On Harish-Chandra induction and restriction for modules of Levi subgroups

R. B. Howlett and G. I. Lehrer

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

§1 Introduction and notation

Let G be a connected reductive algebraic group defined over a finite field F_{q} of characteristic p, and let G be the (finite) group of F_{q}-points of G. The group G has a split BN-pair at characteristic p, as defined in Richen [R] or Carter [C]. In fact B and N may be chosen as the groups of F_{q}-rational elements of, respectively, a rational Borel subgroup B of G, and the normalizer N of a rational maximal torus T with T ⊆ B. Every parabolic subgroup P of G (that is, a subgroup which contains a conjugate of B) has a Levi decomposition P = LU, where U is the unipotent radical (maximal normal p-subgroup) of P, and L∩U = {1}. The groups L which arise in such decompositions are known as Levi subgroups of parabolic subgroups of G; they are the groups of rational points of rational Levi subgroups of rational parabolic subgroups of G. Further details are given below.

If P = LU is a Levi decomposition of P and V is an RL-module, where R is any commutative ring with 1, then we have the notion of Harish-Chandra induction:

R_{L⊂P}^{G} (V) = RG ⊗_{RP} Ṽ

where Ṽ is the lift of V from L to P. (See Springer [S, §4.6] or Digne and Michel [DM, Chapter 4].) It is known that when R is a field of characteristic zero or coprime to |G|, then the RG-module isomorphism class of R_{L⊂P}^{G} (V) depends only on V: it is independent of the particular parabolic subgroup P of which L is a Levi component. This is proved using Harish-Chandra’s theory of cuspidal representations [HC]. However, when R is a field of characteristic p there are simple examples to show that the isomorphism class of R_{L⊂P}^{G} (V) depends on P as well as L.

Our purpose is to extend the independence result to apply whenever p is invertible in R. Moreover, we provide an explicit isomorphism from R_{L⊂P}^{G} (V) to R_{L⊂P′}^{G}(V) whenever P and P′ are two parabolic subgroups which both have L as a Levi factor. This isomorphism defines a natural equivalence between the functors R_{L⊂P}^{G} and R_{L⊂P′}^{G}. En route we prove a corresponding result about the truncation functor, which is dual to R_{L⊂P}^{G}. Our method is similar in spirit and technique to our work on endomorphism algebras [HL].

For the remainder of this paper, R will denote a commutative ring with 1 such that p is invertible in R. Let P = LU as above. We shall work in the slightly more general setting of (RL, A)-bimodules, where A is a fixed R-algebra. If V is an (RL, A)-bimodule, then Ṽ (the lift to P of V) is an (RP, A)-bimodule, and R_{L⊂P}^{G} (V) is an (RG, A)-bimodule in an obvious way. In particular, R_{L⊂P}^{G} (RL) is an (RG, RL)-bimodule. Moreover, for an arbitrary (RL, A)-bimodule V we have that Ṽ ≅ R̃L ⊗_{RL} V, and hence

R_{L⊂P}^{G} (V) = RG ⊗_{RP} Ṽ ≅ RG ⊗_{RP} R̃L ⊗_{RL} V ≅ R_{L⊂P}^{G} (RL) ⊗_{RL} V.

Note that the (RG, RL)-bimodule R_{L⊂P}^{G} (RL) is isomorphic to RG[U], where [U] ∈ RG is defined by [U] = |U|^{−1} ∑_{u∈U} u.

Our main theorem is as follows.

(1.1) Theorem Let P = LU and P′ = LU′ be Levi decompositions of parabolic subgroups P and P′ of G. Let R be a commutative ring with 1 such that p is invertible in R, and let V be an (RL, A)-bimodule, where A is any R-algebra. Then there is an isomorphism of (RG, A)-bimodules

f_{P,P′}: R_{L⊂P}^{G} (V) → R_{L⊂P′}^{G}(V).

