On the integral group algebra of a finite algebraic group

R. B. Howlett and G. I. Lehrer

Department of Pure Mathematics, University of Sydney, N.S.W. 2006, Australia.

§1 Introduction and notation

Let G be the group of F_{q}-rational points of a connected reductive algebraic group defined over F_{q} and let U be a maximal unipotent subgroup of G. In this note we give three explicit embeddings of the ring of endomorphisms E = End_{ℤG}(Ind_{U}^{G}(1)) into the integral group ring ℤG. The embeddings have properties related to certain ℤG modules, and all have the same image. Thus they imply the existence of certain automorphisms of E.

There is a canonical surjection σ: E → \overline{E} = End_{ℤG}(Ind_{B}^{G}(1)), and σ ⊗ 1: E ⊗_{ℤ} ℤ[|H|^{−1}] → \overline{E} ⊗_{ℤ} ℤ[|H|^{−1}] has a left inverse τ. (Here B is a Borel subgroup and H a maximal split torus in G such that B = HU.) One of our embeddings composed with τ gives the embedding of \overline{E} ⊗_{ℤ} ℤ[|H|^{−1}] which we constructed in [3]. Thus the present work may be thought of as an extension of the results of that paper.

Another direction for applications of this work is the construction of idempotents (analogous to Steinberg’s idempotent for St_{G}) in F G (F a field) which are not in the Hecke algebra H(G, B) (or even H(G, U)). This could provide explicit constructions of representations of G in characteristic p (where p|q).

We begin here with some generalities which will establish our notation. Let R be a commutative (unital) ring, A an R-algebra and a ∈ A. Then aA is a (right) A-module, and we write E = End_{A}(aA). We have

(1.1) There is an algebra homomorphism λ: Ã = { x ∈ A | xaA ⊆ aA } → E defined by x ↦ (λ_{x}: y ↦ xy) (x, y ∈ aA).

If a is an idempotent and ϕ ∈ E then ϕ(y) = ϕ(ay) = ϕ(a)y for any y ∈ aA; hence ϕ = λ_{ϕ(a)}, and ϕ(a) = ϕ(a)a ∈ aAa. Thus we have

(1.2) If a is an idempotent then (i) λ is surjective (ii) There is an algebra homomorphism μ: E → aAa ⊆ aA ⊆ A such that

λμ = id_{E}.

(1.3) Definition If there is an algebra homomorphism μ: E → A such that λμ = id_{E}, we say that E is embedded in A, and that μ is an embedding of E into A.

Let G be a finite group and take A = RG, the group ring over R. For any subset S ⊆ G denote by [S] the sum [S] = ∑_{s∈S} s ∈ RG, and consider the case a = [K], where K is a subgroup of G. If G = ∐_{i} Kx_{i}K is the double coset decomposition of G with respect to K, the map T_{i}: [K] ↦ [Kx_{i}K] defines an element of E, and it is well known that

(1.4) E is the free R-module with basis {T_{i}}.

Moreover, in this case we have

(1.5) λ: aA → E is surjective.

For here T_{i} = λ_{y}, where y = ∑_{j} k_{j}x_{j} and Kx_{i}K = ∐_{j} k_{j}x_{i}K. However, in general there is no embedding μ in the sense of (1.3).

Here we shall be concerned with the case R = ℤ, G the group of F_{q}- rational points of a connected reductive F_{q}-group and K = U, its maximal unipotent subgroup. We shall show that there is an embedding of E = End_{ℤG}[U]ℤG into ℤG, and in the course of its study some automorphisms of E will play a role. Our embedding has properties connected with those of the Steinberg representation of ℤG (see §3 below). For this observation we are indebted to Okuyama.

We remark finally that the nature of our formulae are strongly suggestive that there is an underlying geometric explanation for the facts we present here, related to the geometric structure of the Bruhat cells and Schubert varieties of the underlying algebraic group.

