THOMAS J. HAINES. 1. Notation

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1 ON HECKE ALGEBRA ISOMORPHISMS AND TYPES FOR DEPTH-ZERO PRINCIPAL SERIES THOMAS J. HAINES Abstract. These lectures describe Hecke algebra isomorphisms and types for depth-zero principal series blocks, a.k.a. Bernstein components R s(g) for s = s χ = [T, eχ] G, where χ is a depth-zero character on T(O). (Here T is a split maximal torus in a p-adic group G.) We follow closely the treatment of A. Roche [Ro] with input from D. Goldstein [Gol] and L. Morris [Mor]. We give an elementary proof that (I, ρ χ) is a type for s χ, in the sense of Bushnell-Kutzko [BK]. This is a very special case of a result of Roche [Ro]. Our method is to imitate Casselman s proof of Borel s theorem on unramified principal series (the case χ = 1 of the present theorem). In contrast to the situation for general principal series blocks (see [Ro]), in the depth-zero case there is no restriction on the residual characteristic of F. 1. Notation We let F denote an arbitrary p-adic field with ring of integers O, and residue field k F. Let q denote the cardinality of k F. Write for a uniformizer. Let G denote a connected reductive group, defined and split over O. Fix an F-split maximal torus T and a Borel subgroup B containing T; assume T and B are defined over O. Let T = T(O) denote the maximal compact subgroup of T(F). Let Φ X (T) resp. Φ X (T) denote the set of roots resp. coroots for G,T. Let U resp. U denote the unipotent radical of B resp. the Borel subgroup B T opposite to B. The symbol I will stand for an Iwahori subgroup of G(F), which we shall assume it is in good position with respect to T: the alcove a in the building for G(F) which is fixed by I is contained in the apartment corresponding to T. Let dx denote a Haar measure on G. Denote the group of unramified characters of G(F) by X ur (G) (see [BD] or [Be92] for the definition). Let R(G) denote the category of smooth representations of G(F). Let L denote an F-Levi subgroup of G (by definition, L = C G (A L ) for some F-split torus A L in G). Let P = LN denote an F-parabolic subgroup, that is, a parabolic subgroup defined over F, with unipotent radical N and with L as a Levi factor. Let σ denote any smooth representation of L, and define the normalized parabolic induction by i G P(σ) = Ind G P(δ 1/2 P Partially supported by NSF grant FRG , and a University of Maryland Graduate Research Board Semester Award. The author is grateful to the University of Bonn and the University of Chicago for providing financial support and venues for these lectures. 1 σ),

2 2 T. Haines where δ P (l) := det(ad(l);lie(n(f))) F. Here F denotes the normalized absolute value on F. Throughout these notes, we will frequently write G (resp. B,T, etc.) when we really mean G(F) (resp. B(F), T(F), etc.). 2. Bernstein decomposition (review) A cuspidal pair (L,σ) consists of an F-Levi subgroup L of G, together with a supercuspidal representation σ of L(F). The group G = G(F) acts by conjugation on cuspidal pairs: g (L,σ) = ( g L, g σ), where g L = glg 1 and g σ( ) = σ(g 1 g). Denote by (L,σ) G the G-conjugation class of (L,σ). Let (L,σ) denote a cuspidal pair. We say (L 1,σ 1 ) is inertially equivalent to (L 2,σ 2 ) if there exists g G(F) and χ X ur (L 2 ) such that g L 1 = L 2 and g σ 1 χ = σ 2. Let s = [L,σ] G denote the inertial equivalence class of (L,σ) (with respect to G). Note that s depends only on (L,σ) G. Also s is a union of G-conjugacy classes of cuspidal pairs. Fact: For π R(G) irreducible, there exist a (unique