KAZHDAN-LUSZTIG POLYNOMIALS AND A COMBINATORIC FOR TILTING MODULES

WOLFGANG SOERGEL

Abstract. This article gives a self-contained treatment of the theory of Kazhdan-Lusztig polynomials with special emphasis on affine reflection groups. There are only few new results but many new proofs. We close with a conjectural character formula for tilting modules, which formed the starting point of these investigations.

1. Introduction

While trying to write down conjectural character formulas for tilting modules, I dived into the literature on Kazhdan-Lusztig polynomials, notably the works of Kazhdan-Lusztig [KL79], Lusztig [Lus80a], Andersen [And86], Kato [Kat85], Kaneda [Kan87] and Deodhar [Deo87, Deo91]. It seemed reasonable to me, to make a synopsis of all these sources, to make them more easily accessible. That is done in the first sections of this manuscript. The only new result there is Theorem 5.1. However, many proofs and also the presentation as a whole (which fully develops the point of view adopted in [Mil] and [Lus91]) are new. In particular the so-called R-polynomials don’t appear at all in my presentation of the theory. In the last section I finally reach my goal and give conjectural character formulas for tilting modules. After that follows a graphically presented sample computation and an index of notation. For a presentation of the basics of this article including the results of the following section one might consult [Hum90]. For the third section compare also [Deo94].

I thank Henning Haahr Andersen, who showed me his notes with related ideas, and Corinne Blondel, Michèle Couillens, Caroline Gruson, Jens Carsten Jantzen, Friedrich Knop and George Lusztig for their helpful remarks on preliminary versions.

2. The ordinary Kazhdan-Lusztig polynomials

Let (W, S) be a Coxeter system, l : W → ℕ the corresponding length function and ≤ the Bruhat order on W. In particular x < y means x ≤ y, x ≠ y. Let L = ℤ[v, v^{−1}] be the ring of Laurent polynomials with integer coefficients in one variable v. On the free L-module with basis indexed by W,

1991 Mathematics Subject Classification. 05E99, 17B37.

H = H(W, S) = ⨁_{x∈W} LT_{x}

there is exactly one structure of associative L-algebra such that T_{x}T_{y} = T_{xy} if l(x) + l(y) = l(xy) and T_{s}^{2} = v^{−2}T_{e} + (v^{−2} − 1)T_{s} for all s ∈ S, see [Bou81], IV, §2, Exercice 23. This associative algebra H is called the Hecke algebra corresponding to (W, S).

It can also be given as the associative algebra over L with generators {H_{s}}_{s∈S} (for H_{s} = vT_{s}), the quadratic relations H_{s}^{2} = 1 + (v^{−1} − v)H_{s} and the so-called braid relations H_{s}H_{t} … H_{s} = H_{t}H_{s} … H_{t} resp. H_{s}H_{t}H_{s} … H_{t} = H_{t}H_{s}H_{t} … H_{s} if st … s = ts … t resp. sts … t = tst … s for s, t ∈ S. All H_{s} are invertible, more precisely, one checks that H_{s}^{−1} = H_{s} + (v − v^{−1}).

From now on we work with H_{x} = v^{l(x)}T_{x}. Certainly we have H_{x}H_{y} = H_{xy} if l(x) + l(y) = l(xy). Hence with the H_{s} all H_{x} are units in H. There is exactly one ring homomorphism d : H → H, H ↦ \overline{H} such that \overline{v} = v^{−1} and \overline{H}_{x} = (H_{x^{−1}})^{−1}. Certainly d is an involution. We call H ∈ H self-dual if \overline{H} = H.

Theorem 2.1 ([KL79]). For all x ∈ W there exists a unique self-dual element \underline{H}_{x} ∈ H such that \underline{H}_{x} ∈ H_{x} + ∑_{y} vℤ[v]H_{y}.

Remark 2.2. In [KL79] this \underline{H}_{x} is called C′_{x}. Furthermore Kazhdan and Lusztig use the variable q = v^{−2} and the L-basis consisting of the T_{x}.

Proof. As we know already we have \overline{H_{s}} = H_{s}^{−1} = H_{s} + (v − v^{−1}) for all s ∈ S. In particular C_{s} = H_{s} + v is self-dual, \overline{C_{s}} = C_{s}. (The expert reader should be cautioned, that our C_{s} is called C′_{s} in [KL79], and in this source C_{s} means another element of the Hecke algebra. Once the Theorem is established, we could as well write C_{s} = \underline{H}_{s}.)

The multiplication from the right of C_{s} on H is given by the formulas

H_{x}C_{s} = \begin{cases}H_{xs} + vH_{x} & if xs > x; \\ H_{xs} + v^{−1}H_{x} & if xs < x. \end{cases}

We now start proving the existence. To this end we show by induction on the Bruhat order the stronger

Claim 2.3. For all x ∈ W there exists a self-dual \underline{H}_{x} ∈ H such that \underline{H}_{x} ∈ H_{x} + ∑_{y<x} vℤ[v]H_{y}.

Certainly we can start our induction with \underline{H}_{e} = H_{e} = 1. Now let x ∈ W be given and suppose we know the existence of \underline{H}_{y} for all y < x. If x ≠ e we find s ∈ S such that xs < x and by our induction hypothesis we have

\underline{H}_{xs}C_{s} = H_{x} + ∑_{y<x} h_{y}H_{y}

for suitable h_{y} ∈ ℤ[v]. We form

\underline{H}_{x} = \underline{H}_{xs}C_{s} − ∑_{y<x} h_{y}(0)\underline{H}_{y},

and our induction works. Hence there exists \underline{H}_{x} as in the Claim. The unicity of the \underline{H}_{x} follows immediately from

Claim 2.4. For H ∈ ∑_{y} vℤ[v]H_{y} self-duality H = \overline{H} implies H = 0.

Certainly H_{z} ∈ \underline{H}_{z} + ∑_{y<z} L\underline{H}_{y} for \underline{H}_{x} as in the preceding Claim, whence \overline{H_{z}} ∈ H_{z} + ∑_{y<z} LH_{y} for all z ∈ W. If we write H = ∑ h_{y}H_{y} and choose z maximal such that h_{z} ≠ 0,then H = \overline{H} implies h_{z} = \overline{h}_{z} contradicting h_{z} ∈ vℤ[v]. This proves the Claim and the Theorem.

Definition 2.5. For x, y ∈ W we define h_{y,x} ∈ L by the equality

\underline{H}_{x} = ∑_{y} h_{y,x}H_{y}.

Remark 2.6. The h_{y,x} are given in terms of the polynomials P_{y,x} from [KL79] as h_{y,x} = v^{l(x)−l(y)}P_{y,x}. These equations are to be understood in L = ℤ[v, v^{−1}] ⊃ ℤ[q], with q = v^{−2} as before. By induction one may check directly, that v^{l(y)−l(x)}h_{y,x} is even a polynomial in q with constant term 1.

The original definition of the Kazhdan-Lusztig polynomials in [KL79] was along the lines of another characterization we give in the sequel. Let us look once more at Theorem 2.1. Up to the signs there is no reason to prefer v over v^{−1} there.

Theorem 2.7 ([KL79]). For all x ∈ W there exists a unique self-dual \underline{H̃}_{x} ∈ H such that ˜ \underline{H}_{x} ∈ H_{x} + ∑_{y} v^{−1}ℤ[v^{−1}]H_{y}.

Proof. Let us look at the two involutive anti-automorphisms a and i of H given as

\begin{matrix}a(v) = v, & a(H_{x}) = (−1)^{l(x)}H_{x}^{−1} resp. \\ i(v) = v, & i(H_{x}) = H_{x^{−1}}. \end{matrix}

They commute and both of them commute with our involution d : H ↦ \overline{H}. Their composite dia : H → H satisfies dia(H_{y}) = (−1)^{l(y)}H_{y} and dia(v) = v^{−1}.

So we are allowed and forced to take \underline{H̃}_{x} = (−1)^{l(x)}dia(\underline{H}_{x}) and get in addition to the existence of \underline{H̃}_{x} the formula

\underline{H̃}_{x} = ∑_{y}(−1)^{l(x)+l(y)}\overline{h}_{y,x}H_{y}.

Remark 2.8. Instead of dia we could as well use the automorphism b : H → H given by b(H_{x}) = H_{x}, b(v) = −v^{−1}. It commutes with d, and we are allowed and forced to take \underline{H̃}_{x} = b(\underline{H}_{x}). By the way b commutes with i and a, and these four pairwise commuting involutions d, b, i and a define a faithful action of (ℤ/2ℤ)^{4} on H.

Let us finish this section with an explicit formula for finite W.

Proposition 2.9 ([KL79]). Let W be finite, w ∈ W the longest element, and r = l(w) its length. Then we have \underline{H}_{w} = ∑_{y∈W} v^{r−l(y)}H_{y}.

Proof. Let R denote the right hand side. Our formulas for the action of C_{s} show that

{H ∈ H | HC_{s} = (v + v^{−1})H ∀s ∈ S} = LR.

Hence we have \overline{R} ∈ LR and then immediately \overline{R} = R, thus R = \underline{H}_{w}.

The expert reader will miss here the inversion formulas [KL79] for finite Coxeter groups. We treat them in the next section in greater generality (see 3.10).

3. The parabolic case

Let S_{f} ⊂ S be a subset, W_{f} = ⟨S_{f}⟩ ⊂ W the subgroup it generates, W^{f} ⊂ W the set of minimal length representatives for the right cosets W_{f}\W. So multiplication gives a bijection W_{f} × W^{f} →^{∼} W. Let H_{f} = H(W_{f}, S_{f}) ⊂ H be the Hecke algebra of (W_{f}, S_{f}). One sees that the quadratic relation in the Hecke algebra can also be written (H_{s} + v)(H_{s} − v^{−1}) = 0. If we fix u ∈ {−v, v^{−1}}, the prescription H_{s} ↦ u ∀s ∈ S_{f} defines a surjection of L-algebras

φ_{u} : H_{f} ↠ L.

In this way L becomes an H_{f}-bimodule, which we denote L(u). We induce to obtain two right H-modules

\begin{aligned}M & = & M^{f} & = & L(v^{−1}) ⊗_{H_{f}} H, \\ N & = & N^{f} & = & L(−v) ⊗_{H_{f}} H. \end{aligned}

In both these modules the M_{x} = 1⊗H_{x} resp. N_{x} = 1⊗H_{x} with x ∈ W^{f} form an L-basis. The action of C_{s} for s ∈ S is given in these bases as:

M_{x}C_{s} = \{ \begin{matrix}M_{xs} + vM_{x} & if xs ∈ W^{f}, xs > x; \\ M_{xs} + v^{−1}M_{x} & if xs ∈ W^{f}, xs < x; \\ (v + v^{−1})M_{x} & if xs ∉ W^{f}, \end{matrix}

N_{x}C_{s} = \{ \begin{matrix}N_{xs} + vN_{x} & if xs ∈ W^{f}, xs > x; \\ N_{xs} + v^{−1}N_{x} & if xs ∈ W^{f}, xs < x; \\ 0 & if xs ∉ W^{f}. \end{matrix}

To see this one has to use the fact that x ∈ W^{f}, xs ∉ W^{f} implies xs = rx for some r ∈ S_{f}. (In particular xs < x implies xs ∈ W^{f}.) Indeed for arbitrary x ∈ W and r, s ∈ S the relations rx > x and rxs < xs together imply rxs = x.

For all s ∈ S_{f} one easily checks

φ_{u}(C_{s}) = \{ \begin{matrix}(v + v^{−1}) & if u = v^{−1}; \\ 0 & if u = −v. \end{matrix}

Since the C_{s} for s ∈ S_{f} generate H_{f} as an L-algebra, we have φ_{u}(\overline{H}) = \overline{φ_{u}(H)} ∀H ∈ H_{f}. Hence the prescription a ⊗ H ↦ \overline{a} ⊗ \overline{H} defines a homomorphism of additive groups

M → M, M ↦ \overline{M}

such that \overline{M_{e}} = M_{e} and \overline{MH} = \overline{M} \overline{H} for all M ∈ M, H ∈ H. Analogous results hold for N.