§2 The case of a finite algebraic group

For the rest of this note, G, U, B = HU and E will be as in the last section (G the group of rational points of a connected reductive group defined over F_{q}, and so on). If N = N_{G}(H) then N/H = W is the Weyl group of G, with simple reflection set S, length function l(w) (w ∈ W) and longest element w_{0}. Denote by Φ and Π the corresponding root system and set of simple roots (respectively). For elementary facts concerning the structure of G the reader is referred to [2], [1] or [3]. We assume that the representatives ẇ ∈ N (w ∈ W) have been chosen so that if w = w_{1}w_{2} with l(w) = l(w_{1}) + l(w_{2}) then ẇ = ẇ_{1}ẇ_{2}. (See [9] for the proof that this is possible.) We shall require the following well known fact (see, for example, [3] Lemma 2.8):

(2.1) Lemma Let a ∈ Φ with U_{a} ⊆ U the corresponding root subgroup of G, and suppose that n ∈ N maps onto r_{a}, the reflection in W corresponding to a. Then each element x of U_{a}^{#} (= U_{a} − {1}) has a unique expression x = h_{n}(x)f_{n}(x)ng_{n}(x), where h_{n} is a map U_{a}^{#} → H and f_{n}, g_{n} are bijections U_{a}^{#} → U_{−a}^{#}.

An easy computation proves

(2.2) Lemma Use the notation of (2.1) and let t ∈ H. Then for u ∈ U_{a}^{#}, (i) h_{tn}(u) = h_{n}(u)t^{−1} (ii) f_{tn}(u) = tf_{n}(u)t^{−1} and g_{tn}(u) = g_{n}(u).

If r ∈ W is a reflection then we will write h, f and g for hṙ, fṙ and gṙ if there is no danger of ambiguity. We also adopt the notation V = ẇ_{0}^{−1}Uẇ_{0} and V_{a} = U_{−a}.

(2.3) Proposition (i) In the above the notation, if a is a simple root and r = r_{a} then we have (in ℤG)

[B ∩ V ṙV] = ∑_{u∈U_{a}^{#}} k(u)u = ∑_{v∈V_{a}^{#}} vṙf(g^{−1}(v)),

where k(u) = h(u)^{−1}. (ii) We have ṙ^{2} ∑_{u∈U_{a}^{#}} h(u) = ∑_{u∈U_{a}^{#}} k(u).

Proof. Part (i) follows easily from [3,§2] and (2.1).

(ii) Inverting k(u) = f(u)ṙg(u) gives u^{−1}k(u)^{−1} = g(u)^{−1}ṙ^{−1}f(u)^{−1}. Multiplying both sides on the left by k(u) and conjugating by ṙ^{2} we obtain

(2.3.1)  h(ṙ^{−2}k(u)u^{−1}k(u)^{−1}ṙ^{2}) = ṙ^{−2}k(u)  (u ∈ U_{a}^{#}).

Since k(u)uk(u)^{−1} = f(u)ṙg(u)k(u)^{−1} with uniqueness of expression on the right, it follows that k(u)uk(u)^{−1} determines f(u) and hence u; thus as u runs over U_{a}^{#} so does k(u)uk(u)^{−1}. Now sum (2.3.1) over u ∈ U_{a}^{#} to obtain (ii).

Since the sum in (2.3) (ii) will recur later we give it a name: with the notation as in (2.3) write

(2.3.2)  ηṙ = ∑_{u∈U_{a}^{#}} k(u) = ṙ^{2} ∑_{u∈U_{a}^{#}} h(u) ∈ ℤG.

(2.4) Lemma With the notation as in (2.1), let u ∈ U_{a}^{#}. Then we have g(g(u)ṙ)= uṙ (where x^{y} = y^{−1}xy).

Proof. We have u = h(u)f(u)ṙg(u). Make g(u) the subject of this formula, conjugate by ṙ and collect terms as follows:

g(u)ṙ = (ṙ^{−2}h(u)^{−1})(h(u)f(u)^{−1}h(u)^{−1})ṙ(ṙ^{−1}uṙ).

The result follows.

The double coset decomposition of G with respect to U is well-known; we have G = ∐_{n∈N} UnU. Hence it follows from the generalities in §1 above that E = End_{ℤG}([U]ℤG) has a ℤ-module basis { T_{n} | n ∈ N } defined as in (1.3). The rules for multiplying these T_{n} were determined by Yokonuma [10], and may be expressed as follows. (In writing the relations we regard the length function of W as lifted to N—that is, for n ∈ N we define l(n) = l(n̄) where n̄ is the image of n in W.)

(2.5) Proposition (Yokonuma [10]) The algebra E = End_{ℤG}([U]ℤG) has a ℤ-basis { T_{n} | n ∈ N } satisfying

(i) If l(n_{1}n_{2}) = l(n_{1}) + l(n_{2}) then T_{n_{1}}T_{n_{2}} = T_{n_{1}n_{2}}. (ii) If l(n) = 1 then T_{n}^{2} = q_{a}T_{n^{2}} + T_{n} ∑_{u∈U_{a}^{#}} T_{h_{n}(u)n^{2}}, where a ∈ Π is the

root corresponding to n̄ ∈ S and q_{a} = |U_{a}|. (Note that n^{2} ∈ H.)