up to G-conjugacy) cuspidal pair (L,σ) such that π is a subquotient of i G P (σ). Here P = LN is an F-parabolic with unipotent radical N which has L as a Levi factor. We call the class (L,σ) G as above the supercuspidal support of π. Denote by R s (G) the full subcategory of R(G) whose objects are the representations π each of whose irreducible subquotients has supercuspidal support belonging to the inertial class s. Once we fix a cuspidal pair (L,σ) in s, we may reformulate the condition for π to belong to R s (G) as: every irreducible subquotient of π is a subquotient of some i G P (σχ), χ Xur (L). Theorem (Bernstein decomposition). R(G) = s R s(g). Definition An s-type is a pair (K,ρ) consisting of a compact open subgroup K G together with an irreducible smooth representation ρ : K End C (W) such that an irreducible π R(G) belongs to R s (G) iff π K ρ. Now let ρ be any irreducible smooth representation of K, on a vector space W. We define e ρ H(G) = Cc (G,dx) by dx(k) 1 dim(ρ)tr W (ρ(x 1 )), x K e ρ (x) = 0, x / K. For any irreducible smooth representations ρ,ρ of K, we have e ρ dx e ρ = δ ρ,ρ e ρ, where δ ρ,ρ {0,1} vanishes unless ρ and ρ are equivalent. This is an exercise using the Schur orthogonality relations on the group K. In particular, e ρ is an idempotent of the algebra H(G). If ρ = 1 (the trivial character) we write e K in place of e ρ. For any (π,v ) R(G), denote by V ρ the ρ-isotypical component of V. We have V ρ = e ρ V. Also, we let V [ρ] = H(G) V ρ, the G-submodule of V generated by V ρ. Below we will often write π ρ in place of V ρ.

3 Hecke algebra isomorphisms and types for depth-zero principal series 3 We define R ρ (G) to be the full subcategory of R(G) whose objects (π,v ) satisfy V = V [ρ]. There is a functor (2.0.1) R ρ (G) e ρ H(G)e ρ -Mod (π,v ) π ρ. Proposition If (K, ρ) is an s-type, then (2.0.1) is an equivalence of categories. Moreover, in that case R s (G) = R ρ (G) as subcategories of R(G). We will postpone the proof of this proposition to section Depth-zero principal series blocks Example. Consider an Iwahori subgroup I in good position with respect to the torus T (this means that I fixes an alcove a in the apartment of the building for G(F) corresponding to T). Also, for any Borel subgroup B = TU containing T, with opposite Borel B = TU, we have the Iwahori decomposition (3.0.2) I = I U T I U, where I U := U I, I U := U I, and T := T(O) = T I. The inertial class s := [T,1] G indexes the Iwahori block R s (G). A famous theorem of Borel asserts that an irreducible π R(G) is a constituent of an unramified principal series i G B (η), η Xur (T), if and only if π I 0. That is, (I,1) is an s-type. This is a special case of the theorem we will prove below (Theorem 3.0.2). It turns out that e I H(G)e I = H(G,I), the Iwahori-Hecke algebra (see below). In conjunction with the Proposition 2.0.3, we thus recover the finer result of Borel which asserts that π π I gives an equivalence of categories between the Iwahori block and the category H(G,I)-Mod. Fix a character χ : T C. Definition We say χ is depth-zero if χ factors through the quotient T T(k F ) (and we denote the factoring T(k F ) C also by χ). Choose any extension of χ to a character χ : T(F) C. Consider the inertial class s := [T, χ] G. Since s depends only on the W-orbit of χ, we may also write s χ for s. Let I be an Iwahori in good position relative to T, as above. Let I + denote the prounipotent radical of I. There is an obvious isomorphism T/ T I + I/I +