We call an additive map F between two right L- resp. H-modules “L-skew-linear” resp. “H-skew-linear” iff F (MH) = F (M)\overline{H} for all M and all H ∈ L resp. H ∈ H. If a module is given a fixed skew-linear involution, we call the elements stable under this involution “self-dual” For example N ∈ N is self-dual iff \overline{N} = N.

Theorem 3.1 ([Deo87]). 1. For all x ∈ W^{f} there exists a unique

self-dual \underline{M}_{x} ∈ M such that \underline{M}_{x} ∈ M_{x} + ∑_{y} vℤ[v]M_{y}.

2. For all x ∈ W^{f} there exists a unique self-dual \underline{N}_{x} ∈ N such that \underline{N}_{x} ∈ N_{x} + ∑_{y} vℤ[v]N_{y}.

Proof. We show (1), the proof of (2) is identical. To show the existence of \underline{M}_{x} we proceed by induction the length of x and show more precisely, that we can find \underline{M}_{x} of the form

\underline{M}_{x} = M_{x} + ∑_{y<x} m_{y,x}M_{y}.

Certainly we can start our induction with \underline{M}_{e} = M_{e}. Now suppose \underline{M}_{y} is already constructed for all y ∈ W^{f}, y < x, and let s ∈ S be given such that xs < x, xs ∈ W^{f}. Then we have

\underline{M}_{xs}C_{s} = M_{x} + ∑_{z<x} m_{z}M_{z}

for suitable m_{z} ∈ ℤ[v]. By induction possible \underline{M}_{z} for z < x are known already. We form

\underline{M}_{x} = \underline{M}_{xs}C_{s} − ∑_{z} m_{z}(0)\underline{M}_{z}

and find this way a possible \underline{M}_{x}. The existence of these \underline{M}_{x} implies as in the proof of Theorem 2.1 first \overline{M_{x}} ∈ M_{x} + ∑_{y<x} LM_{y} and then the unicity of the \underline{M}_{x}.

Remark 3.2. 1. Let us define m_{y,x} ∈ ℤ[v] by

\underline{M}_{x} = ∑_{y} m_{y,x}M_{y}.

In particular we have m_{x,x} = 1, and m_{y,x} ≠ 0 ⇒ y ≤ x. Again we consider the variable q = v^{−2} ∈ L. By induction we deduce easily that even v^{l(y)−l(x)}m_{y,x} ∈ ℤ[q]. The same holds for the n_{y,x} defined by

\underline{N}_{x} = ∑_{y} n_{y,x}N_{y}.

The v^{l(y)−l(x)}m_{y,x} resp. v^{l(y)−l(x)}n_{y,x} are Deodhar’s [Deo87] parabolic polynomials P_{τ,σ}^{J} for Deodhar’s cases u = −1 resp. u = q, if τ = y^{−1}W_{f}, σ = x^{−1}W_{f}, and W_{J} = W_{f}. The comparison with Deodhar’s definition however will suceed only with the help of Theorem 3.5.

2. Possible interpretations of these parabolic polynomials in a representation theoretic context are summarized in Theorem 3.11.4 of [BGS96]. Up to a transformation v = t and with W_{Q} = W_{f} the polynomials (P^{Q}(t))_{x,y} are precisely the m_{x,y} here, and the (P_{Q}(t))_{x,y} coincide up to a change of parameters with our n_{x,y}, compare 3.10.

3. The proof gives an inductive description of the \underline{M}_{x}. By induction on the length of x we deduce, that for all y ≤ x the leading term of m_{y,x} is v^{l(x)−l(y)}. This statement has no analogue for the \underline{N}_{x}, since N_{y}C_{s} = 0 for certain y and s.

4. To simplify the task of calculating the n_{y,x} one may use the wellknown formula \underline{N}_{x}C_{s} = (v + v^{−1})\underline{N}_{x} for all x ∈ W^{f}, s ∈ S such that xs < x. This is proved by induction on x, where one has to use, that C_{s}^{2} = (v + v^{−1})C_{s} and n_{z}(0) ≠ 0 ⇒ zs < z in the preceeding proof. In particular we have n_{ys,x} = vn_{y,x} if y, x ∈ W^{f}, s ∈ S are such that ys < y, xs < x.

In the same way one proves that \underline{M}_{x}C_{s} = (v + v^{−1})\underline{M}_{x} for all x ∈ W^{f}, s ∈ S such that xs < x or xs ∉ W^{f} and deduces m_{ys,x} = vm_{y,x} for all y, x ∈ W^{f}, s ∈ S such that ys < y, xs < x.

5. For S_{f} = ∅ we certainly have M = N = H, \underline{M}_{x} = \underline{N}_{x} = \underline{H}_{x}, m_{y,x} = n_{y,x} = h_{y,x}.

For an abelian group E with involution d let E^{+} ⊂ E be the subgroup of self-dual elements E^{+} = {e ∈ E | de = e}.

Proposition 3.3. 1. H^{+} ⊂ H is the subalgebra generated over L^{+} =

ℤ[(v + v^{−1})] by the C_{s} with s ∈ S.

2. M^{+} = M_{e}H^{+} and the \underline{M}_{x} form an L^{+}-basis of M^{+}.

3. N^{+} = N_{e}H^{+} and the \underline{N}_{x} form an L^{+}-basis of N^{+}.

Proof. For this proof only let H^{+} ⊂ H be the subalgebra generated over L^{+} = ℤ[(v + v^{−1})] by the C_{s}, s ∈ S. If we show (2) or (3) for this H^{+}, then (1) follows.

We show (2), the proof of (3) being identical. First remark that by the inductive construction of the \underline{M}_{x} all \underline{M}_{x} lie in M_{e}H^{+}. On the other hand the \underline{M}_{x} form an L-basis of M, and M = ∑ m_{x}\underline{M}_{x} is self-dual iff all m_{x} are.

The m_{y,x}, n_{y,x} are related to the ordinary Kazhdan-Lusztig polynomials as follows.

Proposition 3.4 ([Deo87]). Let x, y ∈ W^{f}.

1. If W_{f} is finite and w_{f} ∈ W_{f} is its longest element, we have m_{y,x} = h_{w_{f} y,w_{f} x}.

2. For S_{f} arbitrary we have n_{y,x} = ∑_{z∈W_{f}} (−v)^{l(z)}h_{zy,x}.

Proof. (1) Consider the L-linear embedding

\begin{matrix}L(v^{−1}) & → & H_{f} \\ 1 & ↦ & \underline{H}_{w_{f}} . \end{matrix}

It commutes with the dualities, and by the proof of Proposition 2.9 it is even compatible with the right H_{f} action. Therefore we get by induction an embedding

ζ : M ↪ H

of right H-modules, which is compatible with the dualities as well. We put r = l(w_{f}). By Proposition 2.9 we have

ζ(M_{x}) = ∑_{z∈W_{f}} v^{r−l(z)}H_{zx}

and thus we get ζ(\underline{M}_{x}) = \underline{H}_{w_{f} x}. This even implies m_{y,x} = v^{r−l(z)}h_{zy,w_{f} x} for all y, x ∈ W^{f}, z ∈ W_{f}.

(2) Consider the obvious surjection

ξ : H ↠ N = L(−v) ⊗_{H_{f}} H

with ξ(H) = 1⊗H. It commutes with the dualities, and one may check, that ξ(H_{zx}) = (−v)^{l(z)}N_{x} for all z ∈ W_{f}, x ∈ W^{f}. Thus we get

ξ(\underline{H}_{x}) = \begin{cases}\underline{N}_{x} & if x ∈ W^{f}; \\ 0 & otherwise, \end{cases}

and the Proposition follows.

In the definition of \underline{N}_{x}, \underline{M}_{x} we may ask, whether v couldn’t be replaced by v^{−1}. The answer is given by the following

Theorem 3.5 ([Deo91]). 1. For all x ∈ W^{f} there is a unique selfdual \underline{Ñ}_{x} ∈ N such that \underline{Ñ}_{x} ∈ N_{x} +∑_{y} v^{−1}ℤ[v^{−1}]N_{y}. This \underline{Ñ}_{x} can be given by the formula \underline{Ñ}_{x} = ∑_{y}(−1)^{l(x)+l(y)}\overline{m}_{y,x}N_{y}.

2. For all x ∈ W^{f} there is a unique self-dual \underline{M̃}_{x} ∈ M such that \underline{M̃}_{x} ∈ M_{x} + ∑_{y} v^{−1}ℤ[v^{−1}]M_{y}. This \underline{M̃}_{x} can be given by the formula \underline{M̃}_{x} = ∑_{y}(−1)^{l(x)+l(y)}\overline{n}_{y,x}M_{y}.

Proof. We start with the relation φ_{−v} = φ_{v^{−1}} ∘ ia, in other words the following diagram commutes:

\begin{matrix}H_{f} & ⟶ & L(−v) \\ ia & ↓ & ∥ \\ H_{f} & ⟶ & L(v^{−1}). \end{matrix}

We can thus define an L-skew-linear bijection ϕ : N → M by the formula ϕ(c ⊗ H) = \overline{c} ⊗ dia(H), and clearly ϕ(\overline{N}) = \overline{ϕ(N)} ∀N ∈ N. Certainly we have ϕ(N_{x}) = (−1)^{l(x)}M_{x}. Thus we are allowed and forced to put \underline{M̃}_{x} = (−1)^{l(x)}ϕ(\underline{N}_{x}) and \underline{Ñ}_{x} = (−1)^{l(x)}ϕ^{−1}(\underline{M}_{x}).

Next we discuss inversion formulas. For this we consider the L-modules

\begin{aligned}M^{∗} & = & Hom_{L}(M, L) \\ N^{∗} & = & Hom_{L}(N , L) \end{aligned}

and define on them an L-skew-linear involution F ↦ \overline{F} by the formula \overline{F} (M) = \overline{F (\overline{M})}. Furthermore we define M_{x}^{∗} ∈ M^{∗} by M_{x}^{∗}(M_{y}) = δ_{x,y} and put M^{x} = (−1)^{l(x)}M_{x}^{∗}. Analogously we define N_{x}^{∗} ∈ N^{∗} by N_{x}^{∗}(N_{y}) = δ_{x,y} and put N^{x} = (−1)^{l(x)}N_{x}^{∗}. Why I prefer to work with the M^{x} resp. N^{x} will become clear later. Right now it rather complicates all formulas.

We write the elements of M^{∗} as formal linear combinations F = ∑^{∞} m^{z}M^{z} with m^{z} = (−1)^{l(z)}F (M_{z}) ∈ L. The ∞ sign above the sum should remind us that formal infinite sums are allowed. The elements of N^{∗} are written in the same way. Now we have \overline{M^{x}} ∈ M^{x} + ∑_{z>x}^{∞} LM^{z} and similarly for \overline{N^{x}}, since the matrices of the dualities on M and M^{∗} (resp. N and N^{∗}) are transposed up to signs.

Theorem 3.6. 1. For all x ∈ W^{f} there exists a unique self-dual

\underline{M}^{x} ∈ M^{∗} such that \underline{M}^{x} ∈ M^{x} + ∑^{∞} vℤ[v]M^{z}.

2. For all x ∈ W^{f} there exists a unique self-dual \underline{N}^{x} ∈ N^{∗} such that \underline{N}^{x} ∈ N^{x} + ∑^{∞} vℤ[v]N^{z}.

Proof. We show (1), the proof of (2) is identical. For the unicity we have to show that F = 0 is the only self-dual element of ∑^{∞} vℤ[v]M^{z}. But let F = ∑^{∞} m^{z}M^{z}. If F ≠ 0, we find y minimal such that m^{y} ≠ 0. Then \overline{m^{y}} = m^{y} and this contradicts m^{y} ∈ vℤ[v].

