ≤_{L} y and 𝔞(x) = 𝔞(y) then x ∼_{L} y.

In particular, by (1.10), the 𝔞–function is constant on the two–sided cells of W; given a left cell Γ of W we will sometimes write 𝔞(Γ) for this common value. Theorems (1.8) and (1.10) are used to show that

(1.12) γ_{x,y,z} = γ_{y,z^{−1},x^{−1}} = γ_{z^{−1},x,y^{−1}} for all x, y, z ∈ W and if γ_{x,y,z} ≠ 0 then x ∼_{L} y^{−1}, y ∼_{L} z, and x ∼_{R} z.

Finally, using (1.8) and (1.12), one can show that each left cell Γ contains a unique element d_{Γ} of 𝒟 (this being the element d ∈ 𝒟 occurring in (1.8)).

All of the results above referring to the 𝔞–function are due to Lusztig and may be found in [13] and the references there.

2 Bounding the degrees of structure constants

In this section we give bounds upon the degrees of the polynomials which describe the action of the T̃–basis upon the C and C′–bases. We apply these results in the next section to explicitly describe the action of T̃_{w_{0}} upon the cell representations.

We are interested in the (Laurent) polynomials defined by the equation

T̃_{x}C′_{y} = ∑_{z∈W} g_{x,y,z}C′_{z}.

Thus g_{x,y,z} ∈ 𝒜 and from the action of the T̃–basis upon the C′–basis (see [5, (2.3)]), it is clear that all the terms in g_{x,y,z} have degrees (as a Laurent polynomial in q^{\frac{1}{2}} ), between −ℓ(x) and ℓ(x). We improve upon this bound below.

We begin with an easy but much used lemma:

(2.1) Lemma For all x, y in W we have:

(i) C_{x}C_{y} = ∑_{z} ε(h_{x,y,z})C_{z}. (ii) C′_{x}C_{y} = ∑_{z≤_{L}y} h_{x^{−1},zw_{0},yw_{0}}C_{z}. (iii) C′_{x}C_{y} = ∑_{z≤_{R}x} ε(h_{w_{0}z,y^{−1},w_{0}x})C′_{z}.

Proof: The first statement is obvious, and the remaining statements are essentially known (see, for example, Dyer–Lehrer [2]). For completeness we prove (ii): the coefficient of C_{z} in the product C′_{x}C_{y} is

τ(C′_{x}C_{y}D_{z^{−1}}) = τ(C′_{x}C_{y}T̃_{w_{0}}C′_{w_{0}z^{−1}})
= τ(C′_{w_{0}z^{−1}}C′_{x}D′_{yw_{0}})
= h_{w_{0}z^{−1},x,w_{0}y^{−1}}
= h_{x^{−1},zw_{0},yw_{0}}.

The third statement can now be proved using (i) or (ii).

Recall that w_{0} is the element of longest length in W. Kazhdan and Lusztig [5, Cor 3.2] showed that x ≤_{L} y if and only if w_{0}y ≤_{L} w_{0}x if and only if yw_{0} ≤_{L} xw_{0} for all x, y ∈ W. Consequently, left and right multiplication by w_{0} permutes the left, right, and two–sided cells of W.

For convenience we let f_{x,y} = g_{w_{0},x,y}; that is to say

(2.2)  T̃_{w_{0}}C′_{x} = D_{w_{0}x} = ∑_{y∈W} f_{x,y}C′_{y}.

Given a ∈ 𝒜 we adopt the convention that deg(a) is the degree of a as a polynomial in q^{\frac{1}{2}} and we also define mindeg(a) = − deg(\overline{a}) (ie. the smallest power of q^{\frac{1}{2}} appearing in a).

We begin by generalizing a result of John Graham’s for Hecke algebras of type A. Graham has told me that his proof, although similar, does not fully extend to arbitrary W; it will appear in [3].

(2.3) Proposition Let x, y ∈ W. Then

𝔞(x) − 𝔞(w_{0}x) ≤ mindeg(f_{x,y}) ≤ deg(f_{x,y}) ≤ 𝔞(y) − 𝔞(w_{0}y).

Moreover, both inequalities are strict unless x ∼_{L} y and x ∼_{R} w_{0}yw_{0}.

Proof: Apply the bar involution to (2.2) and multiply on the left by T̃_{w_{0}} to obtain

C′_{x} = ∑_{y} \overline{f_{x,y}} T̃_{w_{0}}C′_{y} = ∑_{y} \overline{f_{x,y}} D_{w_{0}y}.

Therefore, \overline{f_{x,y}} = τ(C′_{x}C_{y^{−1}w_{0}}); equivalently, f_{x,y} = \overline{τ(C′_{x}C_{y^{−1}w_{0}})}. Since the bar involution and ε commute and τ is 𝒜 –linear, Lemma 2.1(iii) gives

(2.4)  f_{x,y} = ∑_{z≤_{R}x} ε(h_{w_{0}z,w_{0}y,w_{0}x})\overline{τ(C′_{z})} = ∑_{z≤_{R}x} ε(h_{w_{0}z,w_{0}y,w_{0}x})qz^{\frac{1}{2}} \overline{P_{1,z}}.

Now for each term upon the right hand side of this equation the smallest degree that can occur is

(2.5)  ℓ(z) − 2δ(z) − 𝔞(w_{0}x) ≥ 𝔞(z) − 𝔞(w_{0}x) ≥ 𝔞(x) − 𝔞(w_{0}x),

where the first inequality comes from (1.7) and the second from (1.10) since z ≤_{R} x. It follows that f_{x,y} cannot contain any terms with degree smaller than 𝔞(x) − 𝔞(w_{0}x). In addition, in order to get equality in (2.5) we need z = d ∈ 𝒟 and γ_{w_{0}d,w_{0}y,w_{0}x} ≠ 0. Therefore, dw_{0} ∼_{R} w_{0}y, w_{0}y ∼_{L} w_{0}x, and w_{0}d ∼_{R} w_{0}x by (1.12). Consequently, x ∼_{L} y and x ∼_{R} d ∼_{R} w_{0}yw_{0} proving the first half of the proposition.