Moreover, f_{P,P′} is independent of V, in the sense that there is an (RG, RL)- bimodule isomorphism ϕ_{P,P′}: RG[U] → RG[U′] which is independent of V, such that

f_{P,P′} = ϕ_{P,P′} ⊗ id_{V}: RG[U] ⊗_{RL} V → RG[U′] ⊗_{RL} V.

It is of course trivial that ϕ_{P,P′} ⊗ id_{V} is an isomorphism if ϕ_{P,P′} is. So it will suffice to prove the existence of ϕ_{P,P′}.

There is a dual version of Theorem (1.1) as follows. For each subgroup H of G let M_{R}(H) be the category of all (RH, A)-bimodules. As well as the Harish-Chandra induction functor R_{L⊂P}^{G}: M_{R}(L) → M_{R}(G), there is the Harish-Chandra restriction functor ^{∗}R_{L⊂P}^{G}: M_{R}(G) → M_{R}(L) defined by

∗R_{L⊂P}^{G} (M) = Inv_{U}(M),

where Inv_{U}(M) = { m ∈ M | um = m for all u ∈ U }. This functor, which is sometimes called “truncation” from G to L, is left adjoint to R_{L⊂P}^{G}.

It is easily seen that the maps f_{P,P′} of Theorem (1.1) define a natural transformation from the functor R_{L⊂P}^{G} to the functor R_{L⊂P′}^{G}, since if ψ: V_{1} → V_{2} is a morphism of (RL, A)-bimodules, then the following diagram commutes:

RG[U] ⊗ V_{1} ⟶^{ϕ_{P,P′} ⊗id_{V_{1}}} RG[U′] ⊗ V_{1}
↓ id_{RG[U]}⊗ψ  ↓ id_{RG[U′]}⊗ψ
RG[U] ⊗ V_{2} ⟶^{ϕ_{P,P′} ⊗id_{V_{2}}} RG[U′] ⊗ V_{2}.

Theorem (1.1) says that this natural transformation of functors is a natural equivalence.

(1.2) Theorem With the notation as above, the functor ^{∗}R_{L⊂P}^{G} is naturally equivalent to^{∗}R_{L⊂P′}^{G}. In particular, the isomorphism class of^{∗}R_{L⊂P}^{G} (V) is independent of P.

This is in fact a consequence of Theorem (1.1) and the adjointness relationship that holds between the induction and restriction functors. Specifically, let R and ^{∗}R be the Harish-Chandra induction and restriction functors corresponding to the inclusion L ⊂ P of a Levi subgroup in a parabolic subgroup, and let R′ and ^{∗}R′ be the Harish-Chandra induction and restriction functors corresponding to the inclusion of the same Levi subgroup L in another parabolic subgroup P′. Let A and A′ be R-algebras, let V be an (RL, A)-bimodule and let M be an (RG, A′)-bimodule. Using the adjoint nature of R and ^{∗}R, one easily checks that the canonical map Hom_{RL}(V,^{∗}R(M))→Hom_{RG}(R(V), M) is an (A,A′)-bimodule isomorphism. Similarly, Hom_{RL}(V,^{∗}R′(M)) ≅ Hom_{RG}(R′(V), M) as (A, A′)-bimodules. But Theorem (1.1) says that R(V) ≅ R′(V) as (RG, A)-bimodules; so we deduce that Hom_{RG}(R(V), M) ≅ Hom_{RG}(R′(V), M) as (A, A′)-bimodules, whence Hom_{RL}(V,^{∗}R(M)) ≅ Hom_{RL}(V,^{∗}R′(M)) as (A, A′)-bimodules. Applying this in the case A = RL and V = RL yields the required (RL, A′)- isomorphism of ^{∗}R(M) and ^{∗}R′(M) (since it is trivial that Hom_{RL}(RL, N) is isomorphic to N whenever N is an (RL, A′)-bimodule).

Since we give an explicit formula for the isomorphism R(V) → R′(V), we can also give the isomorphism ^{∗}R(M) → ^{∗}R′(M) explicitly, and we do so in the remarks following the statement of Theorem (2.5).