To prove existence, we just define \underline{M}^{x} ∈ M^{∗} by the formula

\underline{M}^{x}(\underline{M}_{y}) = (−1)^{l(x)}δ_{x,y}

and only have to check our properties. Certainly this \underline{M}^{x} is self-dual. If we put \underline{M}^{x} = ∑_{z}^{∞} m^{z,x}M^{z}, then clearly

∑_{z}(−1)^{l(z)+l(x)}m^{z,x}m_{z,y} = δ_{x,y}.

However the matrix m_{z,y} is lower triangular with ones on the diagonal and entries from vℤ[v] outside the diagonal, whence the same holds for its inverse and we get m^{z,x} ∈ vℤ[v] if z ≠ x and m^{x,x} = 1 (and even m^{z,x} ≠ 0 ⇒ z ≥ x).

In the same way we introduce the n^{z,x} ∈ ℤ[v] by \underline{N}^{x} = ∑^{∞} n^{z,x}N^{z} and get the inversion formulas

∑_{z}(−1)^{l(z)+l(x)}n^{z,x}n_{z,y} = δ_{x,y}.

In case S_{f} = ∅ we write H^{∗}, H_{x}^{∗}, H^{x}, \underline{H}^{x}, h^{z,x} instead of M^{∗}, M_{x}^{∗}, M^{x}, \underline{M}^{x}, m^{z,x}. Thus the h^{z,x} with \underline{H}^{x} = ∑^{∞} h^{z,x}H^{z} are the renormalized inverse Kazhdan-Lusztig polynomials,

∑_{z}(−1)^{l(z)+l(x)}h^{z,x}h_{z,y} = δ_{x,y}.

As in Proposition 3.4 the parabolic inverse polynomials m^{x,y}, n^{x,y} can be expressed in terms of the ordinary inverse polynomials h^{x,y}. More precisely, we have Proposition 3.7. 1. If W_{f} is finite, w_{f} ∈ W_{f} its longest element

and r = l(w_{f}) its length, then for all x, y ∈ W^{f} we have

m^{y,x} = ∑_{z∈W_{f}}(−v)^{r−l(z)}h^{zy,w_{f} x}.

2. For arbitrary S_{f} we have n^{y,x} = h^{y,x} for all x, y ∈ W^{f}.

Proof. (1) We transpose the map ζ considered in the proof of 3.4 (1) and get

ζ^{∗} : H^{∗} → M^{∗}.

The formula for ζ(M_{x}) implies ζ^{∗}(H^{zx}) = (−v)^{r−l(z)}(−1)^{r}M^{x} for all x ∈ W^{f}, z ∈ W_{f}. Since also ζ(\underline{M}_{x}) = \underline{H}_{w_{f} x} we get

ζ^{∗}(\underline{H}^{tx}) = \begin{cases}(−1)^{r}\underline{M}^{x} & if t = w_{f}; \\ 0 & otherwise, \end{cases}

again for all x ∈ W^{f}, t ∈ W_{f}. If we apply ζ^{∗} to the equation \underline{H}^{w_{f} x} = ∑^{∞} h^{zy,w_{f} x}H^{zy} where the sum runs over z ∈ W_{f}, y ∈ W^{f}, we get

\underline{M}^{x} = ∑_{y}^{∞} ∑_{z∈W_{f}}(−v)^{r−l(z)}h^{zy,w_{f} x}M^{y}

and this proves our claim. By the way we could also apply ζ^{∗} to \underline{H}^{tx} with t ≠ w_{f} to get ∑_{z∈W_{f}} (−v)^{−l(z)}h^{zy,tx} = 0 for all x, y ∈ W^{f}.

(2) We transpose the map ξ from the proof of 3.4 (2) and get

ξ^{∗} : N^{∗} → H^{∗}.

The formula for ξ(H_{zx}) implies

ξ^{∗}(N^{x}) = ∑_{z∈W_{f}} v^{l(z)}H^{zx}

and from the formula for ξ(\underline{H}_{zx}) we get ξ^{∗}(\underline{N}^{x}) = \underline{H}^{x}. When we apply ξ^{∗} to the equation \underline{N}^{x} = ∑_{y}^{∞} n^{y,x}N^{y} we get

\underline{H}^{x} = ∑_{y}^{∞} ∑_{z∈W_{f}} v^{l(z)}n^{y,x}H^{zy}

and deduce even v^{l(z)}n^{y,x} = h^{zy,x} for all x, y ∈ W^{f} and z ∈ W_{f}.

To formulate the next theorem, I have to introduce a convention. Let φ : A → A′ be a ring homomorphism, M an A-module and M′ an A′- module. A homomorphism of additive groups ψ : M → M′ is called “φ-linear” iff ψ(rm) = φ(r)ψ(m) ∀r ∈ A, m ∈ M. Thus for our involution d : H → H with d(H) = \overline{H} a d-linear map is the same thing as an H-skew-linear one.

Certainly M^{∗} is a left H-module via (HF)(M) = F (MH) ∀H ∈ H, F ∈ M^{∗}, M ∈ M. Recall the involutions d, a, and da = d ∘ a on H from the proof of Theorem 2.7.

Theorem 3.8. There exists a da-linear map ψ : N → M^{∗} such that ψ(N_{x}) = M^{x} ∀x.

Proof. Let’s first check the forumlas

C_{s}M_{x}^{∗} = \{ \begin{matrix}M_{xs}^{∗} + vM_{x}^{∗} & if xs ∈ W^{f}, xs > x; \\ M_{xs}^{∗} + v^{−1}M_{x}^{∗} & if xs ∈ W^{f}, xs < x; \\ (v + v^{−1})M_{x}^{∗} & if xs ∉ W^{f}. \end{matrix}

Indeed, the matrix of the right action of C_{s} on M expressed in the basis of the M_{x} decomposes in 1 × 1-blocks and 2 × 2-blocks of the form ( \begin{matrix}v & 1 \\ 1 & v^{−1} \end{matrix} ). Thus this matrix is its own transposed matrix, and this gives the above formulas.

We deduce the existence of an i-linear map M → M^{∗} such that M_{x} ↦ M_{x}^{∗}. On the other hand from the proof of Theorem 3.5 we know there is a dia-linear map N → M such that N_{x} ↦ (−1)^{l(x)}M_{x}. Composing these two maps the Theorem follows.

Certainly all our arguments and results stay valid when we exchange the roles of N, N and M, M. For completeness I finish this section with Douglass’ inversion formulas for finite W.

Proposition 3.9 ([Dou90]). Let W be finite and let w ∈ W resp. w_{f} ∈ W_{f} be the longest elements. Then we have

∑_{z}(−1)^{l(x)+l(z)}m_{z,x}n_{w_{f} zw,w_{f} yw} = δ_{x,y}.

Remark 3.10. In particular this gives the inversion formulas of Kazhdan-Lusztig [KL79]

∑_{z}(−1)^{l(x)+l(z)}h_{z,x}h_{zw,yw} = δ_{x,y}.

Proof. If we put S_{g} = wS_{f}w, the map x ↦ w_{f}wx gives an orderreversing bijection W^{g} →^{∼} W^{f}. Therefore we get an H-skew-linear map

N^{g} → N , N_{x}^{g} ↦ N_{w_{f} wx},

where we put N = N^{f}. Composing this with our ψ : N → M^{∗}, we get an a-linear map

N^{g} → M^{∗}, N_{x}^{g} ↦ M^{w_{f} wx}.

This map even commutes with the dualities on our modules, since N_{e}^{g} is self-dual and so is M^{w_{f} w}, because w_{f}w is the maximal element in W^{f}. But then our map necessarily transforms \underline{N}_{x}^{g} into \underline{M}^{w_{f} wx}, and we deduce

m^{w_{f} wy,w_{f} wx} = n_{y,x}^{g} = n_{wyw,wxw}.

After a transformation of the variables this gives

m^{y,x} = n_{w_{f} yw,w_{f} xw}.

4. Affine reflection groups and the periodic Hecke module

For an explanation of the terminology used in the sequel one may look at [Bou81]. Let V ⊃ R ⊃ R^{+} ⊃ Δ be a vector space over the reals, a root system, a system of positive roots and the corresponding set of simple roots. Let W ⊂ GL(V) be the Weyl group and W = W ⋉ ℤR the affine Weyl group. For μ ∈ V let W_{μ} resp. W_{μ} be its stabilizer in W resp. W. Thus we have W = W_{0}. The group W is generated by its (affine) reflections, and we let F be the set of all its reflection hyperplanes. For F ∈ F let s_{F} ∈ W be the reflection leaving F invariant.

The connected components of the complement of all reflection hyperplanes V − ∪_{F ∈F}F are called “alcoves”. We denote by A the set of all alcoves. The obvious action of W on A is free and transitive. Let

C = {τ ∈ V | ⟨τ, α^{∨}⟩ > 0 ∀α ∈ R^{+}}

be the dominant Weyl chamber. Let A^{+} ∈ A be the unique alcove contained in C and having the null vector in its closure.

Let S ⊂ W be the the set of all reflections, which pointwise fix some wall of A^{+}. Then (W, S) is a Coxeter system. We also consider the bijection W →^{∼} A, w ↦ wA^{+}. The obvious right action of W on itself corresponds under such a bijection to a right action of W on A, denoted A ↦ Aw. For A ∈ A, s ∈ S one may visualize As as follows: Consider the wall of A^{+} fixed by s. Exactly one wall of A is conjugate to this wall of A^{+} under the action of W on V. Then As meets A exactly along this wall of A.

A reflecting hyperplane F ∈ F divides V into two halfspaces

V − F = F^{+} ∪ F^{−},

where we let F^{+} the unique halfspace, which meets every translate of the dominant Weyl chamber, F^{+} ∩ (τ + C) ≠ ∅ ∀τ ∈ V. For A ∈ A, s ∈ S we write As ≻ A (resp. As ≺ A) iff As ⊂ F^{+} (resp. As ⊂ F^{−}) for the reflecting hyperplane F ∈ F separating As and A.

Now we may define the “periodic” Hecke module P. As an L-module P is just free with basis A,

P = ⨁_{A∈A}LA.

Lemma 4.1 ([Lus80a]). On P there can be defined a right H-action such that for all s ∈ S we have:

AC_{s} = \begin{cases}As + vA & if As ≻ A; \\ As + v^{−1}A & if As ≺ A. \end{cases}

Remark 4.2. To identify the M from [Lus80a] with our P, one needs a length function δ : A → ℤ as in [Lus80a]. Our A would be called q^{−δ(A)/2}A in Lusztig’s notation. In addition we let H act from the right.

Proof. First let us consider for s ∈ S the L-linear map ρ_{s} : P → P given by

ρ_{s}(A) = \begin{cases}As + vA & if As ≻ A; \\ As + v^{−1}A & if As ≺ A. \end{cases}

For μ ∈ ℤR we also consider ⟨μ⟩ : P → P, A ↦ μ + A. Certainly we have ⟨μ⟩ ∘ ρ_{s} = ρ_{s} ∘ ⟨μ⟩ for all μ ∈ ℤR, s ∈ S.

In any case we get a right action of H on P by transport of structure via the L-linear bijection H → P given by H_{x} ↦ xA^{+} ∀x ∈ W. Let us denote this right action by P ∗ H for P ∈ P, H ∈ H. The map P → P, P ↦ P ∗ H will be denoted by ρ^{∗}(H).

Let A^{+} ⊂ A be the set of all alcoves contained in the dominant Weyl chamber, A^{+} = {A ∈ A | A ⊂ C}. For x ∈ W, s ∈ S such that xA^{+}, xsA^{+} ∈ A^{+} the relation x > xs is equivalent to xA^{+} ≻ xA^{+}s. Thus for all A ∈ A, s ∈ S such that A, As ∈ A^{+} we have

ρ_{s}(A) = A ∗ C_{s}.

Choose μ ∈ C ∩ ℤR. For any alcove A the translated alcove nμ + A lies inside C, for n ≫ 0. We deduce

ρ_{s}(A) = ⟨−nμ⟩ ∘ ρ_{s} ∘ ⟨nμ⟩(A)
= ⟨−nμ⟩ ∘ ρ^{∗}(C_{s}) ∘ ⟨nμ⟩(A),

if n ≫ 0. Thus for all H ∈ H, P ∈ P the expression ⟨−nμ⟩ ∘ ρ^{∗}(H) ∘ ⟨nμ⟩(P) is independent of n for n ≫ 0. We call this expression P H and have thus defined the looked-for right action of H on P.