4 4 T. Haines so that χ determines a character ρ = ρ χ : I C, which is trivial on I +. In terms of the Iwahori decomposition (3.0.2), ρ is given by for u I U, t 0 T, and u I U. ρ(u t 0 u) = χ(t 0 ), Theorem If s = s χ as above, then (I,ρ) is an s-type. We shall prove this by imitating Casselman s proof of Borel s theorem on unramified principal series. One crucial ingredient is the theory of Hecke algebra isomorphisms for depth-zero principal series types, which we will review in section Proof of Proposition We are in the general situation, where (K, ρ) is a smooth irreducible representation on a vector space W (ie. ρ is not necessarily a character). Lemma Fix an inertial class s. (i) (K,ρ) is an s-type ind ρ := c Ind G K ρ is a generator for R s(g), i.e., ind ρ R s (G) and Hom G (ind ρ, π) 0 for all π 0 in R s (G). (ii) In that case R s (G) = R ρ (G) as subcategories of R(G). In particular R ρ (G) is closed under extensions and subquotients. Proof. First, by Frobenius reciprocity (cf. [Ro],(7.1)) we have Hom G (ind ρ,π) = Hom K (ρ,π). This implies that indρ is a projective object in R(G). (It is also true that indρ is finitelygenerated as a G-module.) Now let us prove (i). ( ): Suppose (π,v ) R s is non-zero. Since all irreducible subquotients of π are also in R s (hence contain ρ) and representations of K are completely reducible, it follows that Hom K (ρ,π) 0 and hence Hom G (ind ρ,π) 0. Next we claim that indρ R s. If not, then ind ρ possesses a non-zero quotient τ in some R t with t s. Since τ is finitely-generated (as ind ρ is), it possesses an irreducible quotient; we may assume τ is itself irreducible. But then Hom K (ρ,τ) 0 implies that τ ρ and this means that (K,ρ) is not an s-type. ( ): Let (π, V ) R(G) be irreducible and non-zero. Then π R s (G) Hom G (ind ρ,π) 0 Hom K (ρ,π) 0 π R ρ (G). The first ( =) holds because ind ρ, hence any of its quotients, lies in R s (G). This completes the proof of (i).

5 Hecke algebra isomorphisms and types for depth-zero principal series 5 Now let us prove (ii). Suppose (π,v ) R s (G). We have (V/V [ρ]) ρ = 0. But then V/V [ρ] = 0, since non-zero objects in R s (G) contain ρ. So V = V [ρ], that is, π R ρ (G). Conversely, if V = V [ρ], then π is a quotient of a direct sum of copies of indρ R s (G), hence π R s (G). Exercise: Since indρ is projective in R(G) and a generator for R s (G) (i.e. ind ρ R s (G) and Hom G (indρ,π) 0 for every π 0 in R s (G)), every π R s (G) is a quotient of a direct sum of copies of indρ. (Consider the maximal subobject in π which is a quotient of a direct sum of copies of indρ.) We have shown that indρ is a f.g. projective generator of R s (G). From this, general categorical arguments ([Ba]) give (Morita) equivalences of categories R s (G) End G (ind ρ) opp -Mod End G (ind ρ) opp End C W-Mod π Hom G (ind ρ,π) Hom G (ind ρ,π) W t.f = f t. Therefore, we need to relate End G (ind ρ) opp End(W) to e ρ H(G)e ρ. First we define H(G,ρ ) = {Φ : G End(W) Φ(k 1 gk 2 ) = ρ(k 1 )Φ(g)ρ(k 2 ), k i K,g G}. Here the functions Φ are assumed to be smooth with compact support. Also, (ρ,w ) is the representation given by ρ (k) := ρ(k 1 ) End(W ). We view H(G,ρ ) as a convolution algebra using the Haar measure dx giving K volume 1. The following lemma is left to the reader. Lemma We have mutually inverse algebra isomorphisms φ t φ : H(G,ρ ) End G (ind ρ) : t φ t, where t φ (f)(g) = G φ(x)(f(x 1 g))dx (f indρ, g G) φ t (g)(w) = t(e w )(g) (g G,w W). Here e w ind ρ is defined by ρ(k)w, g = k K e w (g) = 0, g / K. Furthermore, there is an anti-isomorphism of algebras H(G,ρ ) H(G,ρ) Φ Φ given by Φ (g) := Φ(g 1 ) End(W ).