Before commencing the proof of (1.1), we establish our notation. The Borel subgroup B of G factorizes as T U, where U is a normal p-subgroup of B, and T = B ∩ N has order prime to p. Let Φ be the root system of Weyl group W = N/T (which is a Euclidean reflection group), and for each α ∈ Φ let r_{α} ∈ W be the corresponding reflection. For notational simplicity, whenever w ∈ W we shall also write w for the corresponding element of a fixed set of representatives of the cosets of T in N; this should cause no confusion. Whenever S ⊆ W we shall write S_{N} for the set of representatives in N of the elements w ∈ S. Let Π ⊆ Φ be the set of simple roots corresponding to B, and let Φ = Φ^{+} ∪ Φ^{−} be the corresponding division of the root system into positive and negative roots. The length, l(w), of an element w ∈ W, is the least l such that w = s_{1}s_{2} … s_{l} with s_{i} ∈ { r_{α} | α ∈ Π } for each i. For each J ⊆ Π let W_{J} be the subgroup of W generated by { r_{α} | α ∈ J }, and let w_{J} be the element of W_{J} of maximal length. Thus w_{Π} is the element of W which is usually called w_{0}. For each α ∈ Π the root subgroup, X_{α}, is defined by X_{α} = U ∩ U^{w_{0}r_{α}}. We also define U_{α} = U ∩ U^{r_{α}}. It can then be shown that

U = X_{α}U_{α}  and  X_{α} ∩ U_{α} = {1}.

More generally, for each w ∈ W let X_{w} = U ∩ U^{w_{0}w} and U_{w} = U ∩ U^{w}. Then U = X_{w}U_{w} and X_{w} ∩ U_{w} = {1}. In the case w = w_{J}, where J ⊆ Π, define X_{J} = X_{w} and U_{J} = U_{w}. Define also P_{J} = ⋃_{w∈W_{J}} BwB, and L_{J} = ⋃_{w∈W_{J}} B_{J}wB_{J}, where B_{J} = T X_{J}. Then P_{J} and L_{J} are subgroups of G, and P_{J} = L_{J}U_{J} with L_{J} ∩ U_{J} = {1}. Moreover, U_{J} is the unipotent radical of P_{J}. The P_{J} are the only subgroups of G containing B; the parabolic subgroups of G are, by definition, the conjugates of the P_{J}, and P = LU is a Levi decomposition of the parabolic subgroup P if and only if L = L_{J}^{g} and U = U_{J}^{g} for some g ∈ G and J ⊆ Π (where H^{g} = g^{−1}Hg for H ⊆ G and g ∈ G).

After completing this manuscript the authors learned that a different proof of Theorem (1.1) has been obtained by R. Dipper and J. Du (“Harish-Chandra vertices”, preprint, University of Sydney (1992)).

§2 Initial reductions

Let P = LU be a Levi decomposition of the parabolic subgroup P, and let V be an RL-module. For each g ∈ G we have a Levi decomposition P^{g} = L^{g}U^{g}, and V becomes a module for L^{g} via the composition L^{g} ⟶^{γ} L → End_{R}(V), where γ(x) = gxg^{−1}. If we denote the resulting L^{g}-module by V^{g}, then it is clear that the map ϕ: η ↦ ηg is a linear isomorphism from the (RG, RL)-bimodule F G[U] to the (RG, RL^{g})-bimodule RG[U^{g}], and that ϕ ⊗ id: R_{L⊂P}^{G} (V) → R_{L^{g}⊂P^{g}}^{G}(V^{g}) is an isomorphism of RG-modules. Thus in (1.1) we may assume that P = P_{J}, L = L_{J} and U = U_{J}. Then P′ will be an arbitrary parabolic subgroup of G with L_{J} as a Levi subgroup.

(2.1) Lemma Let P′ be a parabolic subgroup of G such that L_{J} is a Levi subgroup of P′. Then there is an element w ∈ W and a subset K ⊆ Π such that

(i) K = wJ, (ii) P_{K}^{w} = P′, (iii) L_{K}^{w} = L_{J}.