Let X ⊂ V be the lattice of integral weights. For λ ∈ X we define E_{λ} ∈ P by

E_{λ} = ∑_{z∈W} v^{l(z)}(λ + zA^{+}).

Let P^{∘} ⊂ P be the H-submodule generated by all the E_{λ} .

Theorem 4.3 ([Lus80a]). 1. On P^{∘} there exists a unique H-skew-linear involution P^{∘} → P^{∘}, P ↦ \overline{P} such that \overline{E_{λ}} = E_{λ} for all λ ∈ X.

2. For all A ∈ A there exists a unique \underline{P}_{A} ∈ P^{∘} which is self-dual with respect to this involution and such that \underline{P}_{A} ∈ A+∑_{B} vℤ[v]B. The \underline{P}_{A} form an L-basis of P^{∘}.

Remark 4.4. For A, B ∈ A we define p_{B,A} ∈ ℤ[v] by \underline{P}_{A} = ∑ p_{B,A}B. Let d(B, A) ∈ ℤ be the weighted sum of reflecting hyperplanes H ∈ F separating B from A, where we count H with weight 1 resp. (−1) if B ⊂ H^{−} resp. B ⊂ H^{+}. The polynomials Q_{B,A} of Lusztig [Lus80a] are related to these p_{B,A} by p_{B,A} = v^{−d(B,A)}Q_{B,A}. I call the p_{B,A} the “periodic polynomials”.

The proof of the Theorem needs some preparations and will be complete towards the end of this section. We start by repeating Lusztig’s construction of an action of W on P^{∘}.

Proposition 4.5 ([Lus80a]). For all w ∈ W there exists a homomorphism of H-modules ⟨w⟩ : P^{∘} → P^{∘} such that ⟨w⟩E_{λ} = E_{wλ} for all λ ∈ X.

Remark 4.6. Certainly ⟨w⟩ is uniquely defined by this condition and we get thus an action of W on P^{∘}. In addition for w = μ ∈ ℤR this ⟨μ⟩ is obviously the restriction to P^{∘} of our translation ⟨μ⟩ from above.

Proof. For α ∈ R^{+} let F_{α} ⊂ F be the set of reflecting hyperplanes orthogonal to α. Thus F = ∪_{α∈R^{+}}F_{α} is a partition of F. The connected components of V − ∪_{F ∈F_{α}}F are called “α-strips”. Every α-strip U has the form U = F^{+} ∩ G^{−} for unique F, G ∈ F_{α}. We put F = ∂^{−}U and G = ∂^{+}U. For A ∈ A, α ∈ R^{+} we define α↑A = s_{G}A, α↓A = s_{F} A, if A lies in the α-strip U and F = ∂^{−}U, G = ∂^{+}U as above. For a simple root α ∈ Δ let us consider the L-submodule P_{α} ⊂ P generated by all A + v(α ↓ A) with A ∈ A. Certainly these expressions form even an L-basis of P_{α}. So we can define for all F ∈ F_{α} an L-linear map

⟨s_{F} ⟩ : P_{α} → P_{α}

by the prescription ⟨s_{F} ⟩(A + v(α↓A)) = v(s_{F} A) + α↑(s_{F} A).

Lemma 4.7 ([Lus80a]). P_{α} is an H-submodule of P and ⟨s_{F} ⟩ : P_{α} → P_{α} is H-linear.

Proof. We have to show for all A ∈ A, s ∈ S that

(i) (A + v(α↓A))C_{s} ∈ P_{α}. (ii) ⟨s_{F} ⟩{(A + v(α↓A))C_{s}} = {⟨s_{F} ⟩(A + v(α↓A))}C_{s}. Let U be the α-strip of A. Let G ∈ F be the reflecting hyperplane separating As and A. We have to consider three cases.

1. G is not a wall of U. From there G ∈ F_{β} with β ∈ R^{+} − {α}. In particular s_{F} (G^{+}) = (s_{F} G)^{+}. Then (i) and (ii) follow easily.

2. G = ∂^{+}U, left to the reader.

3. G = ∂^{−}U, left to the reader. The Lemma is proved.

Lemma 4.8 ([Lus80a]). We have E_{λ} ∈ P_{α} for all simple roots α ∈ Δ and ⟨s_{F} ⟩E_{λ} = E_{s_{F} λ} for all F ∈ F_{α}.

Proof. Left to the reader.

In particular we have P^{∘} ⊂ P_{α} and ⟨s_{F} ⟩P^{∘} ⊂ P^{∘}. Now for w ∈ W we get ⟨w⟩ : P^{∘} → P^{∘} as follows: We write w = s_{F} ⋯ s_{G} with F, …, G ∈ ∪_{α∈Δ}F_{α} and put ⟨w⟩ = ⟨s_{F} ⟩ ∘ ⋯ ∘ ⟨s_{G}⟩.

It will be important to know, that the H-linear action of W on P^{∘} can be extended to an L-linear action of the “extended affine Weyl group” W̃ = W ⋉ X.

For any μ ∈ X let us consider the L-linear map ⟨μ⟩ : P → P given by A ↦ μ+A ∀A ∈ A. It doesn’t commute with the right action of H in general. If C_{s} denotes for the moment the map P → P, P ↦ P C_{s}, we rather have

⟨μ⟩ ∘ C_{s} = C_{[μ]s} ∘ ⟨μ⟩

for a suitable permutation [μ] : S → S of the simple reflections, and we have [μ + ν] = [μ] ∘ [ν] for all μ, ν ∈ X and [μ] = id for μ ∈ ℤR, hence in particular [wμ] = [μ] = [−μ]^{−1} for all w ∈ W, μ ∈ X.

Lemma 4.9. There exists an L-linear action φ : W̃ → Aut P^{∘} of W̃ on P^{∘} such that φ(w) = ⟨w⟩ for all w ∈ W and φ(μ) = ⟨μ⟩ for all μ ∈ X.

Remark 4.10. Once the Lemma is established, we will abbreviate φ(w) by ⟨w⟩ for all w ∈ W̃.

Proof. It is sufficient to show that the map ⟨w⟩⟨μ⟩⟨w^{−1}⟩⟨−wμ⟩ is the identity on P^{∘}, for all μ ∈ X, w ∈ W. However this map commutes with the right action of C_{s} and maps E_{λ} to itself.

Now we can prove part (1) of Theorem 4.3.

Proposition 4.11 ([Lus80a]). There exists a unique skew-linear map P^{∘} → P^{∘}, P ↦ \overline{P} such that \overline{E_{λ}} = E_{λ} ∀λ ∈ X.

Proof. Certainly by skew-linear we mean H-skew-linear here, but writing this produced an overfull box. Unicity is clear, we only have to construct such a map. Let w_{0} ∈ W be the longest element and r = l(w_{0}) its length. Let c : P → P denote the L-skew-linear map given by c(A) = w_{0}A. For all s ∈ S we have A ≻ As ⇔ w_{0}A ≺ w_{0}As. Hence we have c(AC_{s}) = c(A)C_{s} for all s ∈ S and c is even H-skew-linear. Certainly we have c(E_{λ}) = v^{−r}E_{w_{0}λ}. In particular we get c(P^{∘}) ⊂ P^{∘}. We put \overline{P} = v^{r}c⟨w_{0}⟩P and are done.

This duality even commutes with the W̃-action.

Proposition 4.12. We have \overline{⟨w⟩P} = ⟨w⟩\overline{P} for all w ∈ W̃, P ∈ P^{∘}.

Proof. Let d : P^{∘} → P^{∘} denote our duality P ↦ \overline{P}. We have to show that ⟨w⟩d = d⟨w⟩ for all w ∈ W̃. It will be sufficient to show that ⟨μ⟩d⟨−μ⟩d resp. ⟨w⟩d⟨w^{−1}⟩d are the identity, for all μ ∈ X resp. w ∈ W. However these maps commute with the right action of the C_{s} and map E_{λ} to itself.

We now establish the existence of the \underline{P}_{A}. Let us consider the partial order ⪯ on A generated by the relations

A ⪯ s_{F} A if A ∈ A, F ∈ F, A ⊂ F^{−}.

So A ⪯ B means, that there exists some sequence of alcoves, say A = A_{0}, A_{1}, …, A_{n} = B and some sequence of reflecting hyperplanes F_{i} ∈ F such that A_{i} ⊂ F_{i}^{−} and A_{i+1} = s_{F_{i}}A_{i} for i = 0, …, n − 1. To check that ⪯ is indeed a partial order, we may proceed as follows: Let us denote for an alcove A ∈ A by b(A) ∈ V its barycenter. Then A ⪯ B implies b(A) ∈ b(B) + ℝ_{≤0}R^{+}. Thus B ⪯ A ⪯ B implies b(A) = b(B) and hence A = B.

Obviously our new notation is compatible with our old notation A ≺ As for s ∈ S. Obviously our partial order on A is invariant under translation by μ ∈ X. In addition it has the following property:

Lemma 4.13 ([Lus80a]). Let A, B ∈ A and s ∈ S. Then B ⪯ A ≺ As implies Bs ⪯ As.

Proof. The Bruhat order on W defines via our bijection W → A, w ↦ wA^{+} another partial order ≤ on A. Deep inside C now ⪯ and ≤ coincide. More precisely we have:

Claim 4.14. Let μ ∈ X ∩ C. For A, B ∈ A are equivalent:

1. A ⪯ B.

2. nμ + A ≤ nμ + B for n ≫ 0, i. e. for all n above a suitable lower bound depending on A, B and μ.

This claim follows from the definition of Bruhat order. Indeed A ≤ B means, that there exists a sequence of alcoves A = A_{0}, A_{1}, … A_{n} = B and a sequence of reflecting hyperplanes F_{i} ∈ F such that A_{i+1} = s_{F_{i}}A_{i} and that A_{i} isn’t separated from A^{+} by F_{i}. So for A, B ∈ A^{+} and F ∈ F a hyperplane such that B = s_{F} A, we have

A ⪯ B ⇔ A ⊂ F^{−}
⇔ A and A^{+} are not separated by F
⇔ A ≤ B.

The equivalence of (1) and (2) in general can easily be deduced from this special case. Using the claim we deduce the lemma from the analogous property of the Bruhat order.

Let Π ⊂ V be the fundamental box

Π = {τ ∈ V | 0 < ⟨τ, α^{∨}⟩ < 1 ∀α ∈ Δ}.

For λ ∈ X we abbreviate λ + Π = Π_{λ}. For any alcove A ∈ A there exists a unique λ = λ(A) ∈ X such that A ⊂ Π_{λ}.

Lemma 4.15. Let λ be a dominant weight, i.e. λ ∈ X ∩ \overline{C}. Then we have B ⪯ λ + B for every alcove B.

Proof. We choose τ ∈ B and consider the line segment joining τ and λ+τ. It meets in that order, say, the alcoves B = A_{0}, A_{1}, A_{2}, …, A_{n} = λ + B. Choosing τ properly we can assume that subsequent alcoves in our sequence are separated just by one wall, A_{i+1} = A_{i}s_{i} for s_{i} ∈ S. Now λ ∈ \overline{C} implies A_{i} ≺ A_{i+1}, hence B ⪯ λ + B.

Now we are ready to prove the existence part of Theorem 4.3 (2). More precisely we show

Proposition 4.16 ([Lus80a]). For A ∈ A there exists a self-dual \underline{P}_{A} ∈ P^{∘} such that \underline{P}_{A} ∈ A + ∑_{B≺A} vℤ[v]B and ⟨w⟩\underline{P}_{A} = \underline{P}_{A} for all w ∈ W_{λ(A)}.

Proof. Certainly it will be sufficient to construct \underline{P}_{A} for A ⊂ Π. We proceed by induction on the ordered set of all alcoves in Π and start the induction with

\underline{P}_{A^{+}} = E_{0} = ∑_{z∈W} v^{l(z)}(zA^{+}).