For the other inequality, since f_{x,y} = \overline{τ(C′_{x}C_{y^{−1}w_{0}})}, by Lemma 2.1(ii) we have

(2.6)  f_{x,y} = ∑_{z≤_{L}y^{−1}w_{0}} h_{x^{−1},zw_{s},y^{−1}} \overline{τ(C_{z})} = ∑_{z≤_{L}y^{−1}w_{0}} ϵ_{z}h_{x^{−1},zw_{s},y^{−1}}q_{z}^{− \frac{1}{2}} P_{1,z}.

The largest possible degree of a term on the right hand side is now

(2.7)  𝔞(y^{−1}) − ℓ(z) + 2δ(z) ≤ 𝔞(y^{−1}) − 𝔞(z) ≤ 𝔞(y^{−1}) − 𝔞(y^{−1}w_{0}).

So the right hand inequality of the proposition follows since 𝔞(w) = 𝔞(w^{−1}) for all w ∈ W. As before, we see that in order to achieve this maximum degree we require that z = d ∈ 𝒟 and γ_{x^{−1},dw_{s},y^{−1}} ≠ 0; by (1.12) this forces x ∼_{L} y and x ∼_{R} w_{0}yw_{0}, and the proof is complete.

We shall apply this result in the case where x ∼_{L} y in the next section; however, before we investigate this situation we use the proposition to bound the degrees that can occur in the g–polynomials.

(2.8) Proposition Let x, y, z ∈ W. Then

−𝔞(w_{0}y) ≤ mindeg(g_{x,y,z}) ≤ deg(g_{x,y,z}) ≤ 𝔞(z).

Moreover, if mindeg(g_{x,y,z}) = −𝔞(w_{0}y) then y ∼_{L} z ∼_{R} w_{0}x and y ∼_{R} x^{−1}w_{0}; and, if deg(g_{x,y,z}) = 𝔞(z) then y ∼_{L} z ∼_{R} x and y ∼_{R} x^{−1}. In particular, the left and right inequalities above are strict when y ≁_{L} z.

Proof: Our proof results from a brute force calculation of the g–polynomials using Proposition 2.3 and the D and D′–bases of ℋ . By (1.2) and (1.3), T̃_{x} = qx^{\frac{1}{2}} ∑_{w≥x} q_{w}^{− \frac{1}{2}} P_{x,w}D′_{w}, so

(2.9)  \begin{aligned}T̃_{x}C′_{y} & = & qx^{\frac{1}{2}} ∑_{w≥x} q_{w}^{− \frac{1}{2}} P_{x,w}D′_{w}C′_{y} \\ & = & qx^{\frac{1}{2}} ∑_{w≥x} q_{w}^{− \frac{1}{2}} P_{x,w}C_{ww_{0}} T̃_{w_{0}}C′_{y} \\ & = & qx^{\frac{1}{2}} ∑_{w≥x} q_{w}^{− \frac{1}{2}} P_{x,w} ∑_{v≤_{L}y} f_{y,v}C_{ww_{0}}C′_{v} \\ & = & qx^{\frac{1}{2}} ∑_{z≤_{L}y} ( ∑_{z≤_{L}v≤_{L}y^{w≥x}} q_{w}^{− \frac{1}{2}} P_{x,w}f_{y,v}ε(h_{w_{0}w^{−1},zw_{0},vw_{0}}))C′_{z}, \end{aligned}

where the last line is obtained by applying ε to Lemma 2.1(ii). Now deg(P_{x,w}) ≤ ℓ(w) − ℓ(x) and, by Proposition 2.3, deg(f_{y,v}) ≤ 𝔞(v)−𝔞(w_{0}v). Therefore, the largest possible degree that a term in this sum can have is 𝔞(v) − 𝔞(w_{0}v) + 𝔞(vw_{0}). Now by (1.8) and (1.12) v ∼_{LR} v^{−1}, so 𝔞(vw_{0}) = 𝔞(v^{−1}w_{0}) = 𝔞(w_{0}v). Since z ≤_{L} v, deg(g_{x,y,z}) ≤ 𝔞(v) ≤ 𝔞(z) by (1.10), and we have our upper bound.

In order to attain this upper bound for a term in (2.9) we need x = w, y ∼_{L} v and y ∼_{R} w_{0}vw_{0} (Proposition 2.3), and γ_{w_{0}x^{−1},zw_{0},vw_{0}} ≠ 0. Therefore, by (1.12), x ∼_{R} z, z ∼_{L} v ∼_{L} y, and x^{−1} ∼_{R} w_{0}vw_{0} ∼_{R} y; so we have proven all our claims concerning the upper bound.

The proof of the lower bound is similar; however it does contain a small twist so we give some of the details. Again by (1.2) and (1.3), T̃_{x} = ϵ_{x}q_{x}^{− \frac{1}{2}} ∑_{w≥x} ϵ_{w}qw^{\frac{1}{2}} \overline{P_{x,w}} D_{w} so

T̃_{x}C′_{y} = ϵ_{x}q_{x}^{− \frac{1}{2}} ∑_{w≥x} ϵ_{w}qw^{\frac{1}{2}} \overline{P_{x,w}} D_{w}C′_{y}
= ϵ_{x}q_{x}^{− \frac{1}{2}} ∑_{w≥x} ϵ_{w}qw^{\frac{1}{2}} \overline{P_{x,w}} T̃_{w_{0}}C′_{w_{0}w}C′_{y}
= ϵ_{x}q_{x}^{− \frac{1}{2}} ∑_{z≤_{L}y} ( ∑_{z≤_{L}v≤_{L}y^{w≥x}} ϵ_{w}qw^{\frac{1}{2}} \overline{P_{x,w}} h_{w_{0}w,y,v}f_{v,z})C′_{z}.

An argument similar to that used above now completes the proof.

We now restrict our attention to characters. Recall that 𝔞(Γ) is the common value that the 𝔞–function takes upon the left cell Γ.