6 6 T. Haines Finally, Roche checks in [Ro], p. 390, that there is an algebra isomorphism H(G,ρ) C End(W) e ρ H(G)e ρ Φ (w w ) (g dim ρ w,φ(g)w ) (w W, w W ). In case ρ is a character, the last isomorphism gives H(G,ρ) = e ρ H(G)e ρ and is immediate. Putting these isomorphisms together, we get isomorphisms End G (ind ρ) opp End(W) H(G,ρ) End(W) e ρ H(G)e ρ. In loc. cit. Roche checks that the induced categorical equivalence is R s (G) = R ρ (G) e ρ H(G)e ρ -Mod (π,v ) Hom G (ind ρ,π) = Hom K (ρ,π) = π ρ. (Again, this is quite immediate in the case where ρ is a character.) This completes the proof of Proposition Hecke algebra isomorphisms To prove Theorem 3.0.2, we need to review Hecke algebra isomorphisms. We follow Roche s treatment [Ro] Preliminaries. As before, fix a depth-zero character χ : T C, and let s = [T, χ] G = s χ, for any extension χ : T(F) C of χ. Also, write ρ = ρ χ for the associated character ρ : I = I U T I U C, utu χ(t). Let N denote the normalizer of T in G, let W = N/T = N(F)/T(F) denote the Weyl group, and write W = N(F)/ T for the Iwahori-Weyl group. There is a canonical isomorphism X (T) = T(F)/ T, λ λ := λ( ) (independent of the choice of ). The canonical homomorphism N(F)/ T = W W = N(F)/T(F) has a (non-canonical) section, hence there is a (non-canonical) isomorphism W = X (T) W. Clearly N(F), W and W act on the set of depth-zero characters. We define N χ = {n N(F) nχ = χ} W χ = {w W wχ = χ} W χ = {w W wχ = χ}. There are obvious surjective homomorphisms N χ W χ W χ. Define Φ χ (resp. Φ χ resp. Φ χ,aff ) to be the set of roots α Φ (resp. coroots α Φ resp. affine roots a = α + k, where α Φ, k Z) such that χ α O = 1. Note that W χ F acts in an obvious way on Φ χ,aff. Define the following subgroups of the group of affine-linear automorphisms of V := X (T) R: W χ = s α α Φ χ W χ,aff = s a a Φ χ,aff.

7 Hecke algebra isomorphisms and types for depth-zero principal series 7 Here s a and s α are the reflections on V corresponding to a and α. Let Φ + denote the B-positive roots in Φ, and set Φ + χ = Φ χ Φ +. Then let C χ resp. a χ denote the subsets in V defined by C χ = {v V 0 < α(v), α Φ + χ }, resp. a χ = {v V 0 < α(v) < 1, α Φ + χ }. For a Φ χ,aff we write a > 0 if a(v) > 0 for all v a χ. Similarly, we define an ordering on the set Φ χ,aff. Then let Π χ,aff = {a Φ χ,aff a is a minimal positive element}. Define S χ,aff = {s a a Π χ,aff } Ω χ = {w W χ wa χ = a χ }. It is clear that Φ χ is a root system with Weyl group Wχ, and that W χ W χ. In general, W χ can be larger that Wχ and is not even a Weyl group (see Example 8.3 in [Ro] and Remark below). The following results are contained in [Ro]. Lemma (1) The group W χ,aff is a Coxeter group with system of generators S χ,aff ; (2) there is a canonical decomposition W χ = W χ,aff Ω χ, and the Bruhat order χ and length function l χ on W χ,aff can be extended in an obvious way to W χ such that Ω χ consists of the length-zero elements; (3) if W χ = W χ, then W χ,aff (resp. W χ ) is the affine (resp. extended affine) Weyl group associated to the root system Φ χ V, and C χ resp. a χ is the dominant Weyl chamber resp. base alcove in V corresponding