Now let A ⊂ Π be an alcove and suppose we already know possible \underline{P}_{B} for B ≺ A, B ⊂ Π. If A ≠ A^{+} we find s ∈ S such that As ≺ A and As ⊂ Π. Clearly \underline{P}_{As}C_{s} is self-dual and by Lemma 4.13 and the definition of AC_{s} we get

\underline{P}_{As}C_{s} = ∑_{B⪯A} p_{B}B

with p_{A} = 1 and p_{B} ∈ ℤ[v] for all B. However certain p_{B} with B ≠ A could also have a nonzero constant term. To eliminate these terms, we need a longer argument.

First let us define as in [Lus80a] a new action of W̃ on A, denoted B ↦ w ∗ B, by the formula

w ∗ (λ + A) = (wλ) + A

for all λ ∈ X, A ⊂ Π.

Lemma 4.17. Let P ∈ P^{∘} be of the form P = ∑ p_{B}B with p_{B} ∈ ℤ[v] for all B. Take w ∈ W̃. Then ⟨w⟩P = ∑ q_{B}B with q_{B} ∈ ℤ[v] and p_{B}(0) = q_{w∗B}(0) for all B.

Proof. Without restriction we assume w = s_{F} with F ∈ F_{α} for α ∈ Δ. Then our definitions imply that the claim even holds for all P ∈ P_{α}.

For our special situation this means that p_{B}(0) = p_{z∗B}(0) for all z ∈ W. But for any alcove B we find z ∈ W such that B′ = z ∗ B ∈ A^{+}. Now p_{B} ≠ 0 and B ≠ A imply p_{B′} ≠ 0 and B′ ≠ A, hence B′ ≺ A. On the the other hand B′ ∈ A^{+} implies B′ = λ + B′′ for suitable λ ∈ X ∩ \overline{C} and B′′ ⊂ Π. From Lemma 4.15 we get B′′ ≺ A, so we know a possible \underline{P}_{B′′} by induction, and translating this \underline{P}_{B′′} by λ we get a possible \underline{P}_{B′}. We now consider

\underline{P}_{A} = \underline{P}_{As}C_{s} − ∑_{B∈A^{+},B≠A} p_{B}(0) ∑_{z}⟨z⟩\underline{P}_{B},

where in the second sum z runs over (a set of representatives for) the cosets W/W_{λ(B)}. This completes the induction step and the existence of the \underline{P}_{A} is established.

Next we have to care for unicity of the \underline{P}_{A}. By Proposition 4.16 we even know, that there exists a family of self-dual elements \{\underline{P}_{A}\}_{A∈A} in P^{∘} such that

1. ⟨w⟩\underline{P}_{A} = \underline{P}_{w∗A} ∀A ∈ A, w ∈ W̃.

2. \underline{P}_{A} ∈ A + ∑_{B≺A} vℤ[v]B.

Indeed we get such a family by choosing possible \underline{P}_{A} for A ⊂ Π as in the Proposition and defining the remaining \underline{P}_{A} for A ⊄ Π as translations of these.

Proposition 4.18 ([Lus80a]). 1. Such a family \{\underline{P}_{A}\}_{A∈A} already is

an L-basis of P^{∘}.

2. For P ∈ P^{∘} ∩ ∑ vℤ[v]B self-duality P = \overline{P} implies P = 0.

Remark 4.19. Certainly (2) implies the unicity of the \underline{P}_{A} claimed in Theorem 4.3. The preceeding considerations or Lemma 4.17 then show ⟨w⟩\underline{P}_{A} = \underline{P}_{w∗A}.

Proof. Let us start with (1). Clearly the \underline{P}_{A} are linearly independent. We have to show they generate P^{∘} over L. Thus we have to show, that for all λ ∈ X, H ∈ H the element E_{λ}H lies in the L-submodule generated by \underline{P}_{A}. Without restriction we can assume λ = 0. Clearly it will be sufficient to show that every W -invariant Q ∈ P^{∘} (i.e. ⟨z⟩Q = Q ∀z ∈ W) lies in the L-submodule generated by the \underline{P}_{A}. So suppose

Q = ∑_{B∈A} q_{B}B.

For Q ≠ 0 there exists B ∈ A^{+} such that q_{B} ≠ 0. (To see this one may take the smallest n ∈ ℤ such that v^{n}Q ∈ ∑_{B} ℤ[v]B and apply Lemma 4.17 to v^{n}Q.) Now we proceed by induction on #{A ∈ A^{+} | ∃B ∈ A^{+} such that q_{B} ≠ 0 and B ⪰ A}. Let C ∈ A^{+} be maximal with q_{C} ≠ 0. We consider

Q′ = Q − ∑_{z}⟨z⟩q_{C} \underline{P}_{C},

where z runs over (a system of representations for) the cosets W/W_{λ(C)}. Then Q′ is W -invariant, and by induction Q′ lies in the L-submodule generated by \underline{P}_{A}. This proves (1).

Next we show (2). Certainly any self-dual P has the form P = ∑_{A∈A} c_{A}\underline{P}_{A} with \overline{c}_{A} = c_{A}. On the other hand P = ∑ p_{A}A with p_{A} ∈ vℤ[v] by assumption. If P ≠ 0, there is A maximal such that p_{A} ≠ 0. Then p_{A} = c_{A}, hence \overline{p}_{A} = p_{A} and then p_{A} ∈ vℤ[v] implies p_{A} = 0. This contradiction proves the Proposition and completes the proof of Theorem 4.3.

To simplify the calculation of p_{B,A} one may use

Proposition 4.20 ([Lus80a]). For s ∈ S and A ∈ A such that As ≺ A we have \underline{P}_{A}C_{s} = (v + v^{−1})\underline{P}_{A}. In other words, p_{Bs,A} = vp_{B,A} for all B ∈ A such that Bs ≺ B.

Proof. We put P = \underline{P}_{A}C_{s} − (v + v^{−1})\underline{P}_{A}. From C_{s}^{2} = (v + v^{−1})C_{s} we deduce P C_{s} = 0. By construction P is of the form P = ∑ p_{B}B with p_{B} ∈ ℤ[v], and since P is self-dual we get P = ∑ p_{B}(0)\underline{P}_{B}. In case P ≠ 0 we would find D maximal such that p_{D} ≠ 0, and for this D we would even get p_{D} = p_{D}(0). Now we write

P C_{s} = ∑ q_{B}B

and deduce q_{Ds} = p_{D} ≠ 0 if Ds ≻ D, and q_{D} = p_{Ds} + v^{−1}p_{D} ≠ 0 if Ds ≺ D, contradicting P C_{s} = 0. Thus P = 0 and the Proposition is proved.

For later use we have to discuss an additional symmetry of \underline{P}_{A}. As in the proof of Proposition 4.11 let w_{0} ∈ W be the longest element and r its length. We define a bijection A → A, A ↦ Ǎ as follows: Write A = λ + B with λ ∈ X, B ⊂ Π and put Ǎ = λ + w_{0}B. The inverse bijection is denoted A ↦ Â.

Lemma 4.21 ([Lus80a]). \underline{P}_{A} ∈ v^{r}(Ǎ + ∑_{B≻Ǎ} v^{−1}ℤ[v^{−1}]B).

Proof. We may assume A ⊂ Π. Then \underline{P}_{A} = \overline{\underline{P}_{A}} = v^{r}c⟨w_{0}⟩\underline{P}_{A} = v^{r}c\underline{P}_{A} and the Lemma follows from the definition of c.

We will also need a bound on the support of \underline{P}_{A}.

Proposition 4.22 ([Lus80a]). Let A, B ∈ A be such that p_{B,A} ≠ 0. Then xB ⪯ A ∀x ∈ W_{λ(A)}.

Proof. This is proved by induction using the inductive construction of the \underline{P}_{A} in the proof from Proposition 4.16.

5. Relations between different sorts of polynomials for affine reflection groups

We continue with the notations of the preceding section and put S_{0} = {s ∈ S | s stabilizes zero}. Then we can use our notations M = M^{0}, N = N^{0}, M^{∗}, N^{∗} etc. from Section 3. In particular W_{0} = W is the (finite) Weyl group. The bijection W → A, w ↦ wA^{+} restricts to a bijection W^{0} → A^{+}. We use this bijection to rename our distinguished elements of N , M^{∗}, etc. and put N_{x} = N_{A}, \underline{N}_{x} = \underline{N}_{A}, n_{x,y} = n_{A,B}, M^{x} = M^{A}, etc. if x, y ∈ W^{0} and A, B ∈ A^{+} are given as xA^{+} = A, yA^{+} = B.

Let A^{++} denote the set of all alcoves contained in ρ + C (where ρ is the half-sum of positive roots). Then A ↦ Â is a bijection A^{+} →^{∼} A^{++}. Recall the L-skew-linear map ψ : N → M^{∗} from Theorem 3.8 given by ψ(N_{A}) = M^{A}. The only essential result of this article, which I could’t find in the literature, is the following

Theorem 5.1. For all A ∈ A^{+} we have \underline{M}^{A} = v^{r}ψ\underline{N} Â, in other words m^{B,A} = v^{r}\overline{n}_{B,Â}.

This formula was suggested by the theory of tilting modules, as will be explained in more detail at the end of this article. The proof of the Theorem needs some preparation. We consider in P the H-submodule

P^{sgn} = {P ∈ P^{∘} | ⟨z⟩P = (−1)^{l(z)}P ∀z ∈ W }.

Proposition 5.2. The L-linear “restriction” res : P → N given by res A = N_{A} if A ∈ A^{+} and res A = 0 else induces a homomorphism of right H-modules res : P^{sgn} → N .

Proof. We have to show that res commutes with all C_{s} (s ∈ S). The formula for the action of W on P^{∘} implies that for P = ∑ p_{A}A in P^{sgn} we have p_{A} = −vp_{As} for A ∈ A^{+}, s ∈ S such that As ∉ A^{+}. The Proposition follows.

We define next the H-linear map

\begin{aligned}alt : & P^{∘} & → & P^{sgn} \\ & P & ↦ & ∑_{x∈W}(−1)^{l(x)}⟨x⟩P. \end{aligned}

Now our theorem follows from the following more detailed:

Theorem 5.3. 1. For all A ∈ A^{++} we have \underline{N}_{A} = res alt \underline{P}_{A}. 2. For all A ∈ A^{+} we have \underline{M}^{A} = v^{r}ψ res alt \underline{P} Â.

Remark 5.4. The second statement is a reformulation of the main result in [Kan87] and therefore contains some results of [And86]. Indeed, since ⟨x⟩\underline{P}_{A} = \underline{P}_{x∗A} we have

alt \underline{P}_{A} = ∑_{x∈W}(−1)^{l(x)}\underline{P}_{x∗A}.

Proof. We start with (2) and for simplicity abbreviate the L-skew-linear map (v^{r}ψ res alt) to φ : P^{∘} → M^{∗}. With respect to H both φ, ψ are da-linear. Now \underline{M}^{A} can be characterized by a degree condition and self-duality. By Lemma 4.21 our φ(\underline{P} Â) satisfies the degree condition. We only need to show that all φ(\underline{P} Â) ∈ M^{∗} are self-dual.

By our definitions F ∈ M^{∗} is self-dual iff (\overline{H}F)(M_{A^{+}}) = \overline{(HF)(M_{A^{+}})} for all H ∈ H. We will check this for all F = φ(\underline{P}_{A}) with A ∈ A.

Certainly we can write \underline{P}_{A}H = ∑ c_{B}\underline{P}_{B} and deduce \underline{P}_{A}\overline{H} = ∑ \overline{c}_{B}\underline{P}_{B}. Then we get

(da(H)φ(\underline{P}_{A}))(M_{A^{+}}) = ∑ \overline{c}_{B}φ(\underline{P}_{B})(M_{A^{+}}),
(a(H)φ(\underline{P}_{A}))(M_{A^{+}}) = ∑ c_{B}φ(\underline{P}_{B})(M_{A^{+}}),

and hence only have to show, that all φ(\underline{P}_{B})(M_{A^{+}}) are invariant for the substitution v ↦ v^{−1}. I claim that we have even

φ(\underline{P}_{B})(M_{A^{+}}) = \begin{cases}(−1)^{l(z)} & if B = z ∗ Â^{+} with z ∈ W; \\ 0 & otherwise. \end{cases}

To see this, we consider again \underline{P}_{B} = ∑ p_{C,B}C and prove

Lemma 5.5. Let B ∈ A be such that p_{A^{+},B} ≠ 0. Then either B = Â^{+} or λ(B) lies on a reflecting hyperplane for W.