(2.10) Corollary (cf. Lusztig [11, 3.5(a)]) Let Γ be a left cell of W and let x ∈ W. Then

−𝔞(w_{0}Γ) ≤ mindeg ( tr(T̃_{x}, ℋ′_{Γ})) ≤ deg ( tr(T̃_{x}, ℋ′_{Γ})) ≤ 𝔞(Γ)

and

−𝔞(Γ) ≤ mindeg ( tr(T̃_{x}, ℋ_{Γ})) ≤ deg ( tr(T̃_{x}, ℋ_{Γ})) ≤ 𝔞(w_{0}Γ).

Moreover, the left hand inequalities are strict unless x ∼_{L} w_{0}x^{−1}w_{0} and the right hand inequalities are strict unless x ∼_{L} x^{−1}.

Let E be an irreducible ℋ ⊗_{𝒜} ℚ(q^{\frac{1}{2}} )–submodule of ℋ_{Γ} ⊗_{𝒜} ℚ(q^{\frac{1}{2}} ) and let a_{E} be the largest power of q dividing the generic degree of E. Then it follows from the corollary that a_{E} ≤ 𝔞(Γ) (see [7, Cor. 1.10]). Using the asymptotic Hecke J, Lusztig [11, Prop. 3.3] proved that a_{E} = 𝔞(Γ) for all such E. Corollary 2.10 also follows from this fact together with a little knowledge of generic degrees (see, for example, [7, Prop. 1.9]).

3 The action of T̃_{w_{0}} upon the cell representations

Our main application of Proposition 2.3 is the following result. In the case of Weyl groups of type A the result is due to John Graham [3]. In this case the proof is easier since the cell representations are irreducible.

(3.1) Theorem Let W be a finite Weyl group. Then there exists an involution σ: W ⟶W such that for any left cell Γ of W σ(Γ) = Γ and for all x ∈ Γ

T̃_{w_{0}}C′_{x} ≡ (−1)^{𝔞(w_{0}Γ)}q^{\frac{1}{2} (𝔞(Γ)−𝔞(w_{0}Γ))}C′_{σ(x)} ( mod ∑_{w<_{L}Γ} 𝒜 C′_{w}).

Moreover, x ∼_{R} w_{0}σ(x)w_{0} and σ(x) is the unique element of W such that γ_{σ(x),x^{−1},w_{0}d} ≠ 0, for some Duflo involution d ∈ 𝒟 which is uniquely determined by x.

Proof: First, we note that by Proposition 2.3

(3.2)  T̃_{w_{0}}C′_{x} ≡ (−1)^{𝔞(w_{0}Γ)}q^{\frac{1}{2} (𝔞(Γ)−𝔞(w_{0}Γ))} ∑_{y∈Γ} f′_{x,y}C′_{y} ( mod ∑_{w<_{L}Γ} 𝒜 C′_{w})

where f′_{x,y} ∈ ℤ is the coefficient of (−1)^{𝔞(w_{0}Γ)}q^{\frac{1}{2} (𝔞(Γ)−𝔞(w_{0}Γ))} in f_{x,y}. By (2.6) we see that

(−1)^{𝔞(w_{0}Γ)}q^{\frac{1}{2} (𝔞(Γ)−𝔞(w_{0}Γ))}f′_{x,y} = f_{x,y} = ∑_{z≤_{L}y^{−1}w_{0}} ϵ_{z}h_{x^{−1},zw_{0},y^{−1}}qz^{\frac{1}{2}} P_{1,z}.

By (2.7) the only way that a term on the right hand side can have degree 𝔞(Γ) − 𝔞(w_{0}Γ) is if z ∈ 𝒟 and 𝔞(z) = 𝔞(y^{−1}w_{0}). Since z ≤_{L} y^{−1}w_{0} this, together with (1.11), forces z = d_{y} ∼_{L} y^{−1}w_{0} — the Duflo involution which is in the same left cell as y^{−1}w_{0} — and all other terms in this sum must cancel. By (1.9) the coefficient of q^{δ(d_{y})} in P_{1,d_{y}} is equal to 1; so f′_{x,y} = (−1)^{𝔞(w_{0}Γ)}ϵ_{d_{y}}γ_{x^{−1},d_{y}w_{0},y^{−1}} = γ_{x^{−1},d_{y}w_{0},y^{−1}}; where the signs vanish because 𝔞(w_{0}Γ) = 𝔞(d_{y}) ≡ ℓ(d_{y}) (mod 2) by the definition of 𝒟. Therefore, if f′_{x,y} ≠ 0 then, using (1.12) twice, we see that

(3.3)  f′_{x,y} = γ_{x^{−1},d_{y}w_{0},y^{−1}} = γ_{w_{0}d_{y},x,y} = γ_{y^{−1},w_{0}d_{y},x^{−1}} = γ_{y^{−1},d_{x}w_{0},x^{−1}} = f′_{y,x},

where d_{x} = w_{0}d_{y}w_{0} is the Duflo involution which is in the same left cell as x^{−1}w_{0}. Consequently, f_{x,y} ≠ 0 if and only if f_{y,x} ≠ 0 and f′_{x,y} ∈ ℕ.

We can now show that there is a unique y for which f_{x,y} ≠ 0: multiply (3.2) on the right by C_{x^{−1}w_{0}} and apply τ to both sides. On the left hand side we get τ(T̃_{w_{0}}C′_{x}C_{x^{−1}w_{0}}) = τ(C′_{x}D′_{x^{−1}}) = 1. On the right hand side the product C′_{w}C_{x^{−1}w_{0}} ≠ 0 if and only if x ≤_{L} w by (1.4), so only the terms in ∑_{w<_{L}Γ} 𝒜 C_{w} are annihilated. Therefore, rearranging things a little we obtain

(3.4)  q^{\frac{1}{2} (𝔞(w_{0}Γ)−𝔞(Γ))} = (−1)^{𝔞(w_{0}Γ)} ∑_{y∈Γ} f′_{x,y}τ(C′_{y}C_{x^{−1}w_{0}}).