to a set of simple positive affine roots, which can be identified with Π χ,aff. In the situation of (3), let Π χ denote the set of minimal elements of Φ + χ. This is then a set of simple positive roots for the root system Φ χ. Remark In [Ro], pp , Roche proves that W χ = W χ at least when G has connected center and when p is not a torsion prime for Φ (see loc. cit. p. 396). It is easy to see that W χ = W χ always holds when G = GL d (with no restrictions on p). On the other hand, W χ W χ in general, even for G = SL n. Indeed, suppose G = SL n with n 3. Suppose n q 1 and that χ 1 is a character of F q of order n. Consider χ(a 1,...,a n ) := χ 1 (a 1 )χ 2 1 (a 2) χ n 1 (a n). It is clear that W χ = {1}, but that, since a 1 a n = 1, we have W χ (12 n). In fact W χ is the cyclic group of order n generated by (12 n) Statement. Let H(W χ,aff ) denote the affine Hecke algebra associated to the Coxeter group (W χ,aff,s χ,aff ). It has the usual generators T w, w W χ,aff, and relations T w1 w 2 = T w1 T w2, if l χ (w 1 w 2 ) = l χ (w 1 ) + l χ (w 2 ) T 2 s = (q 1)T s + qt 1. if s S χ,aff.

8 8 T. Haines Let H χ := H(W χ,aff ) C[Ω χ ], where the twisted tensor product is the usual tensor product on the underlying vector spaces, but where multiplication is given by (T w1 e ω1 )(T w1 e ω2 ) = T w1 ω 1 (w 2 ) e ω1 ω 2 where ω( ) refers the conjugation action of ω Ω χ on W χ,aff. We write T wω := T w e ω. The Hecke algebra isomorphism depends on a choice of extension χ : N χ C of χ (this always exists: see [HL] 6.11 and [HR09]). Fix such a χ. Then for any n N χ w W χ, define [InI] χ H(G,ρ) to be the unique element in H(G,ρ) supported on InI and having value χ 1 (n) at n. Note that [InI] χ depends on w W χ but not on the choice of n N χ mapping to w. Theorem (Goldstein [Gol], Morris [Mor], Roche [Ro]). Let χ be a depth-zero character as above. For any extension χ of χ as above, there is an algebra isomorphism which sends q l(w)/2 [InI] χ to q lχ(w)/2 T w. H(G,ρ) H χ, Let Φ n := χ(n)[ini] χ, the unique element in H(G,ρ) supported on InI and having Φ n (n) = 1. Corollary For any n N χ, the element [InI] χ (or equivalently, Φ n ) is invertible in H(G,ρ). 6. The morphism V ρ V χ U We assume B = TU and I are in good position : I fixes an alcove a contained in the apartment corresponding to T, and B is any Borel subgroup containing T. From χ we get ρ as usual. For (π,v ) R(G), let V U R(T) denote the Jacquet module. Proposition Suppose (π, V ) is irreducible (hence, cf. [Be92], admissible). Then the map V V U induces a T-equivariant isomorphism (6.0.1) V ρ V χ U. Remark Since B = TU may be replaced with any w B = T w U (w W), it follows that we may also hold B fixed and replace I with w I. That is, we may replace χ with w χ and ρ with w ρ, where the latter is the character on w I defined by w ρ( ) = ρ(w 1 w). Such a replacement causes no harm for the proof of the main theorem (cf. section 7) because π(w) : V ρ V wρ. We will prove Proposition using only a consequence of the Hecke algebra isomorphism, namely Corollary