Proof. To avoid introducing more notation we rather show the equivalent statement that for B ⊂ Π and λ ∈ X both p_{λ+A^{+},B} ≠ 0 and λ ≠ −ρ together imply W_{λ} ≠ 1. (Here ρ ∈ X denotes as usual the half-sum of positive roots.) Indeed we can take x ∈ W such that x(λ + A^{+}) ⊂ C and get by Proposition 4.22 and the definition of Π the relations

x(λ + A^{+}) ⪯ B ⪯ ρ + w_{0}A^{+}.

But for an alcove C ⊂ C such that C ⪯ ρ + w_{0}A^{+} it is clear that either all its corners from X lie on walls of the dominant chamber or C = ρ + w_{0}A^{+}. For us this means that either W_{xλ} ≠ 1 hence W_{λ} ≠ 1 or λ = −ρ, x = w_{0}.

Now the formula for φ(\underline{P}_{B})(M_{A^{+}}) follows from the observation that alt(\underline{P}_{B}) = 0 and a forteriori φ(\underline{P}_{B}) = 0 if λ(B) lies on a a reflecting hyperplane of W. This proves (2).

Next we attack (1). Let H^{+} ⊂ H be the subring generated by v+v^{−1} and all C_{s} with s ∈ S. By Proposition 3.3 we have H^{+} = {H ∈ H | \overline{H} = H\}. However we will only use the obvious inclusion ⊂. The proof of (1) rests on the following

Lemma 5.6. All \underline{M}^{A} lie in the H^{+}-submodule of M^{∗} generated by the element \underline{M}^{ρ+w_{0}A^{+}}.

Proof. First note that by (2) we have m^{Â,A} = v^{r} and deg_{v}m^{B,A} < r if B ≠ Â. Now consider A ∈ A^{+}, s ∈ S such that As ≺ A (but not necessarily As ∈ A^{+}) and write

a(C_{s})\underline{M}^{A} = ∑ q_{B}M^{B}.

By Theorem 3.8 we know that

a(C_{s})M^{B} = \begin{cases}M^{Bs} + v^{−1}M^{B}, & if Bs ∈ A^{+}, Bs ≻ B; \\ M^{Bs} + vM^{B}, & if Bs ∈ A^{+}, Bs ≺ B; \\ 0 & if Bs ∉ A^{+}. \end{cases}

Thus we have q_{B} ∈ ℤ[v] for all B ∈ A^{+} and we deduce

a(C_{s})\underline{M}^{A} = ∑ q_{B}(0)\underline{M}^{B}.

Indeed any element of M^{∗} can be written uniquely as a formal linear combination of the \underline{M}^{B}, if the element is self-dual all its coefficients are, and for an element from ∑^{∞} ℤ[v]M^{B} all its coefficients lie in ℤ[v]. Thus if ∑^{∞} q_{B}M^{B} is self-dual with all q_{B} in ℤ[v], then we have

∑^{∞} q_{B}M^{B} = ∑^{∞} q_{B}(0)\underline{M}^{B}.

For q ∈ L let q̂(ν) denote the coefficient of v^{ν}, thus q = ∑_{ν} q̂(ν)v^{ν} and q̂(0) = q(0). We can further conclude q̂B̂(r) = q_{B}(0) for all B ∈ A^{+} and q̂_{B}(r) = 0 for B ∉ A^{++}. Thus we get even

a(C_{s})\underline{M}^{A} = ∑ q̂_{B}(r)\underline{M} B̌.

Changing variables, we have proved: If D ∈ A^{++} and s ∈ S are given such that Ds ≻ D, when we write

a(C_{s})\underline{M} Ď = ∑_{B∈A^{+}} q_{B}M^{B},

then we have

a(C_{s})\underline{M} Ď = ∑_{B∈A^{++}} q̂_{B}(r)\underline{M} B̌.

On the other hand we know by (2) that m^{B,Ď} ≠ 0 ⇒ B ⪯ D and m^{D,Ď} = v^{r} for all D ∈ A^{++}. Thus we get more precisely

a(C_{s})\underline{M} Ď = \underline{M}^{(Ds)^{∨}} + ∑_{\begin{matrix}B∈A^{++} \\ B≺Ds \end{matrix}} q̂_{B}(r)\underline{M} B̌.

Using this formula it is easy to show by induction on A^{++}, that all \underline{M} Ǎ with A ∈ A^{++} lie in the H^{+}-submodule of M^{∗} generated by \underline{M}^{(ρ+A^{+})^{∨}}.

To show (1) we still need

Lemma 5.7. \underline{N}_{ρ+A^{+}} = ∑_{z∈W} v^{l(z)}N_{ρ+zA^{+}}.

Proof. Consider more generally for all λ ∈ (ρ + ℤR) ∩ C the expression F_{λ} = ∑_{z∈W} v^{l(z)}N_{λ+zA^{+}}. We have to show \underline{N}_{ρ+A^{+}} = F_{ρ}. Certainly it will be sufficient to show F_{ρ} = \overline{F_{ρ}}. To show this consider the set

S_{ρ} = \{s ∈ S | ρ ∈ \overline{A} ⇒ ρ ∈ \overline{As} ∀A ∈ A\},

where exceptionally \overline{A} resp. \overline{As} means the closure of A resp. As. We claim that F_{ρ} ∈ N is the unique element F = ∑ f_{A}N_{A} of N such that

1. F C_{s} = (v + v^{−1})F ∀s ∈ S_{ρ}

2. f_{A} ≠ 0 ⇒ A ⩽ ρ + A^{+}

3. f_{A} = 1 for A = ρ + A^{+} Here ⩽ means the order we get on A^{+} by transporting the Bruhat order from W^{0}.

First we see that condition (1) is satisfied precisely by all L-linear combinations of the F_{λ} with λ ∈ (ρ+ℤR)∩C. From there we see easily that F_{ρ} is the unique element of N satisfying (1)–(3). However these conditions are self-dual, hence \overline{F_{ρ}} also satisfies (1)–(3) and we deduce \overline{F_{ρ}} = F_{ρ}.

Now by Lemma 5.6 all (res alt \underline{P}_{A}) are contained in the H^{+}-submodule of N generated by (res alt \underline{P}_{ρ+A^{+}}), and (res alt \underline{P}_{ρ+A^{+}}) is self-dual in N by Lemma 5.7. This means all (res alt \underline{P}_{A}) are self-dual in N, and since they satisfy the degree conditions characterizing the \underline{N}_{A}, we deduce \underline{N}_{A} = res alt \underline{P}_{A} ∀A ∈ A^{++}.

6. The generic polynomials

Once we are far enough inside the dominant chamber, the m_{B,A} depend only on the relative position of the alcoves A and B. More precisely we show

Theorem 6.1 ([Lus80a]). 1. For all A, B ∈ A there exists q_{B,A} ∈ ℤ[v] such that q_{B,A} = m_{λ+B,λ+A}, if λ is sufficiently far inside the dominant chamber, i.e. if λ ∈ X ∩ (nρ + C) for suitable n = n(A, B) ∈ ℤ.

2. For the periodic polynomials p_{B,C} we have the inversion formulas

∑_{B}(−1)^{d(A,B)}q_{w_{0}B,w_{0}A} p_{B,C} = δ_{A,C},

where (−1)^{d(A,B)} means the parity of the number of affine reflection hyperplanes separating A and B.

Remark 6.2. The “generic Kazhdan-Lusztig polynomials” P̂_{A,B} of Kato [Kat85] are related to our q_{A,B} by the formula P̂_{A,B} = v^{−d(A,B)}q_{A,B}.

To prove the Theorem we consider the “completed below” Hecke module

P̂ = {f : A → L | there is C ∈ A such that f(A) ≠ 0 ⇒ A ⪯ C}.

For two alcoves C_{1}, C_{2} ∈ A there always exists C ∈ A such that C_{1} ⪯ C, C_{2} ⪯ C. Thus P̂ is an L-submodule of the space of all maps from A to L. We write elements f ∈ P̂ as formal linear combinations f = ∑^{∞} f_{A}A with f_{A} = f(A), where the upper index ∞ again should remind us that also certain infinite formal linear combinations are permitted. We extend the right action of H on P to P̂ in the obvious way. For λ ∈ ℤR we extend also ⟨λ⟩ : P → P to a map ⟨λ⟩ : P̂ → P̂ in the obvious way. For α ∈ R^{+} we define the operator

ϑ_{α} : P̂ → P̂

as the formal sum

ϑ_{α} = (1 + v^{2}⟨−α⟩ + v^{4}⟨−2α⟩ + v^{6}⟨−3α⟩ + ⋯).

Certainly ϑ_{α} commutes with the right H-action. Also the ϑ_{α} commute among themselves. We put

η = ∏_{α∈R^{+}} ϑ_{α} : P̂ → P̂.

This η is closely related to Kostant’s partition function. It gives another relation between the periodic and the generic polynomials, namely the following

Theorem 6.3 ([Kat85]). η\underline{P}_{A} = ∑_{B} q_{B,A}B.

Proof. Will be given later.

Finally we could also ask whether one could define alternative periodic polynomials by changing v to v^{−1} in the definition of \underline{P}_{A}. It turns out that the \underline{P̃}_{A} so defined exist only in P̂. More precisely we extend our skew-linear duality P ↦ \overline{P} to P̂ as follows: We can write P uniquely as a formal sum P = ∑_{A}^{∞} p_{A}\underline{P}_{A} with p_{A} ∈ L, where we start in the highest alcoves where P has a nonzero coefficient, and then work our way down. Then we define \overline{P} = ∑_{A}^{∞} \overline{p}_{A}\underline{P}_{A}. It is easy to see that \overline{A} ∈ A + ∑_{B≺A}^{∞} LB for A ∈ A.

Theorem 6.4. 1. \underline{P}_{A} is the unique self-dual element of P̂ contained

in A + ∑_{B}^{∞} vℤ[v]B.

2. \underline{P̃}_{A} = ∑(−1)^{d(A,B)}\overline{q}_{B,A}B is the unique self-dual element of P̂ contained in A + ∑_{B}^{∞} v^{−1}ℤ[v^{−1}]B.

Remark 6.5. Here (1) comes from [Lus80a] and (2) from [Kat85].

Now we prove the three preceding Theorems. Let Alt = ⟨−ρ⟩ ∘ alt ∘⟨ρ⟩ : P^{∘} → P^{∘} be antisymmetrization around −ρ, thus

Alt(P) = ∑_{x∈W_{−ρ}}(−1)^{l(x)}⟨x⟩P.

We also define the L-linear restriction

\begin{aligned}Res : & P̂ & → & M \\ & ∑_{A}^{∞} f_{A}A & ↦ & ∑_{A∈A^{+}} f_{A}M_{A}. \end{aligned}

This restriction doesn’t commute with the H-actions on our spaces. However we have

Proposition 6.6. The composition Res ∘η ∘Alt : P^{∘} → M is a homomorphism of right H-modules.

Proof. It will be sufficient to show that this map commutes with all C_{s}. With our definitions this is easily deduced from the following

Claim 6.7. Let A, B ∈ A be neighbouring alcoves such that A ∈ A^{+}, B ∉ A^{+}. Then for f = ∑^{∞} f_{C}C ∈ η ∘ Alt(P^{∘}) we have f_{B} = vf_{A}.