Proof. (i) follows easily from the definition of T_{n} (namely, T_{n}[U] = [UnU]) and Chevalley’s refinement of the Bruhat decomposition for G, by induction on l(n_{2}). Note that the case l(n_{2}) = 0 reads T_{n}T_{h} = T_{nh} (h ∈ H, n ∈ N). For (ii) one uses the Chevalley decomposition together with (2.1) above. We leave the details to the reader.

(2.6) Definition For n ∈ N define the element γ_{n} ∈ ℤG by

γ_{n} = ∑_{t∈W}(−1)^{l(t)}[Bt ∩ V nV].

Note that from (2.3) of [3] it follows that Bt ∩ V nV = ∅ unless t ≤ n̄ (in the Bruhat order on W). Hence (2.6) may be written

(2.6)′  γ_{n} = ∑_{\begin{matrix}t≤n̄ \\ t∈W \end{matrix}}(−1)^{l(t)}[Bt ∩ V nV].

(2.7) Theorem The elements γ_{n} form a Z-basis of a subalgebra Γ of ℤG, and the map ϕ: T_{n} → γ_{n} is an isomorphism of Z-algebras.

Proof. Since the support of γ_{n} is contained in V nV and G is the disjoint union of { V nV | n ∈ N }, the γ_{n} are clearly Z-linearly independent. In view of (2.5) it therefore suffices to prove

(2.7.1) γ_{n_{1}}γ_{n_{2}} = γ_{n_{1}n_{2}} if n_{1}, n_{2} ∈ N with l(n_{1}n_{2}) = l(n_{1}) + l(n_{2}),

and

(2.7.2) γ_{n}^{2} = q_{a}γ_{n^{2}} + γ_{n} ∑_{u∈U_{a}^{#}} γ_{h(u)n^{2}} if a ∈ Π and n = ṙ_{a}, where h = h_{n} is the function defined in (2.1).

We first prove (2.7.1). If l(n_{2}) = 0 then n_{2} ∈ H and it follows that γ_{n_{2}} = [B ∩ V n_{2}V] = [B ∩ n_{2}V] = n_{2}; thus γ_{n_{1}}γ_{n_{2}} = γ_{n_{1}}n_{2} = γ_{n_{1}n_{2}}, since [Bt ∩ V n_{1}V]n_{2} = [Bt ∩ V n_{1}n_{2}V] for all t ∈ W. Observe now that if we can prove (2.7.1) for n_{2} = ṙ_{a} with a ∈ Π then we will be finished; for in general we may write n_{2} = hṙ_{1}ṙ_{2} … ṙ_{l} with h ∈ H, r_{i} ∈ S for each i and l = l(n_{2}), and then induction on l will give γ_{n_{2}} = γ_{h}γṙ_{1}γṙ_{2} … γṙ_{l} and γ_{n_{1}}γ_{n_{2}} = γ_{n_{1}}γ_{h}γṙ_{1} … γṙ_{l} = γ_{n_{1}hṙ_{1}...r_{l}} = γ_{n_{1}n_{2}} as required.

Thus we turn to the proof of (2.7.1) in the case where n_{2} = ṙ with r = r_{a} ∈ S and n_{1} = n ∈ N with l(nṙ) = l(n) + 1. First observe that γṙ = [B ∩ V ṙV] − [Br ∩ V ṙV] (by (2.6)′), and from (2.3) we have [B ∩ V ṙV] = ∑_{u∈U_{a}^{#}} k(u)u. Moreover, by a slight extension of Lemma 2.4 of [3], [Br ∩ V ṙV] = [U_{a}ṙ] = [ṙV_{a}]. Hence

(2.7.3) γ_{n}γṙ = ∑_{l(tr)>l(t)}(−1)^{l(t)}([Bt∩V nV]−[Btr∩V nV])( ∑_{u∈U_{a}^{#}}k(u)u−[U_{a}ṙ]).

Now writing B_{a} = HU_{a}, we have BtB_{a} = Bt since l(tr_{a}) > l(t). Hence

[Bt ∩ V nV]( ∑_{u∈U_{a}^{#}} k(u)u) = ∑_{u∈U_{a}^{#}}[Bt ∩ V nV k(u)u]

= ∑_{v∈V_{a}^{#}}[Bt ∩ V nV vṙf(g^{−1}(v))] (by (2.3))

= [Bt ∩ V nV ṙV_{a}^{#}]

since the sets V nV ṙv are disjoint for distinct v. Hence we have

(2.7.4)  [Bt ∩ V nV]( ∑_{u∈U_{a}^{#}} k(u)u) = [Bt ∩ V nṙV] − [Bt ∩ V nV ṙ].