9 Hecke algebra isomorphisms and types for depth-zero principal series 9 Proof. We change notation slightly and write the Iwahori decomposition as I = U 0 T U 0 where U 0 := I U and U 0 := I U. For any (π,v ) R(G), we define a projector P χ I : V V ρ by P χ I (v) = 1 ρ(k) 1 π(k)v dk. I I It is clear that P χ I really is a projector V V ρ. Write V χu 0 for the set of v V which are fixed by π(u 0 ) and transform under π(t), t T, by the scalar χ(t). Recall that we define P U0 (v) := 1 U 0 U 0 π(k)v dk. Lemma (Jacquet s Lemma I). Let v V χu 0. Then P χ I (v) = P U 0 (v) and has the same image in V U as v. Proof. Writing the integral over I = U 0 T U 0 as an iterated integral proves the desired equality. The rest follows from a basic property of the operator P U0. Recall we assume (π,v ) R(G) is irreducible, hence admissible. V ρ V χ U is surjective: The T-morphism V χ V χ U is surjective. Since V χ U is finitedimensional, there is a finite-dimensional subspace W V χ which still surjects onto V χ U. Choose a compact open subgroup U 1 U 0 such that W V χu 1. Let T + denote the monoid of positive elements in T(F), i.e., those in a subset of the form ν T where ν is B-dominant. (This notion does not depend on the choice of.) Choose a T + such that a 1 U 0 a U 1. Then π(a)w V χu 0, and π(a)w has image π(a)v χ U = V χ U. So, V χu 0 V χ U. We need to prove the smaller subset V ρ V χu 0 still surjects onto V χ U. But this follows using Lemma 6.0.3: for v V χu 0, the element P χ I (v) belongs to V ρ and has the same image in V U as v. This completes the proof of the surjectivity. V ρ V χ U is injective: Lemma For v V ρ = e ρ V, and a T +, we have π(φ a )v = IaI P χ I (π(a)v). Here the action of H(G,ρ) on V ρ is defined using the Haar measure dg which gives I measure 1, and IaI := vol dg (IaI). Proof. Let S a denote any set of representatives in U 0 for a 1 U 0 a\u 0. There is a natural bijection S a (a 1 Ia I)\I I\IaI

10 10 T. Haines (we used a 1 U 0 a U 0 and U 0 a 1 U 0 a). We have π(φ a )v = Φ a (g)π(g)v dg IaI = Φ a (g)π(g)v dg s S a = S a Ias I ρ 1 (k)π(k)π(a)v dk = S a P χ I (π(a)v). Suppose U 1 U is a compact open subgroup. Let V (U 1 ) = {v V P U1 (v) = 0}. It is easy to see that ker(v V U ) = U 1 V (U 1 ). Lemma (Jacquet s Lemma II). Suppose v V ρ V (U 1 ) for some compact open subgroup U 1 U. Suppose a T + satisfies U 1 a 1 U 0 a. Then P U0 (π(a)v) = 0. Proof. The vanishing of π(a) U 0 π(a 1 ua)v du follows from the vanishing of U 1 π(u)v du, since U 1 is a subgroup of a 1 U 0 a. Now we can complete the proof of the injectivity. Suppose v V ρ maps to zero in V χ U. Choose U 1 and a T + satisfying the hypotheses of Lemma Note that π(a)v V χu 0. Then using Jacquet s Lemmas I and II together with Lemma 6.0.4, we see 0 = P U0 (π(a)v) = P χ I (π(a)v) = IaI 1 π(φ a )v. Since Φ a is invertible (Corollary 5.2.2), this implies v = 0, which is what we needed to show. This completes the proof of Proposition Proof of Theorem using Proposition Let (π, V ) R(G) be irreducible. Replacing χ with a Weyl-conjugate if necessary (cf. Remark 6.0.2), we see that π R s (G) iff there exists some η X ur (T) such that π i G B ( χη). By Frobenius reciprocity Hom G (V,i G B ( χ η)) = Hom T(V U, C 1/2 δ B eχ η) this is equivalent to: non-zerov U C 1/2 δ eχη, for some η Xur (T) B (V U χ 1 ) has a T-invariant vector which is an eigenvector for T/ T (V U χ 1 ) has a T-invariant vector (since T/ T is abelian) (V U χ 1 ) T 0 V χ U 0 ( ) V ρ 0 π R ρ (G),