Let us check this claim. By our assumptions A and B meet along a wall of the dominant chamber. Let β ∈ Δ be the corresponding simple root. By a β-string in A we mean a minimal nonempty subset containing with A also β ↑ A and β ↓ A. Now let us consider in P̂ the H-submodule

P̂_{β} = {∑_{A∈A}^{∞} f_{A}A ∈ P̂ | for every β-string K we have ∑_{A∈K}^{∞} f_{A}A ∈ P_{β}}.

Certainly ϑ_{α}P̂_{β} ⊂ P̂_{β} for α ∈ R^{+}, α ≠ β. Let s̃_{β} ∈ W_{−ρ} be the reflection along the β-wall passing through (−ρ). Then ⟨s̃_{β}⟩ : P_{β} → P_{β} can be extended to ⟨s̃_{β}⟩ : P̂_{β} → P̂_{β} in an obvious way, and we have ⟨s̃_{β}⟩∘ϑ_{α} = ϑ_{s_{β}(α)} ∘⟨s̃_{β}⟩ for all α ∈ R^{+}, α ≠ β. Now we choose a system of representatives Rep ⊂ W_{−ρ} for the cosets {e, s̃_{β}}\W_{−ρ} and get

η ∘ Alt = ∏_{α∈R^{+}} ϑ_{α} ∘ (1 − ⟨s̃_{β}⟩) ∏_{x∈Rep}(−1)^{l(x)}⟨x⟩
= ϑ_{β} ∘ (1 − ⟨s̃_{β}⟩) ∏_{\begin{matrix}α∈R^{+} \\ α≠β \end{matrix}} ϑ_{α} ∏_{x∈Rep}(−1)^{l(x)}⟨x⟩.

Thus our claim will follow immediately from the much more elementary

Claim 6.8. Let A, B ∈ A be neighbouring alcoves separated only by the β-wall of the dominant chamber. If A lies above this wall, then for all f = ∑^{∞} f_{C}C ∈ ϑ_{β}(1 − ⟨s̃_{β}⟩)P̂_{β} we have f_{B} = vf_{A}.

This claim can be checked separately for every β-string, thus we have only to check the case

f = ϑ_{β}(1 − ⟨s̃_{β}⟩)(C + v(β ↓C))

with C ∈ A. But this case is clear from the definitions.

Corollary 6.9. \underline{M}_{A} = Res ∘η ∘ Alt \underline{P}_{A} for all A ∈ A^{+}.

Remark 6.10. This is Theorem 4.2 of Kato [Kat85]. Remark that Alt \underline{P}_{A} = ∑_{x∈W_{−ρ}}(−1)^{l(x)}⟨xλ(A)⟩\underline{P}_{A} and thus

η Alt \underline{P}_{A} = ∑_{x∈W_{−ρ}}(−1)^{l(x)}⟨xλ(A)⟩η\underline{P}_{A}.

Thus the Corollary implies in particular part (1) of Theorem 6.1 (where the generic polynomials q_{B,A} are defined) and Theorem 6.3.

Proof. For A = A^{+} both sides equal M_{A^{+}} = \underline{M}_{A^{+}} and our formula is true. But by Theorem 5.3 and Lemma 5.6 we know already that Alt \underline{P}_{A} ∈ (Alt \underline{P}_{A^{+}})H^{+} for all A ∈ A. Indeed, translating by ρ it will be sufficient to show that alt \underline{P} Â ∈ (alt \underline{P}_{ρ+A^{+}})H^{+} for all A ∈ A^{+}. By Theorem 5.3 (2) and its proof the map v^{r}ψ res defines a da-linear injection alt P^{∘} ↪ M^{∗} with alt \underline{P} Â ↦ \underline{M}^{A} for all A ∈ A^{+}, thus it will be sufficient to show that \underline{M}^{A} ∈ (\underline{M}^{ρ+w_{∘}A})H^{+} for all A ∈ A^{+}. But this is precisely Lemma 5.6.

From Alt \underline{P}_{A} ∈ (Alt \underline{P}_{A^{+}})H^{+} we deduce immediately that

Res ∘η ∘ Alt \underline{P}_{A} ∈ \underline{M}_{A^{+}}H^{+}

is self-dual for all A ∈ A. On the other hand certainly

Res ∘η ∘ Alt \underline{P}_{A} ∈ M_{A} + ∑_{B} vℤ[v]M_{B}

for all A ∈ A^{+} and the Corollary is established.

Next we show Theorem 6.4.

Proof. Here everything is left to the reader, except the proof that the formula claimed for \underline{P̃}_{A} gives indeed a self-dual element of P̂. Let ξ : P̂ → P̂ be the L-skew-linear map such that

ξ(∑_{A}^{∞} p_{A}A) = ∑_{A}^{∞}(−1)^{d(A^{+},A)}\overline{p}_{A}A.

We have to show that ξη\underline{P}_{A} ∈ P̂ is self-dual. We prove this by contradiction. Let us write

ξη\underline{P}_{A} − \overline{ξη\underline{P}_{A}} = ∑^{∞} f_{C}C

and choose B ∈ A such that f_{B} ≠ 0. Moving the pair (A, B) sufficiently far inside the dominant chamber we may assume that η Alt \underline{P}_{A} and η\underline{P}_{A} coincide on all alcoves C such that C ⪰ B. By Corollary 6.9 on these alcoves also Res η\underline{P}_{A} coincides with \underline{M}_{A} and res ξη\underline{P}_{A} with ϕ^{−1}\underline{M}_{A}, where ϕ as in Section 3 denotes the L-skew-linear map ϕ : N → M such that ϕ(N_{x}) = (−1)^{l(x)}M_{x}, and we extended our old res : P → N to P̂ in the obvious way.

Now by Theorem 5.3 we know that \underline{N}_{D} = res alt \underline{P}_{D} for D ∈ A^{++}, in particular we have \underline{N}_{D} = res \underline{P}_{D} for all D ⊂ C which are sufficiently far from all walls of the dominant chamber. Moving (A, B) if necessary still further inside the dominant chamber we can assume in addition that all alcoves C such that A ⪰ C ⪰ B are already so far from the walls, that \underline{N}_{C} = res \underline{P}_{C}. Now we can write

ξη\underline{P}_{A} = ∑_{C}^{∞} p_{C} \underline{P}_{C},
ϕ^{−1}\underline{M}_{A} = ∑ n_{C} \underline{N}_{C},

and deduce p_{C} = n_{C} for A ⪰ C ⪰ B. But since ϕ^{−1}\underline{M}_{A} = ±\underline{Ñ}_{A} is self-dual (by the end of the proof of Theorem 3.5), all n_{C} have to be self-dual, hence all p_{C} for A ⪰ C ⪰ B have to be self-dual as well, and this finally leads to the contradiction f_{B} = 0.

We are left with proving part (2) of Theorem 6.1.

Proof. Let us start with the almost tautological formula

∑_{B}(−1)^{d(A,B)}m_{B,A}m^{B,C} = δ_{A,C}.

If the pair (C, A) is sufficiently far inside the dominant chamber, we have m_{B,A} = q_{B,A} for all B ⪰ C, thus for all B such that m^{B,C} ≠ 0. On the other hand we also get m^{B,C} = v^{r}\overline{p}_{B,Ĉ} = p_{w_{0}B,w_{0}C}. Here the first equality follows from Theorem 5.3, and the last equality follows from the fact that \underline{P} Ĉ is self-dual. Indeed from there we get \underline{P} Ĉ = v^{r}c⟨w_{0}⟩\underline{P} Ĉ = v^{r}c\underline{P}_{w_{0}C} by the construction of the duality on P^{∘}, where we use the formula w_{0} ∗ Ĉ = w_{0}C.

All three Theorems of this section are established.

7. Relation with tilting modules

Let h be the Coxeter number of our root system R, and let l ⩾ h be odd. For a primitive l-th root of unity ζ we form following Lusztig the quantum group with divided powers U_{ζ}. Let U_{ζ}-mof be the category of all finite dimensional U_{ζ}-modules, and let B ⊂ U_{ζ}-mof be the principal block, i. e. the smallest direct summand containing the trivial representation. Certainly B is a k-category for k = ℚ(ζ).

The simple objects of B are parametrized in a natural way by the set A^{+} of alcoves in the dominant chamber. For A ∈ A^{+} let L_{A} ∈ B be the corresponding simple object. L_{A} is the socle resp. the unique simple quotient of the standard modules ∇_{A} resp. Δ_{A}. For example L_{A^{+}} = ∇_{A^{+}} = Δ_{A^{+}} = k is the trivial representation. In B there are enough projectives. The projective cover of L_{A} is denoted P_{A}. By a ∇-flag (resp. Δ-flag) of an object of B we mean a filtration such that all subquotients are of the form ∇_{A} resp. Δ_{A} for suitable A ∈ A^{+}.

Definition 7.1. An object T ∈ B is called a tilting module if and only if T admits a ∇-flag and a Δ-flag.

We recall without proof some facts from the theory of tilting modules. As standard reference for the completely analogous case of algebraic groups in finite characteristic compare [Don93]. First of all a direct summand of a tilting module is also tilting. Furthermore for an A ∈ A^{+} there exists a unique indecomposable tilting module T_{A}, which admits a Δ-flag starting with Δ_{A} ⊂ T_{A}. Here unicity follows easily from the following property of the standard objects:

Ext_{B}^{n}(Δ_{A}, ∇_{B}) = \begin{cases}k & if A = B and n = 0; \\ 0 & otherwise. \end{cases}

One may define a duality on B, i.e. an involutive exact contravariant k-functor d : B → B fixing the simple objects and exchanging the standard objects, dΔ_{A} ≅ ∇_{A}. It is known that tilting modules are self-dual for such a duality, dT_{A} ≅ T_{A}. In particular T_{A} is the unique indecomposable tilting module which admits a ∇-flag ending with a surjection T_{A} ↠ ∇_{A}.

The present work was motivated by the problem to determine the multiplicity (T_{A} : ∇_{B}) of ∇_{B} as a subquotient in a ∇-flag of T_{A}, for all A, B ∈ A^{+}. For this problem I propose the following

Conjecture 7.1. (T_{A} : ∇_{B}) = n_{B,A}(1).

Remark 7.2. 1. Recently I found a proof for this conjecture. However it is quite far from the reason for the conjecture explained below. An interpretation of the coefficients of the n_{B,A} was proposed by [And96].

2. The conjecture implies also character formulas for indecomposable tilting modules “on the walls”. More precisely we will show that for an indecomposable tilting module T on walls and Ψ the translation from the walls ΨT is indecomposable as well.

To see this, let Φ be the translation onto the walls, i.e. the adjoint of Ψ. Let A(1), …, A(r) be the alcoves containing the highest weight of T in their closure. Then we have ΦΨT ≅ T ⊕ … ⊕ T (r copies), since both sides are tilting and have the same character. This already means that only the T_{A(i)} are possible direct summands of ΨT.

Now let A(1) be maximal among the A(i). Since we know the highest weight of ΨT, we also know that T_{A(1)} has to be a summand of ΨT. Using remark 3.2 (4), the conjecture implies that (T_{A(1)} : ∇_{A(i)}) = 1 for i = 1, …, r. But on the other hand we know that (ΨT : ∇_{A(i)}) = 1 for i = 1, …, r. Thus in ΨT there is no room for other summands T_{A(i)}, and we deduce ΨT ≅ T_{A(1)}.

Most indecomposable tilting modules are indeed projective, more precisely we have P_{A} ≅ TÂ by [And92], 5.8. For projective objects the looked-for multiplicities are given already by the reciprocity formulas (P_{A} : ∇_{B}) = [∇_{B} : L_{A}]. Now the Lusztig conjecture says that in the Grothendieck group of B we have

L_{A} = ∑_{B}(−1)^{d(B,A)}m_{B,A}(1)∇_{B}.

(Use Proposition 3.4 to see that this coincides with the conjecture formulated by Lusztig in [Lus80b].) We can invert this equality to get

∑_{A} m^{C,A}(1)L_{A} = ∇_{C}.

Thus the Lusztig conjecture implies (P_{A} : ∇_{B}) = m^{B,A}(1), and as a first test of our conjecture we should check for m^{B,A}(1) = n_{B,Â}(1). This is an easy consequence of Theorem 5.1.