Now by Lemma 2.1(ii),

τ(C′_{y}C_{x^{−1}w_{0}}) = ∑_{z≤_{L}x^{−1}w_{0}} h_{y^{−1},zw_{0},x^{−1}}τ(C_{z}) = ∑_{z≤_{L}x^{−1}w_{0}} h_{y^{−1},zw_{0},x^{−1}}ϵ_{z}qz^{\frac{1}{2}} \overline{P_{1,z}}.

As in the proof of Proposition 2.3, the terms in the polynomials on the right hand side have degrees (in q^{\frac{1}{2}} ) greater than or equal to the required degree of 𝔞(w_{0}Γ) − 𝔞(Γ) in (3.4). This smallest degree can occur only if z ∈ 𝒟 and 𝔞(z) = 𝔞(x^{−1}w_{0}), forcing z = d_{x} by (1.11) (the Duflo involution in the same left cell as x^{−1}w_{0} as above). Consequently the only contribution from τ(C′_{y}C_{x^{−1}w_{0}}) to the sum in (3.4) is (−1)^{𝔞(d_{x})}γ_{y^{−1},d_{x}w_{0},x^{−1}}. Therefore, looking at the coefficient of q^{\frac{1}{2} (𝔞(w_{0}Γ)−𝔞(Γ))} in (3.4) and using (3.3), we see that

1 = ∑_{y∈Γ} f′_{x,y}γ_{y^{−1},d_{x}w_{0},x^{−1}} = ∑_{y∈Γ} (f′_{x,y})^{2}.

Since f′_{x,y} ∈ ℕ it follows that this sum has exactly one non–zero term and therefore that there is a unique y ∈ Γ for which f_{x,y} ≠ 0. Moreover, for this y we must have f′_{x,y} = f′_{y,x} = 1; and so f_{x,y} = (−1)^{𝔞(w_{0}Γ)}q^{\frac{1}{2} (𝔞(Γ)−𝔞(w_{0}Γ))} as claimed.

To complete the proof we need only show that the involution defined by σ(x) = y, where γ_{x^{−1},d_{y}w_{0},y^{−1}} ≠ 0, has the required properties. The uniqueness statement for y follows by (3.3) since γ_{x^{−1},d_{y}w_{0},y^{−1}} = γ_{y,x^{−1},w_{0}d_{y}} by (1.12); this also shows that d_{y} is the Duflo involution right equivalent to xw_{0}, and hence that d_{y} is completely determined by x. Finally, x ∼_{R} w_{0}σ(x)w_{0} by Proposition 2.3, completing the proof.

(3.5) Corollary Let Γ be a left cell of a finite Weyl group W. Then T̃_{w_{0}} acts upon the cell representations ℋ_{Γ} and ℋ′_{Γ} via

C_{x} + ∑_{w<_{L}Γ} 𝒜 C_{w} ⟼^{T̃_{w_{0}}} (−1)^{𝔞(Γ)}q^{\frac{1}{2} (𝔞(w_{0}Γ)−𝔞(Γ))}C_{σ(x)} + ∑_{w<_{L}Γ} 𝒜 C_{w}
C′_{x} + ∑_{w<_{L}Γ} 𝒜 C′_{w} ⟼^{T̃_{w_{0}}} (−1)^{𝔞(w_{0}Γ)}q^{\frac{1}{2} (𝔞(Γ)−𝔞(w_{0}Γ))}C′_{σ(x)} + ∑_{w<_{L}Γ} 𝒜 C′_{w}

respectively, for all x ∈ Γ. In particular, the central element (T̃_{w_{0}})^{2} acts as q^{𝔞(w_{0}Γ)−𝔞(Γ)} times the identity on ℋ_{Γ} and as q^{𝔞(Γ)−𝔞(w_{0}Γ)} times the identity on ℋ′_{Γ}.

Given an irreducible cell representation ℋ_{Γ} ⊗_{𝒜} ℚ(q^{\frac{1}{2}} ) this result is not unexpected since the central element (T̃_{w_{0}})^{2} must act as a scalar in this case. However, since the cell representations are not in general irreducible Theorem 3.1 is, I think, surprising.

(3.6) Suppose that W is a Weyl group of type A. Then Kazhdan and Lusztig showed that the left cells are completely determined by the Robinson–Schensted correspondence; namely, if x =^{RS} (P, Q), via Robinson–Schensted, and y =^{RS} (P′, Q′) then x ∼_{L} y if and only if Q = Q′ (see [5, section 5]). Consequently, the condition that x ∼_{R} w_{0}σ(x)w_{0} completely determines σ. Explicitly, if x =^{RS} (P, Q) and w_{0}xw_{0} =^{RS} (P′, Q′) then σ(x) =^{RS} (P′, Q). It is known that the tableau P′ can be obtained from P via the combinatorial operation of evacuation (see, for example, [17]). In particular we see that the trace of w_{0} upon an irreducible representation of W is, up to sign, the number of standard tableaux stable under evacuation.

In this guise the result is known being discovered independently by Graham [3] (who first told me of it) and Stembridge (in the specialization q = 1; see [17, Theorem 5.1]).

(3.7) Corollary Given a left cell Γ of W let Γ^{σ} = { x ∈ Γ : σ(x) = x }. Then (i) tr(T̃_{w_{0}}, ℋ_{Γ}) = (−1)^{𝔞(Γ)}q^{\frac{1}{2} (𝔞(w_{0}Γ)−𝔞(Γ))}|Γ^{σ}|. (ii) tr(T̃_{w_{0}}, ℋ′_{Γ}) = (−1)^{𝔞(w_{0}Γ)}q^{\frac{1}{2} (𝔞(Γ)−𝔞(w_{0}Γ))}|Γ^{σ}|.

The proof of the theorem also gives a property for the γ–coefficients analogous to (1.8):

(3.8) Corollary Let x ∈ W and d ∈ 𝒟. Then γ_{y,x^{−1},w_{0}d} ≠ 0 if and only if y = σ(x) and d ∼_{R} xw_{0}. In this case, γ_{y,x^{−1},w_{0}d} = 1.