11 Hecke algebra isomorphisms and types for depth-zero principal series 11 where of course ( ) comes from Proposition This completes the proof. 8. Remarks on constructing the Hecke algebra isomorphisms 8.1. Intertwining sets. Let ρ : K C be a smooth character. Definition We define the intertwining set I G (ρ) G by requiring that g I G (ρ) iff ρ K g K = g ρ K g K. Equivalently, there exists φ 0 in H(G,ρ) supported on KgK. [For one direction, if such a φ exists, note that for k K g K we have ρ(k) 1 φ(g) = φ(kg) = φ(g g 1 k) = φ(g)ρ( g 1 k) 1.] Lemma ([Ro], Prop. 4.1). Let K = I and ρ = ρ χ. Then (i) I G (ρ) N = N χ ; (ii) I G (ρ) = IN χ I. The lemma shows that the set {[InI] χ, n N χ /N χ I = W χ } forms a C-basis for H(G,ρ). Proof. (i): If n N I G (ρ), then n ρ I n I = ρ I n I, which implies that n χ T = χ T, hence n χ = χ, i.e., n N χ. Conversely, suppose n N χ maps to w N/T = W. We want to show: for i I n I, we have ρ(i) = ρ(n 1 in). Write i = i i 0 i + I U T I U. Then I n 1 in = n 1 i n n 1 i 0 n n 1 i + n U T U, for U := w 1 Uw, and U := w 1 Uw. Since n 1 in I can also be expressed using the Iwahori decomposition as an element in I U T I U, and the expressions in U TU are unique, we see that n 1 i n I U, n 1 i + n I U, and in particular these elements belong to I +. Using this, we see ρ(n 1 in) = χ(n 1 i 0 n) = n χ(i 0 ) = χ(i 0 ) = ρ(i). This completes part (i), and (ii) is a consequence of (i) Presentation for End(ind ρ 1 ). Recall there is a canonical isomorphism H(G,ρ) = End G (indρ 1 ) (Lemma 4.0.2). Therefore, we just need to find generators and relations for the right hand side. Fix an extension χ : N χ C of χ. For w W χ, choose an element n N χ mapping to it. We consider the element Θ n End G (ind ρ 1 ) defined by (8.2.1) Θ n (f)(x) = 1 I + I + f(n 1 ux)du, (f ind ρ 1 ).

12 12 T. Haines Here, I + := vol du (I + ). Write Iw := I + w I + \I + and Iw := I + / I + w I + (the ratio of the volumes). Since ρ is trivial on I +, we see that (8.2.2) Θ n (f)(x) = 1 Iw f(n 1 ux)du. Note that since I = TI + = (I w I)I + and I + w I = I + w I +, there is a canonical isomorphism and I w I w I w I\I, (8.2.3) I w = [I : I w I] = q l(w). Lemma Let n N χ and let w denote its image in W χ (and write n = n w ). (i) Θ n End G (indρ 1 ). (ii) For n N χ, let Φ n denote the unique element in H(G,ρ) which is supported on InI and takes value 1 at n. Then t Φn = q l(w) Θ n. (iii) {Θ nw } w f Wχ is a C-basis for End G (ind ρ 1 ). (iv) Let n i = n wi for i = 1,2. If l(w 1 w 2 ) = l(w 1 ) + l(w 2 ), then Θ n1 n 2 = Θ n1 Θ n2. Proof. (i): We need to check that Θ n (f) ind ρ 1. Write i I as i = ti + for t T and i + I + (not a unique expression). Then since Ad(t) is a measure-preserving automorphism of I +, we have I + θ n (f)(ix) = f(n 1 uti + x)du = I + f(n 1 tn n 1 ui + x)du I + since n χ(t) = χ(t) = ρ(i). = n χ(t) 1 I + f(n 1 ux)du = ρ(i) 1 I + f(n 1 ux)du, (ii): By Lemma 4.0.2, it is enough to prove φ Θn = q l(w) Φ n. Recalling W = C and letting w = 1 C, we have φ Θn (g)(w) = Θ n (e w )(g) = 1 I + e w (n 1 ug)du. I + This is non-zero only if n 1 ug I for some u I +, i.e., only if g InI. Therefore φ Θn H(G,ρ) is supported on InI. It remains to check its value at g = n. We find it is 1 I + e w (n 1 un)du = I+ w I + I + = q l(w) (cf. (8.2.3)). I + w I