The true motivation for our conjecture however comes from the philosophy of “ℤ-graded representation theory,” as explained in the sequel. One expects, that the category B admits a ℤ-graded version B̃ in the meaning of [BGS96], 4.3. The objects ∇_{A}, Δ_{A}, L_{A} should admit ℤgraded versions ˜∇_{A}, Δ̃_{A}, L̃_{A} ∈ B̃, there should exist on B̃ a “shift of ℤ-grading” M ↦ M⟨i⟩ (for i ∈ ℤ), the duality should lift to a duality d̃: B̃ → B̃ such that d̃(M⟨i⟩) = (d̃M)⟨−i⟩, d̃( ˜∇_{A}) = Δ̃_{A}, and we would have

Extn ; B̃(Δ̃_{A}, ˜∇_{B}⟨i⟩) = \begin{cases}k & if A = B and i = n = 0; \\ 0 & otherwise. \end{cases}

Recall every s ∈ S gives an exact functor Θ_{s} : B → B, the so called “translation through the wall”. It commutes with the duality Θ_{s}d = dΘ_{s} and is easily determined on standard modules: For A ∈ A^{+} we have short exact sequences

\begin{matrix}∇_{A} & ↪ & Θ_{s}∇_{A} ↠ & ∇_{As} & if As ≻ A, As ∈ A^{+}; \\ ∇_{As} & ↪ & Θ_{s}∇_{A} ↠ & ∇_{A} & if As ≺ A, As ∈ A^{+}, \end{matrix}

and Θ_{s}∇_{A} = 0 if As ∉ A^{+}.

One may hope that Θ_{s} admits a graded version Θ̃_{s} : B̃ → B̃ as well, which commutes with d̃ and is such, that again for A ∈ A^{+} there are short exact sequences

\begin{matrix}˜∇_{A}⟨1⟩ & ↪ & Θ̃_{s} ˜∇_{A} ↠ & ˜∇_{As} & if As ≻ A, As ∈ A^{+}; \\ ˜∇_{As} & ↪ & Θ̃_{s} ˜∇_{A} ↠ & ˜∇_{A}⟨−1⟩ & if As ≺ A, As ∈ A^{+}, \end{matrix}

resp. that Θ̃_{s} ˜∇_{A} = 0 if As ∉ A^{+}. These expectations are supported by the fact that up to existence of d̃ they can be proved in the analogous situation concerning G_{1}T -modules, see [AJS94].

Now we try to inductively build up graded tilting modules T̃_{A}. Thus a ˜∇-flag of T̃_{A} should end with a surjection T̃_{A} ↠ ˜∇_{A}. We start with T̃_{A^{+}} = ˜∇_{A^{+}} = L̃_{A^{+}}. If T̃_{A} is constructed already, we choose s ∈ S such that As ≻ A and form Θ̃_{s}T̃_{A}. Certainly this is tilting and has even a ˜∇-flag finishing with ˜∇_{As}. However it should not be indecomposable in general, but should rather decompose as

Θ̃_{s}T̃_{A} = T̃_{As} ⊕ ⨁_{B}T̃_{B},

where the sum runs over a suitable multiset of alcoves B ≺ As. Now one might expect, that all homomorphisms in B̃ (i.e. all B-homomor-phisms “of degree zero”) from T̃_{B} to Θ̃_{s}T̃_{A} split, and this assumption leads precisely to the conjecture above. Indeed let us consider the Grothendieck group [B̃] of B̃ and define the homomorphism h : [B̃] ⟶ N by h( ˜∇_{A}⟨i⟩) = v^{i}N_{A}. By our formulas we have h(Θ̃_{s}M̃) = h(M̃)C_{s} for all s ∈ S. Now assume we already know by induction that h(T̃_{A}) = \underline{N}_{A}. Certainly we have

dim HomB̃(T̃_{B}, Θ̃_{s}T̃_{A}) = ∑_{i,C}(T̃_{B} : Δ̃_{C}⟨i⟩)(Θ̃_{s}T̃_{A} : ˜∇_{C}⟨i⟩)
= (Θ̃_{s}T̃_{A} : ˜∇_{B}),

since by induction T̃_{B} = d̃T̃_{B}, thus (T̃_{B} : Δ̃_{C}⟨i⟩) = (T̃_{B} : ˜∇_{C}⟨−i⟩) = 0 if i > 0 or i = 0, B ≠ C. But we have h(Θ̃_{s}T̃_{A}) = \underline{N}_{A}C_{s} = ∑_{B} m_{B}N_{B} and by our definitions (Θ̃_{s}T̃_{A} : ˜∇_{B}) = m_{B}(0), thus

h(T̃_{As}) = h(Θ̃_{s}T̃_{A}) − ∑_{B≺As} dim HomB̃(T̃_{B}, Θ̃_{s}T̃_{A})h(T̃_{B})
= \underline{N}_{A}C_{s} − ∑_{B≺As} m_{B}(0)\underline{N}_{B}
= \underline{N}_{As},

as I wished to explain.

I want to add that the formula h(T̃_{A}) = \underline{N}_{A} also implies T_{A} is indecomposable. Indeed one may check as above that under our assumptions E = End_{B} T_{A} admits a ℤ-grading

E = ⨁_{i} HomB̃(T̃_{A}⟨i⟩, T̃_{A})

which starts in degree zero with E_{0} = k and has no components of negative degree. However a finite dimensional k-algebra which admits such a grading is necessarily local. Hence under our assumptions T_{A} is indecomposable.

8. The example B_{2}

In the sequel I want to show for B_{2} the algorithm computing the \underline{N}_{A}. An element ∑ n_{A}N_{A} ∈ N will be represented by a picture, where the Laurent polynomial n_{A} is written inside the alcove A. We put S = {k, l, a} with a for affine, l long and k for short, as in the picture

BILD FEHLT

We start our computation with a picture of \underline{N}_{A^{+}}. A picture where only one alcove A contains a 1 gives the corresponding \underline{N}_{A}. Right multiplication by C_{s} will be written ⟶^{C_{s}}. In the element \underline{N}_{A}C_{s} thus obtained there could be additional ones, which have to be eliminated by subtraction of suitable \underline{N}_{B} with B ≺ As. This is symbolized by a dotted arrow − − − →.

BILD FEHLT

The last picture represents res alt E_{ρ} and thus illustrates 5.3 (1).

9. Notations

(W, S) a Coxeter group,

in Section 3 and after the affine Weyl group

W̃ the extended affine Weyl group W ⋉ X (W_{f}, S_{f}) a parabolic subgroup of W W^{f} the shortest representatives of

the right cosets W_{f}\W W_{λ} the isotropy group of λ in W W = W_{0} the finite Weyl group W_{λ} the isotropy group of λ in W ρ the half sum of positive roots C the dominant Weyl chamber A the set of all alcoves A^{+} the set of all alcoves

in the dominant chamber C

A^{++} the set of all alcoves in ρ + C A^{+} the fundamental dominant alcove F the set of all affine reflection hyperplanes F^{+}, F^{−} the open positive and negative halfspace

in the complement of F ∈ F

⪯ Lusztig’s partial order on the alcoves β ↑ A, β ↓ A defined in the proof of 4.5 X the lattice of integral weights l(x) the length of the element x of a Coxeter group w_{0} the longest element of W r the length of w_{0} Π the fundamental box Π_{λ} the translated box λ + Π λ(A) the lower left corner of the box containing A Ǎ is obtained by moving A with w_{0} around λ(A) Â A ↦ Â is the inverse of A ↦ Ǎ H the Hecke algebra d its standard involution d : H ↦ \overline{H} i, a two involutive antiautomorphisms of H,

defined in the proof of Theorem 2.7

φ-linear defined right above Theorem 3.8 L the ring of Laurent-polynomials L = ℤ[v, v^{−1}] C_{s} self-dual generators C_{s} = v(T_{s} + 1) of H

Some Hecke modules with duality, standard basis, self-dual basis and transition matrix

(H, H_{x}, \underline{H}_{x}, h_{x,y}) the Hecke algebra itself, H_{x} = v^{l(x)}T_{x} (M, M_{x}, \underline{M}_{x}, m_{x,y}) Deodhar’s parabolic analogs, (N , N_{x}, \underline{N}_{x}, n_{x,y}) for x, y ∈ W^{f} (M^{∗}, M^{x}, \underline{M}^{x}, m^{x,y}) The dual Hecke-modules, (N^{∗}, N^{x}, \underline{N}^{x}, n^{x,y}) for x, y ∈ W^{f}

From Section 4 we identify W^{0} ≅ A^{+} and write M_{A}, \underline{M}_{A}, m_{A,B} … for M_{x}, \underline{M}_{x}, m_{x,y} … (P, A, \underline{P}_{A}, p_{A,B}) Lusztig’s periodic Hecke module P^{∘} submodule of P admitting a duality q_{A,B} the (renormalized) generic polynomials w ∗ A a new action of W̃ on A,

w ∗ (λ + B) = (wλ) + B for λ ∈ X, B ⊂ Π

⟨w⟩ action of w ∈ W̃ on P^{∘}, see 4.10

References

[AJS94] Henning Haahr Andersen, Jens Carsten Jantzen, and Wolfgang Soergel, Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: Independence of p, Astérisque 220 (1994), 1–320.

[And86] Henning Haahr Andersen, An inversion formula for the Kazhdan-Lusztig polynomials for affine Weyl groups, Advances in Mathematics 60 (1986), 125–153.

[And92] Henning Haahr Andersen, Tensor products of quantized tilting modules, Comm. Math. Physics 149 (1992), 149–159.

[And96] Henning Haahr Andersen, Filtrations and tilting modules, Aarhus Preprint No 7, 1996.

[BGS96] Alexander A. Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, JAMS 9 (1996), no. 2, 473–527.

[Bou81] Nicolas Bourbaki, Groupes et algèbres de Lie, vol. 4-6, Masson, 1981.

[Deo87] Vinay V. Deodhar, On some geometric aspects of Bruhat orderings ii. the parabolic analogue of Kazhdan-Lusztig polynomials, Journal of Algebra 111 (1987), 483–506.

[Deo91] Vinay V. Deodhar, Duality in parabolic set up for questions in Kazhdan-Lusztig theory, Journal of Algebra 142 (1991), 201–209.

[Deo94] Vinay V. Deodhar, A brief survey of Kazhdan-Lusztig theory and related topics, Proceedings of Symposia in Pure Mathematics 56, Part 1, AMS, 1994, pp. 105–124.

[Don93] Stephen Donkin, On tilting modules for algebraic groups, Mathematische Zeitschrift 212 (1993), 39–60.

[Dou90] J.M. Douglass, An inversion formula for relative Kazhdan-Lusztig polynomials, Comm. Algebra 18 (1990), 371–387.

[Hum90] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge studies in advanced mathematics, vol. 29, Cambridge University Press, 1990.

[Kan87] M. Kaneda, On the inverse Kazhdan-Lusztig polynomials for affine Weyl groups, J. Reine Angew. Math. 381 (1987), 116–135.

[Kat85] S.I. Kato, On the Kazhdan-Lusztig polynomials for affine Weyl groups, Adv. in Math. 55 (1985), 103–130.

[KL79] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Inventiones 53 (1979), 191–213.

[Lus80a] George Lusztig, Hecke algebras and Jantzen’s generic decomposition patterns, Advances in Mathematics 37 (1980), 121–164.

[Lus80b] George Lusztig, Some problems in the representation theory of finite Chevalley groups, Proceedings of Symposia in Pure Mathematics 37, AMS, 1980, pp. 313–317.

[Lus91] George Lusztig, Intersection cohomology methods in representation theory, Proceedings of the International Congress of Mathematicians 1990 (Ichiro Satake, ed.), Springer, 1991, pp. 155–174.

[Mil] Dragan Miličić, Localization and representation theory of reductive Lie groups, Book in preparation.

Universität Freiburg, Mathematisches Institut, Eckerstraße 1, D-79104 Freiburg, Germany

E-mail address: soergel@mathematik.uni-freiburg.de