Similarly,

[Bt ∩ V nV][U_{a}]ṙ = ∑_{u∈U_{a}}[Bt ∩ V nV u]ṙ

= ∑_{u∈U_{a}}[Btr ∩ V nV uṙ]

= ∑_{v∈V_{a}}[Btr ∩ V nV ṙv]

= [Btr ∩ V nV ṙV_{a}].

Thus

(2.7.5)  [Bt ∩ V nV][U_{a}]ṙ = [Btr ∩ V nṙV].

The next term arising in the expansion of (2.7.3) is

[Btr ∩ V nV]( ∑_{v∈V_{a}^{#}} vṙf(g^{−1}(v))) = ∑_{v∈V_{a}^{#}}[Btrv ∩ V nV v]ṙf(g^{−1}(v))

= ∑_{v∈V_{a}^{#}}[Btr ∩ V nV]ṙf(g^{−1}(v))

= [Btr ∩ V nV]ṙ[V_{a}^{#}]. (2.7.6) The last term is

(2.7.7)  [Btr ∩ V nV][U_{a}ṙ] = [Btr ∩ V nV]ṙ[V_{a}].

Adding the right hand sides of (2.7.4)–(2.7.7) with appropriate signs gives [Bt ∩ V nṙV] − [Btr ∩ V nṙV], and substituting into (2.7.3) completes the proof of (2.7.1).

To complete the proof of Theorem (2.7) we now have to prove the formula (2.7.2). For this, we have

γ2 ; ṙ = [B∩V ṙV]^{2}−[B∩V ṙV][Br∩V ṙV]−[Br∩V ṙV][B∩V ṙV]+[Br∩V ṙV]^{2},

and we proceed to compute the four terms on the right.

[B ∩ V ṙV]^{2} = [B ∩ V ṙV_{a}] ∑_{u∈U_{a}^{#}} k(u)u

= ∑_{u∈U_{a}^{#}}[B ∩ V ṙV_{a}k(u)u]

= ∑_{u∈U_{a}^{#}}[B ∩ V ṙV_{a}f(u)ṙg(u)] (by (2.1))

= ∑_{v∈V_{a}^{#}}[B ∩ V ṙV_{a}ṙv] (since g is a bijection from U_{a}^{#} to V_{a}^{#})

= ∑_{v∈V_{a}^{#}}[B ∩ V ṙ^{2}v] + ∑_{v′,v∈V_{a}^{#}}[B ∩ V ṙv′ṙv]

= (q_{a} − 1)ṙ^{2} + ∑_{\begin{matrix}u∈U_{a}^{#} \\ v∈V_{a}^{#} \end{matrix}} ṙ^{2}[B ∩ V uv]

= (q_{a} − 1)ṙ^{2} + ∑_{\begin{matrix}u∈U_{a}^{#} \\ v∈V_{a} \end{matrix}} ṙ^{2}[B ∩ V h(u)f(u)ṙg(u)v] − ∑_{u∈U_{a}^{#}} ṙ^{2}[B ∩ V u]

= (q_{a} − 1)ṙ^{2} + ṙ^{2} ∑_{u∈U_{a}^{#}} h(u)[B ∩ V ṙV] − ṙ^{2}[U_{a}^{#}]

= (q_{a} − 1)ṙ^{2} + ∑_{u∈U_{a}^{#}} k(u)[B ∩ V ṙV] − ṙ^{2}[U_{a}^{#}] (by (2.3) (ii)). (2.7.8)

Next we have

(2.7.9)  [B ∩ V ṙV][Br ∩ V ṙV] = ∑_{u∈U_{a}^{#}} k(u)u[U_{a}ṙ] = ∑_{u∈U_{a}^{#}} k(u)[U_{a}]ṙ

and

[Br ∩ V ṙV][B ∩ V ṙV] = [ṙV_{a}] ∑_{u∈U_{a}^{#}} f(u)ṙg(u)

= [ṙV_{a}][ṙV_{a}^{#}], (2.7.10)

while finally

(2.7.11)  [Br ∩ V ṙV]^{2} = [ṙV_{a}][ṙV_{a}].