We now investigate the involution σ and describe some of its properties. First we show how σ interacts with multiplication by w_{0}. For the proof of the next result it will be convenient to introduce another involution α: ℋ ⟶ ℋ which is the 𝒜 –linear map determined by α(T̃_{x}) = T̃_{x^{−1}} for all x ∈ W. It is easily seen that α is an anti–isomorphism and that α(C′_{x}) = C′_{x^{−1}} for all x ∈ W.

(3.9) Proposition Let σ: W ⟶W be the involution of Theorem 3.1 and let x ∈ W. Then: (i) σ(x^{−1}) = w_{0}σ(x)^{−1}w_{0}. (ii) σ(w_{0}x) = w_{0}σ(x). (iii) σ(xw_{0}) = σ(x)w_{0}.

Proof: Let Γ be the left cell containing x, and write y = σ(x). Considering all terms modulo ∑_{w<_{LR}Γ} 𝒜 C′_{w}, we have:

(−1)^{𝔞(w_{0}Γ)}q^{\frac{1}{2} (𝔞(Γ)−𝔞(w_{0}Γ))}C′_{x} ≡ T̃_{w_{0}}C′_{y} ≡ D_{w_{0}y} ≡ C′_{w_{0}yw_{0}} T̃_{w_{0}},

by Theorem 3.1. Applying α yields (i).

For (ii), recall equation (3.3) for f′_{x,y} from the proof of Theorem 3.1. This was obtained by an argument using (2.6) and (2.7); a similar calculation using (2.4) and (2.5) shows that f′_{x,y} = γ_{w_{0}d,w_{0}y,w_{0}x} = γ_{w_{0}y,x^{−1}w_{0},dw_{0}} ≠ 0, where x ∼_{R} d ∈ 𝒟. Hence, by Corollary 3.8 we get (ii).

The third statement is now a consequence of (i) and (ii):

σ(xw_{0}) = σ((w_{0}x^{−1})^{−1}) = w_{0}(σ(w_{0}x^{−1}))^{−1}w_{0} = w_{0}σ(x^{−1})^{−1} = σ(x)w_{0}.

We can now say a little bit more about Corollary 2.10 (note however that this is mostly a consequence of Proposition 2.8 and Theorem 3.1):

(3.10) Corollary Let Γ be a left cell of W. Then for all x ∈ W the coefficient of q^{\frac{1}{2} 𝔞(Γ)} in tr(T̃_{x}, ℋ′_{Γ}) is ∑_{y∈Γ} γ_{σ(yw_{0}),w_{0}y^{−1},w_{0}x^{−1}}; in particular, it is non–negative. Similarly, the coefficient of q^{− \frac{1}{2} 𝔞(w_{0}Γ)} in tr(T̃_{x}, ℋ′_{Γ}) is equal to ∑_{y∈Γ} γ_{σ(y),y^{−1},w_{0}x} ≥ 0. Moreover, if these coefficients are non–zero then x ∼_{L} x^{−1} and x ∼_{L} w_{0}x^{−1}w_{0} respectively.

Proof: The proof of Proposition 2.8 (and in particular equation (2.9)), combined with Theorem 3.1 shows that the coefficient of q^{\frac{1}{2} 𝔞(Γ)} in tr(T̃_{x}, ℋ′_{Γ}) is ∑_{y∈Γ} γ_{w_{0}x^{−1},yw_{0},σ(y)w_{0}}. By part (iii) of the proposition σ(y)w_{0} = σ(yw_{0}) so an application of (1.12) gives the first result. The proof of the second claim is similar.

This enables us to prove:

(3.11) Theorem Let Γ be a left cell of W. Then the following are equivalent: (i) ℋ_{Γ} ⊗_{𝒜} ℚ(q^{\frac{1}{2}} ) is an irreducible ℋ ⊗_{𝒜} ℚ(q^{\frac{1}{2}} )–module.

(ii) For all y ∈ Γ, γ_{σ(y),y^{−1},w_{0}x} ≠ 0 if and only if x ∈ 𝒟 and xw_{0} ∼_{R} y.

Before we can prove the theorem we need to recall some facts concerning generic degrees. The generic degree of an irreducible ℋ ⊗_{𝒜} ℚ(q^{\frac{1}{2}} )–module E is the polynomial (see [7, section 1]),

D_{E}(q) = \frac{W (q) dim E}{∑_{x∈W} tr(T̃_{x}, E)^{2}} ,  where  W (q) = ∑_{x∈W} q_{x}

(that D_{E}(q) is a polynomial in ℚ[q] is not obvious, but nevertheless true). Extend this definition to arbitrary ℋ ⊗_{𝒜} ℚ(q^{\frac{1}{2}} )–modules by linearity. Since the Poincaré polynomial W (q) is a product of cyclotomic polynomials it follows that D_{E}(q) is of the form λ_{E}q^{a_{E}}Φ_{E}(q) where λ_{E} ∈ ℚ, a_{E} ∈ ℕ, and Φ_{E}(q) is also a product of cyclotomic polynomials.

Let D_{Γ}(q) denote the generic degree of ℋ_{Γ}⊗_{𝒜} ℚ(q^{\frac{1}{2}} ) and write (Γ : E) for the multiplicity of the irreducible E in ℋ_{Γ} ⊗_{𝒜} ℚ(q^{\frac{1}{2}} ). Then

D_{Γ}(q) = ∑_{E}(Γ : E)D_{E}(q) = ∑_{E}(Γ : E)λ_{E}q^{a_{E}}Φ_{E}.

Lusztig [7, Cor. 1.10] has shown that if (Γ : E) > 0 then a_{E} = 𝔞(Γ); so the coefficient of the leading term of D_{Γ}(q) is ∑_{E}(Γ : E)λ_{E}. However, it is known that D_{Γ}(q) is equal to the “fake degree” of ℋ_{Γ} ⊗_{𝒜} ℚ(q^{\frac{1}{2}} ) (by results of Lusztig in [8, 12]); consequently, the leading coefficient of D_{Γ}(q) is 1 and therefore 1 = ∑_{E}(Γ : E)λ_{E}. Hence, λ_{E} ≤ 1 for all E and λ_{E} = 1 if and only if E ≅ ℋ_{Γ} ⊗_{𝒜} ℚ(q^{\frac{1}{2}} ) for some left cell Γ.