Proof. Let P′ be the parabolic subgroup of G of which P′ is the group of rational points, and define L_{J} similarly. Then L_{J} is a Levi subgroup of P′. Since the maximal torus T is contained in L_{J} we have T ⊆ P′, and hence there exists an element w′ ∈ W such that P′ = P_{K}^{w′} for some (unique) K ⊆ Π. But then L_{K} and w′L_{J}(w′)^{−1} are both Levi subgroups of P_{K} which contain T; hence they are equal, and it follows that w′Φ_{J} = Φ_{K}.

Now w′J is a simple system in Φ_{K}, whence there is an element w′′ ∈ W_{K} such that w′′w′J = K. Defining w = w′′w′, the properties (i), (ii) and (iii) are easily verified.

We have now reduced (1.1) to the following:

(2.2) Theorem Let J ⊆ Π and w ∈ W be such that K = wJ ⊆ Π. Then there is an (RG, RL_{J})-bimodule isomorphism ψ: RG[U_{J}] → RG[U_{K}^{w}].

The composite of right multiplication by w followed by the inverse of the map ψ of (2.2) yields an isomorphism ϕ: RG[U_{K}] → RG[U_{J}]. Translating the bimodule condition on ψ into a condition on ϕ, we obtain the following equivalent version of (2.2):

(2.3) Theorem Let J ⊆ Π and w ∈ W be such that K = wJ ⊆ Π. Then there is a linear isomorphism ϕ: R[U_{K}] → RG[U_{J}] satisfying

ϕ(η[U_{K}]θ) = ηϕ([U_{K}])θ^{w}  (2.3.1)

for all η ∈ RG and θ ∈ RL.

We prove (2.3) by producing an explicit isomorphism ϕ, namely, right multiplication by [U_{K}]w[U_{J}]:

(2.4) Theorem Let the notation be as in Theorem (2.3). Then the map ϕ: RG[U_{K}] → RG[U_{J}] defined by

ϕ(ξ) = ξ[U_{K}]w[U_{J}] (for all ξ ∈ RG[U_{K}])

is a linear isomorphism satisfying (2.3.1).

It is easily seen that the map ϕ defined in (2.4) satisfies the condition (2.3.1), since clearly ϕ(ηξ) = ηϕ(ξ) (for η ∈ RG, ξ ∈ RG[U_{K}]) as ϕ is a right multiplication, while for θ ∈ RL_{K} we have θ^{w} ∈ RL_{J}, whence θ^{w} commutes with [U_{J}]. Thus

ϕ(ξθ) = ξθ[U_{K}]w[U_{J}] = ξ[U_{K}]wθ^{w}[U_{J}] = ξ[U_{K}]w[U_{J}]θ^{w} = ϕ(ξ)θ^{w}

(for all ξ ∈ RG[U_{K}] and θ ∈ RL_{K}).

Thus the proof of Theorem (2.4) reduces to showing that ϕ is a linear isomorphism, and for this it suffices to prove that right multiplication by a suitable ξ_{0} ∈ [U_{J}]RG[U_{K}] is inverse to ϕ. So Theorem (2.4) reduces to the following:

(2.5) Theorem With notation as in (2.4), there exists ξ_{0} ∈ [U_{J}]RG[U_{K}] such that ξ_{0}[U_{K}]w[U_{J}] = [U_{J}] and [U_{K}]w[U_{J}]ξ_{0} = [U_{K}].

Note also that Theorem (1.2) is a direct consequence of Theorem (2.5). With the notation as above, let M be an (F G, A)-bimodule and let P = P_{J} and P′ = P_{K}^{w} be two parabolic subgroups having L = L_{J} as a Levi subgroup (cf. (2.1)). Then clearly ^{∗}R_{L⊂P}^{G} (M) ≅ [U_{J}]M and ^{∗}R_{L⊂P′}^{G}(M) ≅ [U_{K}^{w}]M (as (RL, A)-bimodules), and to prove Theorem (1.2) we require a linear isomorphism ψ: [U_{J}]M → [U_{K}^{w}]M which respects the actions of L and A. Let ψ be left multiplication by [U_{K}^{w}]. This respects the L-action since L centralizes [U_{K}^{w}] in RG, and (as a left multiplication) obviously respects the right action of A. But it is clear from (2.5) that the map from [U_{K}^{w}]M to [U_{J}]M given by left multiplication by ξ_{0}w (with ξ_{0} as in (2.5)) is inverse to ψ.