Now add the terms (2.7.8)–(2.7.11) with appropriate signs, obtaining

\begin{aligned}γ2 ; ṙ & = q_{a}ṙ^{2} + ∑_{u∈U_{a}^{#}} k(u)γṙ \\ & = q_{a}ṙ^{2} + ηṙγṙ \end{aligned}

in the notation of (2.3.2). But γṙ trivially commutes with ṙ^{2} and with γ2 ; ṙ. It follows (since γṙ is invertible in ℚG) that γṙ commutes with ηṙ. Hence again using (2.3) (ii), we have

γ2 ; ṙ = q_{a}γ_{}_{ṙ^{2}} + γṙ ∑_{u∈U_{a}^{#}} γ_{h(u)ṙ^{2}},

which is (2.7.2). This completes the proof of Theorem (2.7).

The map ϕ of (2.7) is not an embedding in the sense of (1.3). This follows easily from the next proposition, which deals with the effect of left multiplication by γ_{n} on the module [U]ℤG.

(2.8) Proposition Let a be a simple root and r = r_{a}. Then left multiplication by γṙ is the endomorphism ∑_{u∈U_{a}^{#}} T_{k(u)} − Tṙ of [U]ℤG.

Proof. We have

γṙ[U] = ( ∑_{u∈U_{a}^{#}} k(u)u − [U_{a}ṙ] )[U] = ∑_{u∈U_{a}^{#}} k(u)[U] − [UṙU]

= ( ∑_{u∈U_{a}^{#}} T_{k(u)} − Tṙ)[U].

Since left multiplication obviously commutes with the ℤG action on [U]ℤG, the result follows.

(2.8)′ Corollary The endomorphism Tṙ of [U]ℤG is given by left multiplication by ∑_{u∈U_{a}^{#}} k(u) − γṙ ∈ Γ.

The results of this section may be summarized in

(2.9) Theorem Let E = End_{ℤG}[U]ℤG where U is a maximal unipotent subgroup of the finite algebraic group G. (For a more precise definition of G see the beginning of this section.) Then

(i) There is a map μ: E → ℤG satisfying T_{h} ↦ γ_{h} (for h ∈ H) and Tṙ ↦ ηṙ − γṙ (for r = r_{a} ∈ S, with ηṙ and γṙ as defined in (2.3.2) and (2.6) respectively), such that μ is an embedding of E in ℤG in the sense of (1.3) (that is, T ∈ E is given by left multiplication by μ(T)), and μ(E) = Γ, the Z-linear span of the elements γ_{n} (n ∈ N).

(ii) There is an (involutory) automorphism α of E defined by α(T_{h}) = T_{h} for h ∈ H and α(Tṙ) = ∑_{u∈U_{a}^{#}} T_{k(u)} − Tṙ for r = r_{a} ∈ S.

Proof. (i) By Proposition (2.8) we see that the elements γ_{h} and γṙ induce endomorphisms of [U]ℤG by left multiplication, and since these elements generate Γ the same holds for all γ ∈ Γ. Thus there is a map λ: Γ → E such that λ(γ) is left multiplication by γ. It follows from (2.8) and (2.8)′ that T_{h}, Tṙ ∈ E are respectively given by left multiplication by μ(T_{h}), μ(Tṙ) ∈ Γ as defined in the statement. Now since the T_{h} and the Tṙ generate E, it follows that every element of E is realized by left multiplication by some element of Γ; that is, λ is surjective. Since Γ and E both have Z-rank |N| it follows that λ is injective too, whence μ(T) = λ^{−1}(T) defines our embedding μ. (ii) The map ϕ: T_{n} ↦ γ_{n} is an isomorphism E → Γ (Theorem (2.7)) and the automorphism α is simply ϕ^{−1}μ. It is trivially an involution.

Remark. Notice that in the proof of (2.9) (i) we have only used the fact that Γ is a subalgebra of ℤG. This is the thrust of Theorem (2.7).

§3 Connection with Steinberg’s representation

Maintaining the notation of §2, let B = HU and write St = ∑_{w∈W} ε_{w}[Bw] where ε_{w} = (−1)^{l(w)}. The elements { (St)u | u ∈ U } are linearly independent since the coefficient of ẇ_{0}u_{1} in ∑_{u∈U} λ_{u}(St)u is λ_{u_{1}}ε_{w_{0}} (where w_{0} is the longest element of W). Moreover, Steinberg’s method (see Theorem 1 of [8]) may be applied to show

(3.1) Proposition The set { (St)u | u ∈ U } is a Z-basis for the right ℤG-module (St)ℤG.