All of these facts are known and with them we may begin our proof.

Proof of Theorem 3.11: By the orthogonality relations for ℋ ⊗_{𝒜} ℚ(q^{\frac{1}{2}} ) and the definitions above

∑_{x∈W} tr(T̃_{x}, ℋ_{Γ} ⊗_{𝒜} ℚ(q^{\frac{1}{2}} ))^{2} = ∑_{E} (Γ : E)^{2} ∑_{x∈W} tr(T̃_{x}, E)^{2}
= ∑_{E} (Γ : E)^{2} \frac{W (q) dim E}{D_{E}(q)}
= ∑_{E} (Γ : E)^{2} \frac{dim E}{λ_{E}} \frac{W (q)}{q^{𝔞(Γ)}Φ_{E}(q)}.

Therefore the leading term of this polynomial has coefficient ∑_{E}(Γ : E)^{2} \frac{dim E}{λ_{E}} ≥ |Γ| with equality if and only if ℋ_{Γ} ⊗_{𝒜} ℚ(q^{\frac{1}{2}} ) is irreducible since otherwise λ_{E} < 1. Using Corollary 3.10 we can compute this coefficient in another way; namely it is

∑_{x∈W} ( ∑_{y∈Γ} γ_{σ(y),y^{−1},w_{0}x})^{2}.

Now for each y ∈ Γ there is a unique d ∈ 𝒟 such that γ_{σ(y),y^{−1},w_{0}d} = 1; so this sum is greater than or equal to |Γ| with equality if and only if for each y ∈ Γ there is a unique x ∈ W such that γ_{σ(y),y^{−1},w_{0}x} ≠ 0 (and then x ∈ 𝒟 by Corollary 3.8). The result follows.

(3.12) Proposition Let x ∈ W and let d_{x} ∈ 𝒟 be the unique Duflo involution such that d_{x} ∼_{R} xw_{0}. Then

(i) If w_{0}d_{x} ∈ 𝒟 then σ(x) = x. (ii) If σ(x) = x then w_{0}d_{x} is an involution. Moreover, if ℋ_{Γ} ⊗_{𝒜} ℚ(q^{\frac{1}{2}} ) is irreducible then σ(x) = x if and only if d_{x}w_{0} ∈ 𝒟.

Proof: Let σ(x) = y then γ_{y,x^{−1},w_{0}d_{x}} = 1 by Corollary 3.8. Hence, by (1.8), if w_{0}d_{x} ∈ 𝒟 then y = x. Conversely, if σ(x) = x then γ_{x,x^{−1},w_{0}d_{x}} = 1. Therefore, by (1.12), x ∼_{R} w_{0}d_{x} and x^{−1} ∼_{L} w_{0}d_{x}. Consequently w_{0}d_{x} ∼_{R} d_{x}w_{0} so that d_{x} ∼_{R} w_{0}d_{x}w_{0}. However, it easy to see that w_{0}d_{x}w_{0} ∈ 𝒟 so this forces d_{x} = w_{0}d_{x}w_{0} and it follows that w_{0}d_{x} is an involution.

Now suppose that ℋ_{Γ} ⊗_{𝒜} ℚ(q^{\frac{1}{2}} ) is irreducible and that σ(x) = x. Then by Theorem 3.11 there is a unique w ∈ W such that γ_{x,x^{−1},w_{0}w} ≠ 0. However, γ_{x,x^{−1},w_{0}d_{x}} = 1 and γ_{x,x^{−1},w_{0}(w_{0}d_{Γ})} = 1 by Corollary 3.8 and (1.8) respectively, so d_{x} = w_{0}d_{Γ} and we are done.

This result suggests that σ(x) = x if and only if w_{0}d_{x} ∈ 𝒟. This is certainly true in all examples I know of and I expect it to be true in general.

(3.13) Proposition Let Γ be a left cell of W and let d ∈ 𝒟 be the Duflo involution in w_{0}Γ. Then σ(d_{Γ}) = w_{0}d. Moreover, the following are equivalent:

(i) σ(d_{Γ}) = d_{Γ}. (ii) d_{Γ}w_{0} ∈ 𝒟. (iii) σ(x) = w_{0}xw_{0} for all x ∈ Γ.

Proof: Since d ∼_{L} w_{0}d_{Γ}, if σ(d_{Γ}) = y then γ_{y,d_{Γ},w_{0}d} = 1 by Corollary 3.8. So by (1.12), γ_{dw_{0},y,d_{Γ}} ≠ 0 and therefore y = w_{0}d by (1.8).

Now d_{Γ}w_{0} ∈ 𝒟 if and only if w_{0}d_{Γ} ∈ 𝒟 which is if and only if w_{0}d_{Γ} = d since w_{0}Γ contains a unique element of 𝒟. It follows that (i) and (ii) are equivalent. Further, if (iii) holds, then in particular w_{0}d = w_{0}d_{Γ}w_{0} so we see that (iii) implies (ii).

It remains to show that (ii) implies (iii): Let x ∈ Γ and suppose that y = σ(x). By Proposition 3.9, σ(x^{−1}w_{0}) = σ(x^{−1})w_{0} = w_{0}y^{−1} (note that x^{−1}w_{0} ∼_{L} w_{0}y^{−1} since x ∼_{R} w_{0}yw_{0} by Theorem 3.1). Therefore, γ_{w_{0}y^{−1},w_{0}x,w_{0}d} ≠ 0 for some d ∈ 𝒟 by Theorem 3.1. However by (1.12), w_{0}x ∼_{L} w_{0}d, so in fact d = d_{Γ}. Therefore, by (1.8) if w_{0}d_{Γ} ∈ 𝒟 then w_{0}x = yw_{0}, so y = w_{0}xw_{0} as claimed.