For all J, K ⊆ Π, define

D(K, J) = { w ∈ W | wJ ⊆ Φ^{+} and w^{−1}K ⊆ Φ^{+} },

and let D(J) = D(∅, J). Clearly, if J, K and w satisfy wJ = K then w ∈ D(J, K). The next two results are well-known (see [C]):

(2.6) Lemma Each (W_{K}, W_{J}) double coset in W contains a unique element of D(K, J), and each (P_{K}, P_{J}) double coset in G a unique element of D(K, J)_{N}. Moreover, if t ∈ D(K, J) then

(i) W_{K} ∩ tW_{J}t^{−1} = W_{K∩tJ}, and (ii) each element w ∈ W_{K}tW_{J} is uniquely expressible in the form w_{1}tw_{2},

with w_{1} ∈ W_{K} ∩ D(K ∩ tJ) and w_{2} ∈ W_{J}. Furthermore, when w is so expressed, l(w) = l(w_{1}) + l(t) + l(w_{2}).

(2.7) Lemma Let w ∈ W. Then (i) l(w_{0}w) = l(w_{0}) − l(w), (ii) each element of UwU is uniquely expressible in the form uwx with

u ∈ U and x ∈ X_{w}, and (iii) if v ∈ W is such that l(vw) = l(v)+l(w) then U_{vw} ⊆ U_{w} and U_{vw} ⊆ U_{v}^{w}.

§3 Proof of the main lemma

This section contains the heart of the proof. We start with a technical lemma.

(3.1) Lemma Let J, K ⊆ Π and w ∈ D(K, J), and let M = K ∩ wJ and M′ = w^{−1}M. Then [U_{K}]w[U_{J}] = [U_{M}]w[U_{M′}].

Proof. Since M ⊆ K it follows from Lemma (2.7) that U_{K} ⊆ U_{M}. Hence U_{M} = U_{K}X_{K} ∩ U_{M} = U_{K}(X_{K} ∩ U_{M}). Since w_{0}w_{K}(M) ⊆ Π we see that

l(w_{0}w_{K}w_{M}) = l(w_{0}w_{K}w_{M}w_{K}^{−1}w_{0}^{−1}) + l(w_{0}w_{K}),

and, by Lemma (2.7), U_{w_{0}w_{K}w_{M}} is contained in U^{w_{0}w_{K}}. Thus

X_{K} ∩ U_{M} = U ∩ (U ∩ U^{w_{0}w_{K}w_{M}})^{w_{M}} ⊆ U ∩ (U^{w_{0}w_{K}})^{w_{M}} = U_{w_{0}w_{K}w_{M}} .

Clearly w_{K}w_{M} ∈ D(M) ∩ W_{K}; so by Lemma (2.6),

l(w_{K}w_{M}w) = l(w_{K}w_{M}) + l(w).

Hence l(w_{0}w_{K}w_{M}) = l(w_{0}w_{K}w_{M}w) + l(w^{−1}), and Lemma (2.7) yields

U_{w_{0}w_{K}w_{M}}^{w} ⊆ U_{w_{0}w_{K}w_{M} w}.

Moreover, Lemma (2.6) also gives l(w_{K}w_{M}ww_{J}^{−1}) = l(w_{K}w_{M}w) + l(w_{J}), and hence l(w_{0}w_{K}w_{M}w) = l(w_{0}w_{K}w_{M}ww_{J}^{−1}) + l(w_{J}). So by Lemma (2.7), U_{w_{0}w_{K}w_{M} w} ⊆ U_{J}, and combining the various containments we have proved we find that (X_{K} ∩ U_{M})^{w} ⊆ U_{w_{0}w_{K}w_{M}}^{w} ⊆ U_{w_{0}w_{K}w_{M} w} ⊆ U_{J}.