We shall begin by proving

(3.2) Lemma Let V = ẇ_{0}^{−1}Uẇ_{0} as in §2. Then the right ℤG-module [V]ℤG ⊗_{ℤ} (St)ℤG is cyclic, generated by [V] ⊗ St.

Proof. Consider the submodule M generated by [V] ⊗ St. Applying ẇ_{0} we obtain ε_{w_{0}}[V]ẇ_{0} ⊗ St, and since for u ∈ U we have [V]ẇ_{0}u = [V]ẇ_{0} we may apply elements of U and obtain that [V]ẇ_{0} ⊗ (St)ℤG ⊆ M (by (3.1)). The result now follows trivially.

(3.2)′ Corollary There is a ℤG-isomorphism ℤG → [V]ℤG ⊗_{ℤ} (St)ℤG defined by 1 ↦ [V] ⊗ St.

Proof. Since ℤG is free 1 ↦ [V] ⊗ St defines a map. It is surjective by (3.2), and an isomorphism since the two modules have the same Z-rank and are Z-free.

Now End_{ℤG} ℤG = { λ_{β} | β ∈ ℤG }, where λ_{β}(ξ) = βξ for ξ ∈ ℤG, and β ↦ λ_{β} is an isomorphism ℤG → End_{ℤG} ℤG. Hence from (3.2)′ we deduce

(3.3) There is an isomorphism ℤG → End_{ℤG}([V]ℤG ⊗_{ℤ} (St)ℤG) given by β ↦ ν(β), where ν(β): ([V] ⊗ (St))ξ ↦ ([V] ⊗ (St))βξ for all ξ, β ∈ ℤG.

Let E′ = End_{ℤG}[V]ℤG. This has Z-basis { T′_{n} | n ∈ N }, where T′_{n}([V]) = [V nV] for each n ∈ N. Since ẇ_{0}[V] = [U]ẇ_{0} there is an isomorphism ω: [V]ℤG → [U]ℤG given by ω(x) = ẇ_{0}x for all x ∈ [V]ℤG, and therefore

(3.4) There is an isomorphism ω̃: E → E′ given by

ω̃(ζ) = ω^{−1}ζω (ζ ∈ End_{ℤG}[U]ℤG)

where (as above) ω is left multiplication by ẇ_{0}.

A simple calculation shows that

(3.4.1)  ω̃(T_{n}) = T′_{˙w_{0}^{−1}nẇ_{0}}.

Now there is a natural embedding E′ → End_{ℤG}([V]ℤg ⊗_{ℤ} (St)ℤG) given by θ ↦ θ ⊗ id_{(St)ℤG}. So by (3.3) there is an algebra monomorphism E′ → ℤG such that θ ↦ β, where θ ⊗ id_{(St)ℤG} = ν(β). Pulling back via ω̃ gives the following statement.

(3.5) Theorem We maintain the above notation. Then

(i) We have T ′ ; ṙ ⊗ id_{(St)ℤG} = ν(γṙ); that is,

(T ′ ; ṙ[V] ⊗ St)ξ = ([V] ⊗ St)γṙξ

for ξ ∈ ℤG, a ∈ Π and r = r_{a} ∈ S. (ii) The map T′_{n} ↦ γ_{n} defines an isomorphism E′ = End_{ℤG}[V]ℤG → Γ

which pulls back via ω̃ (see (3.4)) to the isomorphism η: E → Γ, where η(T_{n}) = γ_{ẇ_{0}nẇ_{0}}.

Proof. (i) It suffices to show that ([V] ⊗ St)γṙ = [V ṙV] ⊗ St. Now

(3.5.1)  ([V] ⊗ St)γṙ = ∑_{u∈U_{a}^{#}}([V] ⊗ St)k(u)u − ∑_{u∈U_{a}}([V] ⊗ St)uṙ.

Using (2.1)we have

(3.5.2)  ∑_{u∈U_{a}^{#}}([V] ⊗ St)k(u)u = ∑_{u∈U_{a}^{#}}[V]ṙg(u) ⊗ (St)u.

The second term on the right side of (3.5.1) is

(3.5.3)  −([V]ṙ ⊗ St) + ∑_{u∈U_{a}^{#}}[V]uṙ ⊗ (St)uṙ.

Now to compute (St)uṙ for u ∈ U_{a}, take w ∈ W with l(wr) > l(w). Then BwB_{a} = Bw, and so [Bw] = [X][B_{a}] for some transversal X. Thus

([Bw] − [Bwr])uṙ = [X]([B_{a}r] − [B_{a}])uṙ
= [X]([B_{a}r] − [B_{a}rf(u)ṙg(u)ṙ])
= [X][B_{a}r](1 − g(u)ṙ)
= [X]([B_{a}r] − [B_{a}])(1 − g(u)ṙ)
= ([Bw] − [Bwr])(g(u)ṙ − 1).