(3.14) Corollary Suppose that w_{0} is central in W and let Γ be a left cell of W such that d_{Γ}w_{0} ∈ 𝒟. Then σ fixes each element of Γ. Consequently,

(i) tr(T̃_{w_{0}}, ℋ_{Γ}) = (−1)^{𝔞(Γ)}q^{\frac{1}{2} (𝔞(w_{0}Γ)−𝔞(Γ))}|Γ|.

(ii) tr(T̃_{w_{0}}, ℋ′_{Γ}) = (−1)^{𝔞(w_{0}Γ)}q^{\frac{1}{2} (𝔞(Γ)−𝔞(w_{0}Γ))}|Γ|.

If w_{0} is not central in W then there exist left cells Γ for which d_{Γ}w_{0} ∈ 𝒟 and σ is not trivial when restricted to Γ (for example, this is true for the left cell containing the involution (2, 3) in S_{4}).

The proposition also allows us to relate the left cells of a Weyl group of type A to Lusztig’s [9] ϕ–cells in type B (see also Stembridge [17, Theorem 4.2]).

(3.15) Corollary Let W ≅ S_{n} be a Weyl group of type A and let W_{1} be the subgroup of W fixed by conjugation by w_{0} (so W_{1} is a Weyl group of type B). Let Γ be a left cell of W. Then tr(T̃_{w_{0}}, ℋ_{Γ}) ≠ 0 if and only if there exists a left cell Λ such that ℋ_{Γ} ≅ ℋ_{Λ} and w_{0}Λw_{0} = Λ (set wise). Moreover, if tr(T̃_{w_{0}}, ℋ_{Γ}) ≠ 0 then

tr(T̃_{w_{0}}, ℋ_{Γ}) = (−1)^{𝔞(Γ)}q^{\frac{1}{2} (𝔞(w_{0}Γ)−𝔞(Γ))}|Λ ∩ W_{1}|

and |Λ∩W_{1}| is the dimension of an irreducible ℚW_{1}–module (which is completely determined by the isomorphism type of ℋ_{Γ}).

Proof: Firstly note that tr(T̃_{w_{0}}, ℋ_{Γ}) ≠ 0 if and only if there exists x ∈ Γ such that σ(x) = x. Write x =^{RS} (P, Q) as in (3.6); then the tableau P is stable under evacuation so the involution d =^{RS} (P, P) is also fixed by σ. Let Λ be the left cell containing d; then ℋ_{Γ} ≅ ℋ_{Λ} and, by Proposition 3.13, w_{0}Λw_{0} = Λ and Λ^{σ} = { y ∈ Λ : σ(y) = y } = Λ ∩ W_{1}. So the trace formula above is a special case of Corollary 3.7. The remaining statements follow directly from Lusztig [9, Theorem 11].

(3.16) Remark Lusztig’s result [9, Theorem 11] states that the ϕ–cells of a certain Hecke algebra of W_{1} give all of the irreducible representations of W_{1} over a field of characteristic zero. The proof of Corollary 3.15 shows that there is a one–to–one correspondence between the fixed points of σ in Γ and the left cells Λ which are in the same two–sided cell as Γ and for which Λ ∩ W_{1} ≠ ∅ (since each non–empty intersection contains an involution by (3.6)). By [9] the left ϕ–cells of W_{1} are precisely the non–empty intersections Λ ∩ W_{1} as Λ ranges over the left cells of W.

Lastly, motivated by a question of Matthew Douglass, we show that σ “preserves” the graph structure of the W –graphs determined by the cell representations. Recall that if y ≤ x then μ(y, x) is the coefficient of q^{\frac{1}{2} (ℓ(x)−ℓ(y)−1)} in P_{y,x} and set μ(x, y) = μ(y, x) if y ≤ x and μ(x, y) = 0 otherwise. Also let ℒ_{x} = { s ∈ S : sx > s } be the left ascent set of x ∈ W.

(3.17) Proposition Let Γ be a left cell of W and let x and y ∈ Γ. Then ℒ_{σ(x)} = w_{0}ℒ_{x}w_{0} and if ℒ_{x} ≠ ℒ_{y} then μ(x, y) = μ(σ(x), σ(y)).

Proof: Throughout the proof for any y ∈ Γ we let C′_{y} denote the corresponding basis element of ℋ_{Γ} and let λ = (−1)^{𝔞(w_{0}Γ)}q^{\frac{1}{2} (𝔞(Γ)−𝔞(w_{0}Γ))}. Fix x ∈ Γ and r ∈ S and let s = w_{0}rw_{0} ∈ S. If sx < x (ie. s ∉ ℒ_{x}), then, recalling the multiplication formulae for the C′–basis from [5],

T̃_{r}C′_{σ(x)} = λ^{−1}T̃_{r}T̃_{w_{0}}C′_{x} = λ^{−1}T̃_{w_{0}} T̃_{s}C′_{x} = q^{\frac{1}{2}} λ^{−1}T̃_{w_{0}}C′_{x} = q^{\frac{1}{2}} C′_{σ(x)}.

It follows that sx < x if and only if rσ(x) < σ(x). Similarly, if sx > x then

T̃_{r}C′_{σ(x)} = −q^{− \frac{1}{2}} C_{σ(x)} + ∑_{s∉ℒ_{y}^{y∈Γ}} μ(x, y)C′_{σ(y)}.

Using the linear independence of the C′_{y}–basis of ℋ_{Γ} to compare coefficients shows that μ(x, y) = μ(σ(x), σ(y)).

Note that the restriction that ℒ_{x} ≠ ℒ_{y} is not important because if an edge joins x and y when ℒ_{x} = ℒ_{y} then this edge does not contribute to the representation carried by the W –graph.

4 The cell representations as submodules of ℋ

In this section we show how to realize any cell representation as a submodule of ℋ generalizing [14, Theorem 5.3].