13 Hecke algebra isomorphisms and types for depth-zero principal series 13 (iii): This is proved in greater generality in [Mor], 5.4, 5.5. Alternatively, we can use the fact that we have proved {Φ nw } w fw χ is a basis for H(G,ρ) (Lemma 8.1.2), together with part (ii). (iv): This is proved in [Mor], Prop Alternatively, it is an easy consequence of (8.2.2) and standard calculations. From now on we want to choose the family {n w } in a compatible way: we require that n w1 w 2 = n w1 n w2 whenever l(w 1 w 2 ) = l(w 1 ) + l(w 2 ). It is always possible to do this (see [Mor], 5.2). Note that Θ n depends on n w and not just on w. So, we define a new basis element in H(G,ρ) by B w := χ(n) 1 Θ n. This indeed depends just on w (and χ, of course). We also define T w := q (lχ(w)+l(w))/2 B w = q (lχ(w) l(w))/2 χ(n w ) 1 t Φnw for w W χ. The main computation in this subject shows that these elements T w generate the algebra H χ : Theorem (Goldstein [Gol], Morris [Mor]). The elements T w, w W χ satisfy the following relations: (i) T w1 w 2 = T w1 T w2, if l χ (w 1 w 2 ) = l χ (w 1 ) + l χ (w 2 ) (ii) T 2 s = (q 1)T s + qt 1, if s S χ,aff. Thus, the algebra End G (ind ρ 1 ) is isomorphic to H χ 1 = H χ, by an isomorphism which depends only on the choice of χ Proof of Theorem We can now see that the isomorphism t of Lemma gives the desired algebra isomorphism H(G,ρ) H χ. Indeed, by Lemma and our definitions, t takes to q l(w)/2 [InI] χ = q l(w)/2 χ 1 (n)φ n q l(w)/2 χ 1 (n)θ n q l(w) = q l(w)/2 B w = q lχ(w)/2 T w. This completes the (sketch of the) proof of Theorem

14 14 T. Haines References [Ba] H. Bass, Algebraic K-theory, New York, [BD] J.-N. Bernstein, rédigé par P. Deligne, Le centre de Bernstein, in Représentations des groupes réductifs sur un corps local, Hermann (1984). [Be92] J. Bernstein, Representations of p-adic groups, Notes taken by K. Rumelhart of lectures by J. Bernstein at Harvard in the Fall of [BK] C. J. Bushnell and P. C. Kutzko, Smooth representations of reductive p-adic groups: structure theory via types, Proc. London Math. Soc. (3) 77 (1998), [Gol] D. Goldstein, Hecke algebra isomorphisms for tamely ramified characters. PhD thesis, University of Chicago, [HR09] T. Haines, M. Rapoport, Shimura varieties with Γ 1(p)-level structure via Hecke algebra isomorphisms: the Drinfeld case. In preparation. [HL] R. B. Howlett and G. I. Lehrer, Induced cuspidal representations and generalised Hecke rings, Invent. Math. 58 (1980), [Mor] L. Morris, Tamely ramified intertwining algebras, Invent. Math. 114, (1993), [Ro] A. Roche, Types and Hecke algebras for principal series representations of split reductive p-adic groups, Ann. Sci. École Norm. Sup. 4e e série, tome 31, n o 3 (1998), University of Maryland Department of Mathematics College Park, MD U.S.A. tjh@math.umd.edu