Therefore

(3.5.4)  (St)uṙ = (St)(g(u)ṙ − 1).

Thus, using Lemma (2.4), we obtain

∑_{u∈U_{a}^{#}}[V]uṙ ⊗ (St)(g(u)ṙ − 1) = ∑_{u∈U_{a}^{#}}[V]ṙg(u) ⊗ (St)(u − 1).

Hence the expression (3.5.3) becomes

−([V]ṙ ⊗ St) − ∑_{u∈U_{a}^{#}}[V]ṙg(u) ⊗ St + ∑_{u∈U_{a}^{#}}[V]ṙg(u) ⊗ (St)u

(3.5.5) = −[V ṙV] ⊗ St + ∑_{u∈U_{a}^{#}}[V]ṙg(u) ⊗ (St)u.

Now substitute into (3.5.1) using (3.5.2), obtaining

([V] ⊗ Stγṙ = ∑_{u∈U_{a}^{#}}[V]ṙg(u) ⊗ (St)u + [V ṙV] ⊗ St − ∑_{u∈U_{a}^{#}}[V]ṙg(u) ⊗ (St)u

= [V ṙV] ⊗ St

= (T ′ ; ṙ ⊗ id)([V] ⊗ St)

which completes the proof of (i). (ii) Since T′_{h} = ν(h) = ν(γ_{h}) (for h ∈ H) and E′ is generated by the T′_{h} together with the T ′ ; ṙ for r ∈ S, the result follows from (i) and the discussion immediately preceding (3.5) (and Theorem (2.7)).

(3.6) Corollary The algebra E has another automorphism, namely, the map β defined by β(T_{n}) = T_{˙w_{0}^{−1}nẇ_{0}}. Moreover, if the transversal {ẇ | w ∈ W } is chosen so that ẇ = ẇ_{1}ẇ_{2} whenever w = w_{1}w_{2} with l(w) = l(w_{1}) + l(w_{2}) then β is involutory.

Proof. The automorphism β is given by β = ϕ^{−1}η where ϕ is the isomorphism E → Γ of (2.7). The involutory nature of β follows from the fact that the stated assumptions on the ẇ imply that ẇ_{0}^{2} is in the centre of N. This is proved as follows. Let r ∈ S and write r′ = w_{0}rw_{0}^{−1} ∈ S. Then w_{0} = w′r′ = rw′ where w′ = rw_{0} has length l(w_{0}) − 1. By assumption ẇ_{0} = ẇ′ṙ′ = ṙẇ′, whence ẇ_{0}(ṙ′)^{−1} = (ṙ)^{−1}ẇ_{0} and ẇ_{0}^{−1}ṙẇ_{0} = ṙ′. By symmetry ẇ_{0}^{−1}ṙ′ẇ_{0} = ṙ, and hence ẇ_{0}^{2} commutes with ṙ. Since ẇ_{0}^{2} lies in H (which is abelian) it also commutes with all elements of H, and hence with all elements of N, since N is generated by H and the elements ṙ for r ∈ S.

Remark. The algebra Γ ⊆ ℤG appears in yet a third way, since another computation (different from the one above) shows that in the left ℤG-module ℤG[V] ⊗ ℤG(St) = ℤG([V] ⊗ St) ≅ ℤG we have

(3.7)  γṙ([V] ⊗ St) = [V ṙV] ⊗ St.

§4 Connection with principal series

The principal series of G is realized by the module [B]ℤG, which is a submodule of [U]ℤG and is fixed by each of the endomorphisms in E (since T_{n}[B] = [UnU][H] = [BnB]); moreover, each endomorphism in the algebra \overline{E} = End_{ℤG}[B]ℤG is the restriction of an endomorphism in E. Thus

(4.1) There is an epimorphism ρ: E → \overline{E} given by ρ(T_{n}) = \overline{T}_{n} = T_{n}|_{[B]ℤG}.

Clearly \overline{T}_{n} = \overline{T}_{n′} if and only if n̄ = \overline{n′} ∈ W, and \overline{E} has Z-basis consisting of \{ T_{w} | w ∈ W \}, where T_{w} = \overline{T}_{n} for any n ∈ N such that n̄ = w.