It follows now that

[U_{M}]w[U_{J}] = [U_{K}][X_{K} ∩U_{M}]w[U_{J}] = [U_{K}]w[(X_{K} ∩U_{M})^{w}][U_{J}] = [U_{K}]w[U_{J}],

and by this same result applied to w^{−1} ∈ D(J, K) we can also conclude that [U_{M′}]w^{−1}[U_{K}] = [U_{J}]w^{−1}[U_{K}]. Applying the antiautomorphism of RG defined by g ↦ g^{−1} for all g ∈ G, we see that [U_{K}]w[U_{M′}] = [U_{K}]w[U_{J}], and hence

[U_{K}]w[U_{J}] = [U_{M}]w[U_{J}] = [U_{M}][U_{K}]w[U_{J}] = [U_{M}][U_{K}]w[U_{M′}] = [U_{M}]w[U_{M′}]

as required.

We also need the following elementary result:

(3.2) Lemma Let K ⊆ Π, and let w, v ∈ W be such that J′ = wK and J = v^{−1}K are both subsets of Π. Suppose also that l(wv) = l(w) + l(v). Then U_{K} = (U_{J′}^{w} ∩ U_{K})(U_{K} ∩ U_{J}^{v^{−1}}).

Proof. Since l(ww_{K}) = l(w)+l(w_{K}) we have U_{ww_{K}} ⊆ U_{K}, by Lemma (2.7). So U_{K} = U_{K} ∩ U_{ww_{K}} X_{ww_{K}} = U_{ww_{K}} (U_{K} ∩ X_{ww_{K}}). Since wv(J) ⊆ Φ^{+} it follows that

l(ww_{K}v) = l(wvw_{J}) = l(wv) + l(w_{J}) = l(w) + l(v) + l(w_{K}),

and therefore that l(w_{0}ww_{K}) = l(w_{0}ww_{K}v) + l(v^{−1}). So Lemma (2.7) yields that X_{ww_{K}} ⊆ U^{v^{−1}}. Hence U_{K} ⊆ U_{ww_{K}} (U_{K} ∩ U^{v^{−1}}).

Again by Lemma (2.7),

U_{ww_{K}} = U_{K} ∩ U_{ww_{K}} = U_{K} ∩ U_{w_{J′} w} ⊆ U_{K} ∩ U^{w},

and so we have shown that U_{K} ⊆ (U_{K} ∩ U^{w})(U_{K} ∩ U^{v^{−1}}). But

U_{K}^{w^{−1}} ∩ U ⊆ U ∩ U^{w^{−1}w_{J′}} ⊆ U_{J′},

and so U_{K} ∩U^{w} = U_{K} ∩U_{J′}^{w}. Similar reasoning gives U_{K} ∩U^{v^{−1}} = U_{K} ∩U_{J}^{v^{−1}}, completing the proof.

We now come to the main lemma.

(3.3) Lemma Let J ⊆ Π and let w ∈ W be such that wJ ⊆ Π. Then [U_{J}] ∈ [U_{J}]RG[U_{wJ}]w[U_{J}].

Proof. We prove this by induction on the cardinality of J. Let n be a nonnegative integer such that (3.3) holds for all J ⊆ Π with |J| < n; we shall show that (3.3) also holds when |J| = n. Observe that if J = ∅ (that is, n = 0) then the induction hypothesis automatically applies, and so the proof below starts the induction as well as providing the induction step.