(4.1) Theorem Let W be a finite Weyl group and let Γ be a left cell of W with Duflo involution d_{Γ}. Then there are canonical isomorphisms of ℋ –modules

ℋ′_{Γ} ≅ ∑_{x∈Γ} 𝒜 C′_{x}C_{d_{Γ}w_{0}} and ℋ_{Γ} ≅ ∑_{x∈Γ} 𝒜 C_{x}C′_{d_{Γ}w_{0}}.

Proof: Now C′_{w}C_{d_{Γ}w_{0}} ≠ 0 if and only if d_{Γ} ≤_{L} w by (1.5). Consequently ∑_{x∈Γ} 𝒜 C′_{x}C_{d_{Γ}w_{0}} is an ℋ –module (by the definition of the preorder ≤_{L}). This also shows that the 𝒜 –linear map ϕ: ℋ′_{Γ} ⟶∑_{x∈Γ} 𝒜 C′_{x}C_{d_{Γ}w_{0}} defined by

ϕ(C′_{x} + ∑_{w<_{L}Γ} 𝒜 C′_{w}) = C′_{x}C_{d_{Γ}w_{0}} for all x ∈ Γ

is both a well defined and surjective ℋ –homomorphism. We need to show that ϕ is injective; or, equivalently, that the elements C′_{x}C_{d_{Γ}w_{0}} are linearly independent as x runs through Γ.

We will show that given scalars λ_{x} ∈ 𝒜 (x ∈ Γ), not all zero, Λ = ∑_{x∈Γ} λ_{x}C′_{x}C_{d_{Γ}w_{0}} ≠ 0. Without loss of generality we may assume that λ_{x} ∈ ℤ[q^{\frac{1}{2}} ] for all x ∈ Γ and that there exists at least one x ∈ Γ such that λ_{x} has non–zero constant term. By Lemma 2.1(ii) we have

∑_{x∈Γ} λ_{x}C′_{x}C_{d_{Γ}w_{0}} = ∑_{x∈Γ} λ_{x} ∑_{z≤_{L}d_{Γ}w_{0}} h_{x^{−1},zw_{0},d_{Γ}}C_{z} = ∑_{z≤_{L}d_{Γ}w_{0}} ( ∑_{x∈Γ} λ_{x}h_{x^{−1},zw_{0},d_{Γ}})C_{z}.

Consequently if z ≤_{L} d_{Γ}w_{0} then the coefficient of C_{z} in Λ is ∑_{x∈Γ} λ_{x}h_{x^{−1},zw_{0},d_{Γ}}. Let w ∈ Γ, then ww_{0} ∼_{L} d_{Γ}w_{0}; so letting z = ww_{0} the coefficient of C_{ww_{0}} in Λ is ∑_{x∈Γ} λ_{x}h_{x^{−1},w,d_{Γ}}. Using (1.8) we see that the coefficient of q^{− \frac{1}{2} 𝔞(d_{Γ})} in this sum is

∑_{x∈Γ} λ_{x}(0)γ_{x^{−1},w,d_{Γ}} = λ_{w}(0),

where λ_{x}(0) is the polynomial λ_{x}(q^{\frac{1}{2}} ) evaluated at 0 (ie. its constant term). Since λ_{w}(0) ≠ 0 for some w ∈ Γ it follows that Λ ≠ 0 as claimed. This completes the proof in the case of ℋ′_{Γ}; applying ε we obtain the result for ℋ_{Γ}.

(4.2) Remark Essentially the same argument shows that if W is an affine Weyl group, or more generally any Coxeter group for which the positivity results (1.1) and (1.6) are known to hold, and Γ is a left cell of W then ℋ_{Γ} ≅ ∑_{x∈Γ} C_{x}D_{d_{Γ}}. However it is necessary to work in the ℋ –bimodule ℋ̂ of formal (ie. possibly infinite) 𝒜 –linear combinations of the form ∑_{x∈W} a_{x}T̃_{x} (a_{x} ∈ 𝒜).

Lusztig [10, Lemma 5.4] has shown that if Γ is a left cell of a finite Weyl group then the ℋ –module ∑_{x,y∈Γ} 𝒜 C_{x}D_{y^{−1}} is a two–sided ideal of ℋ which contains a submodule isomorphic to ℋ_{Γ}. Since D_{y^{−1}} = C′_{y^{−1}w_{0}} T̃_{w_{0}} our result is a refinement of Lusztig’s.

Given an arbitrary element w ∈ Γ it does not seem unreasonable to hope that the two ℋ –modules ℋ_{Γ} and ℋ_{Γ,w} = ∑_{x∈Γ} 𝒜 C_{x}C′_{w^{−1}w_{0}} would be isomorphic. Indeed, if the cell representation ℋ_{Γ} ⊗_{𝒜} ℚ(q^{\frac{1}{2}} ) is irreducible this is follows by Schur’s lemma since ℋ_{Γ,w} is a non–zero homomorphic image of ℋ_{Γ} by the argument above. In general it appears that ℋ_{Γ} ≅ ℋ_{Γ,w} for all w ∈ Γ if and only if ℋ_{Γ} ⊗ ℚ(q^{\frac{1}{2}} ) is irreducible. It is not hard to see that ℋ_{Γ} ⊗ ℚ(q^{\frac{1}{2}} ) is irreducible if and only if I_{Γ} is free of rank |Γ|^{2}; however, I have been unable to exploit this.

Acknowledgments

I would like to thank John Graham for numerous helpful conversations which have added to my understanding of the cells and cell representations. I am also grateful for the computer algebra package Gap [15], and in particular Meinolf Geck’s code for working with Hecke algebras. This has been a welcome tool in the calculation of many examples. Finally, this work was completed during a year at the University of Notre Dame and I would like to thank the mathematics department for its hospitality. In particular, I am grateful to Matthew Dyer for several conversations.

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[15] M. Schönert et al., eds., Gap: groups, algorithms, and programming, Manual, release 3.2, Lehrstuhl D für Mathematik, RWTH Aachen, 1993.

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Department of Mathematics, Imperial College, Queen’s Gate, London SW7 2BZ

                                e-mail:  a.mathas@ic.ac.uk