For any integer a and any Z-module M denote by M_{a} the ℤ[a^{−1}]-module M ⊗_{ℤ} ℤ[a^{−1}].

(4.2) Theorem (i) The map τ: \overline{T}_{n} ↦ |H|^{−1} ∑_{h∈H} T_{h}T_{n} defines a mono-

morphism \overline{E}_{|H|} → E_{|H|}. (ii) For w ∈ W let γ_{w} = |H|^{−1}[H]γ_{n}, where n is any element of N such

that n̄ = w. Then γ_{w} = |H|^{−1} ∑_{t∈W} ε_{w}[Bt ∩ DwD], where D = HV, and γ_{w} = h_{w} in the notation of [3], (1.8).

(iii) The map (ϕ ⊗ id)τ: \overline{E}_{|H|} → Γ_{|H|} is the isomorphism of [3], Theo-

rem (1.7). (iv) The map (μ⊗id)τ: \overline{E}_{|H|} → ℤG_{|H|} is an embedding in the sense of (1.3),

and coincides with the embedding of (1.11′) of [3].

(v) The involution T_{w} ↦ T̂_{w} of [3] is realized as the restriction of α ⊗ id (see (2.7) above) to τ(\overline{E}).

All the above statements are straightforward consequences of the results of §2 above.

This puts the results of [3] into a natural setting, and shows why the stipulation that one should be able to divide by |H| in the case of Ind_{B}^{G}(1) is a natural one. It also makes explicit that the obstruction to carrying out the embeddings of [3] in arbitrary characteristic are cohomological (cf. Tits [9]).

§5 Concluding remarks

(5.1) In §4 of [3] certain almost-idempotents e_{J} ∈ \overline{E} ⊗_{ℤ} ℤ(|H|^{−1}) were constructed (for each J ⊆ Π), which, over a field k of characteristic p (where p|q), were used to give an algorithmic decomposition of Ind_{B}^{G}(1) into indecomposables, each of which has irreducible socle. We hope in a later work to produce idempotents e_{J}(λ) ∈ E ⊗_{ℤ} k, one for each character λ: H → k^{∗}, which will play an analogous role for the representation Ind_{U}^{G}(1). Since the socle of this representation is the direct sum of all the irreducible representations of G, each occurring with multiplicity one, this would be a significant construction.

(5.2) The sets Bt ∩ V nV in the overlying algebraic group G are locally closed in G, and one might ask whether our multiplication formulae for the alternating sums of their “characteristic functions” reflect some geometric facts concerning composition of sheaves on “Schubert-like varieties”, in analogy with the case (cf. [7]) of the ordinary Hecke algebra (cf. [4]).

(5.3) The indecomposable summands of Ind_{B}^{G}(λ) (or Ind_{U}^{G}(1)) in characteristic p can be constructed homologically from coefficient systems on the Tits building (see [5] and [6]). A knowledge of the e_{J}(λ) ((5.1) above) would assist in the analysis of these direct summands.

References

1. N. Bourbaki, Groupes et algèbres de Lie, Chap. IV, V et VI, Hermann, Paris (1968).

2. C. Chevalley, Séminaire sur le classification des groupes de Lie algébriques, Paris (1956–58).

3. R. B. Howlett and G. I. Lehrer, ‘Embeddings of Hecke Algebras in group algebras’, J. Alg. 105 (1987) 159–174.

4. N. Iwahori, ‘On the structure of the Hecke ring of a Chevalley group over a finite field’, J. Fac. Sci. Univ. Tokyo 10 (1964) 214–236.

5. G. I. Lehrer, ‘The spherical building and regular semisimple elements’, Bull. Austral. Math. Soc. 27 (1983) 361–379.

6. M-T. Schmidt, ‘Beziehungen zwischen Homologie-Darstellungen und der Hauptserie endlicher Chevalley-Gruppen’, Bonner Math. Schr. Bonn (1986).

7. T. A. Springer, ‘Quelques applications de la cohomologie d’intersection’, Astérisque 92–93 (Sém. Bourbaki 589) (1982) 249–273.

8. R. Steinberg, ‘Prime power representations of finite linear groups II’, Can. J. Math. 9 (1957) 347–351.

9. J. Tits, ‘Normalisateurs de tores I. Groupes de Coxeter étendus’, J. Alg. 4 (1966) 96–116.

10. T. Yokonuma, ‘Sur le commutant d’une représentation d’un groupe de Chevalley finis’, J. Fac. Sci. Univ. Tokyo 15 (1968) 115–129; II ibid. 16 (1969), 65–82.