Let J ⊆ Π with |J| = n and w ∈ W with wJ ⊆ Π, and let M be a proper subset of J. Then wM ⊆ Π, and since |M| < n we have [U_{M}] ∈ [U_{M}]RG[U_{wM}]w[U_{M}], by the induction hypothesis. But (3.1) gives U_{wM}wU_{M} = U_{wM}wU_{J}, so that

\begin{aligned}[U_{M}] ∈ [U_{M}]RG[U_{wM}]w[U_{J}] & = [U_{J}][U_{M}]RG[U_{wM}][U_{wJ}]w[U_{J}] \\ & ⊆ [U_{J}]RG[U_{wJ}]w[U_{J}]. \end{aligned}

Furthermore, if l ∈ L_{J} is arbitrary, then

[U_{J}]RG[U_{wJ}]w[U_{J}]l = [U_{J}]RG[U_{wJ}]l^{w^{−1}}w[U_{J}] = [U_{J}]RG[U_{wJ}]w[U_{J}],

and we have established the following:

(3.3.1) With n as above, if J ⊆ Π with |J| = n, and if w ∈ W satisfies wJ ⊆ Π, then we have [U_{M}]l ∈ [U_{J}]RG[U_{wJ}]w[U_{J}] whenever l ∈ L_{J} and M is properly contained in J.

Following Howlett [H], we define v(α, K) = w_{K∪{α}}w_{K} when K ⊂ Π and α ∈ Π − K. It is shown in [H] that v(α, K)K ⊆ K ∪ {α}, and, furthermore, whenever wK ⊆ Π there exist v(α_{i}, K_{i}) such that K = K_{1} and v(α_{i}, K_{i})(K_{i}) = K_{i+1} for all i, and

w = v(α_{n}, K_{n})v(α_{n−1}, K_{n−1}) … v(α_{1}, K_{1})

with lengths adding.

It is clear that to complete the proof of (3.3) it suffices to prove the following:

(3.3.2) Let n be as above. Then [U_{J}] ∈ [U_{J}]RG[U_{wJ}]w[U_{J}] whenever J ⊆ Π with |J| = n, and w is such that wJ ⊆ Π.

Proof of (3.3.2). Our proof of (3.3.2) proceeds by induction on l(w).

The case l(w) = 0 is trivial. Consider next the case w = v(α, J), where α ∈ Π − J. Put K = v(α, J)J and S = [U_{J}]RG[U_{K}]v(α, J)[U_{J}]. Let β ∈ Π be such that K ∪ {β} = J ∪ {α}, so that v(β, K) = v(α, J)^{−1}, and consider the element ξ = [U_{J}]v(β, K)[U_{K}]v(α, J)[U_{J}] ∈ S. Observe that

ξ = |U_{K}|^{−1} ∑_{u∈U_{K}}[U_{J}]v(β, K)uv(α, J)[U_{J}].

We investigate the individual summands in this expression.

Fix an element u ∈ U_{K}. Since v(β, K)uv(α, J) ∈ P_{J∪{α}} there exists a unique element t ∈ W_{J∪{α}} ∩ D(J, J) and elements l_{1}, l_{2} ∈ L_{J} such that v(β, K)uv(α, J) ∈ U_{J}l_{1}tl_{2}U_{J}. We show that

[U_{J}]v(β, K)uv(α, J)[U_{J}] \{ \begin{matrix}∈ S & if t ≠ 1 \\ = [U_{J}] & if t = 1. \end{matrix}  (∗)

Since v(β, K)uv(α, J) ∈ U_{J}l_{1}tl_{2}U_{J} we have

[U_{J}]v(β, K)uv(α, J)[U_{J}] = l_{1}[U_{J}]t[U_{J}]l_{2}.

By Lemma (3.1), [U_{J}]t[U_{J}] = [U_{J}][U_{wM}]t[U_{M}], where M = J ∩ t^{−1}J. If J ≠ tJ it follows from (3.3.1) that [U_{M}]l_{2} ∈ S. Multiplying on the left by [U_{J}]l_{1}[U_{wM}]t we conclude that [U_{J}]l_{1}[U_{wM}]t[U_{M}]l_{2} ∈ [U_{J}]RGS ⊆ S, and we have proved (∗) in case tJ ≠ J.

Consider next those cases for which tJ = J. (Note that this holds in particular if J = ∅; hence the following argument starts the induction for (3.3).) Since t ∈ W_{J∪{α}} it follows from the results of [H] that either t = 1 or t = v(α, J). In the latter case, K = v(α, J)J = tJ = J, and

[U_{J}]v(β, K)uv(α, J)[U_{J}] = [U_{J}]l_{1}l_{2}^{v(α,J)}[U_{J}]v(α, J)[U_{J}] ∈ S,

so that (∗) holds when t = v(α, J). This leaves just the case t = 1, which occurs when u ∈ U_{K} ∩ v(α, J)P_{J}v(α, J)^{−1}. However, if z ∈ W_{J} is not 1 then U_{K} has empty intersection with v(α, J)BzBv(α, J)^{−1}, since U_{K}v(α, J) ⊆ Bv(α, J)B and v(α, J)BzB ⊆ Bv(α, J)zB. Hence

U_{K} ∩ v(α, J)P_{J}v(α, J)^{−1} = U_{K} ∩ v(α, J)Bv(α, J)^{−1}

= U ∩ U^{w_{K}} ∩ U^{w_{L}w_{K}} (where L = J ∪ {α})

= U_{w_{L}},

so that there are exactly |U_{wL}| elements u ∈ U_{K} for which t = 1. Moreover, we have v(β, K)uv(α, J) ∈ U_{J} for all these elements u, and it follows that [U_{J}]v(β, K)uv(α, J)[U_{J}] = [U_{J}], which establishes (∗). We deduce also that

ξ = \frac{|U_{w_{L}}|}{|U_{K}|}[U_{J}] + \frac{1}{|U_{K}|} ∑[U_{J}]v(β, K)uv(α, J)[U_{J}]

where the sum is over those u ∈ U_{K} for which t ≠ 1. Since ξ and all the terms in the sum are in S, we conclude that [U_{J}] ∈ S. This proves (3.3.2) in the case w = v(α, J).

To complete the proof of (3.3.2), let w ∈ W be such that wJ ⊆ Π, where J ⊆ Π and |J| = n, and assume that [U_{K}] ∈ [U_{K}]RG[U_{w′K}]w′[U_{K}] whenever l(w′) < l(w) and K is a subset of Π of cardinality n such that w′K ⊆ Π. We may write w = w′v(α, J), with lengths adding. If we define K = v(α, J)J then the induction hypothesis gives [U_{K}] ∈ [U_{K}]RG[U_{wJ}]w′[U_{K}], and so

[U_{K}]v(α, J)[U_{J}] ∈ [U_{K}]RG[U_{wJ}]w′[U_{K}]v(α, J)[U_{J}].

But it follows from Lemma (3.2) that

w[U_{K}]v(α, J) = [U_{wJ} ∩ U_{K}^{w^{−1}}]wv[U_{K}^{v(α,J)} ∩ U_{J}],

and therefore

[U_{K}]v(α, J)[U_{J}] ∈ [U_{K}]RG[U_{wJ}]w′v(α, J)[U_{J}] = [U_{K}]RG[U_{wJ}]w[U_{J}]. It follows that

\begin{aligned}[U_{J}] ∈ [U_{J}]RG[U_{K}]v(α, J)[U_{J}] & ⊆ [U_{J}]RG[U_{K}]RG[U_{wJ}]w′[U_{K}]v(α, J)[U_{J}] \\ & ⊆ [U_{J}]RG[U_{wJ}]w[U_{J}], \end{aligned}

completing the proof of (3.3.2) and hence that of (3.3).

§4 Completion of the proof

It follows from Lemma (3.3) that whenever w ∈ W and J, K ⊆ Π with wJ = K, there exists ξ_{0} ∈ [U_{J}]RG[U_{K}] with ξ_{0}[U_{K}]w[U_{J}] = [U_{J}]. By analogous reasoning, or by making use of the antiautomorphism g ↦ g^{−1}, we may also conclude that there exists ξ_{1} ∈ [U_{J}]RG[U_{K}] with [U_{K}]w[U_{J}]ξ_{1} = [U_{K}]. Clearly ξ_{1} = ξ_{0}, and Theorem (2.5) follows. As noted above, this implies (2.4), (2.2), (1.1) and (1.2). This completes our proof.

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