Compositio Mathematica 118: 135–157, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands. 135 Level Zero G-Types ⋆ LAWRENCE MORRIS Department of Mathematics, Clark University, Worcester, MA 01610, U.S.A. e-mail: lmorris@black.clark.edu (Received: 23 December 1996; accepted in final form: 13 March 1998) Abstract. Let G be a connected reductive group defined over a local non-Archimedean field F with residue field Fq ; let P be a parahoric subgroup with associated reductive quotient M. If σ is an irreducible cuspidal representation of M(Fq ) it provides an irreducible representation of P by inflation. We show that the pair (P , σ ) is an S-type as defined by Bushnell and Kutzko. The cardinality of S can be bigger than one; we show that if one replaces P by the full centraliser P̂ of the associated facet in the enlarged affine building of G, and σ by any irreducible smooth representation σ̂ of P̂ which contains σ on restriction then (P̂ , σ̂ ) is an s-type for a singleton set s. Our methods employ invertible elements in the associated Hecke algebra H (σ ) and they imply that the appropriate parabolic induction functor and its left adjoint can be realised algebraically via pullbacks from ring homomorphisms. Mathematics Subject Classification (1991): 22E50. Key words: local non-Archimedean field, connected reductive group, parahoric subgroup, reductive quotient, irreducible cuspidal representation, type, Hecke algebra, parabolic induction, Jacquet functor. Introduction Let G = G(F ) be the group of rational points of a connected reductive group defined over a local non-Archimedean field F . Let B(G) be the set of classes of irreducible supercuspidal representations of rational Levi components of rational parabolic subgroups of G under the equivalence relation arising from Gconjugation and twisting by unramified quasicharacters of Levi components. If π is an irreducible representation of G, then it determines a unique element of B(G) which we denote by I(π ) and call the inertial equivalence class of π . (This notation and definition is taken from [BK2].) Now let S be a subset of B(G). In [BK2] the authors define the notion of an S-type. This is an ordered pair (K, ρ) where K is a compact open subgroup of G and ρ is an irreducible smooth representation of K with the following property: an irreducible smooth representation π of G contains ρ if and only if the inertial equivalence class I(π ) of π belongs to S. The authors show that S-types have many remarkable properties. In particular, if V denotes the space of ρ and (π, V) ⋆ Research partially supported by NSF grant DMS 90-03213. PDF VS O/s disc INTERPRINT ...... Article COMP4186 (compkap:mathfam) v.1.15 165882.tex; 23/08/1999; 9:09; p.1 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 136 LAWRENCE MORRIS is a smooth representation of G let Vρ = HomK (V , V). Then the functor V 7 → Vρ induces an equivalence of categories between the category SRρ (G) of smooth representations of G generated by their ρ-isotypic component and the category of (unital) H (ρ)-modules. Here H(ρ) = H (G, ρ) denotes the algebra of ρ̌-spherical functions on G with compact support, where ρ̌ is the contragredient of ρ. The prototype (due to Borel [B]) of all S-types is the pair (B, 1) where B is the centraliser of an alcove of the enlarged building for G, and 1 denotes the trivial representation of B. (In general the full centraliser of an alcove may be larger than the Iwahori subgroup that it contains.) Bushnell and Kutzko provide many other non trivial examples of S-types in [BK2] arising from their work on GLn and SLn . The prototype in the preceding paragraph can be generalised substantially in the following manner. Let P be a parahoric subgroup of G with reductive quotient M, and let σ be an irreducible cuspidal representation of M. One can view σ as a representation of P by inflation. Theorem 4.8 of this paper asserts that the pair (P , σ ) is an S-type where S is a finite set; in fact S = {[L, ρ1], . . . , [L, ρn ]} where L = L(F ) is the group of rational points of a canonically chosen Levi component. We remark that the number n can be larger than 1. The proof proceeds by associating to P a Levi component L in a canonical way; in fact if P is any parabolic subgroup containing L with a Levi decomposition P = L · U there is an Iwahori decomposition (P ∩ U(F )) ×(P ∩ L) ×(P ∩ U− (F )) → P . Further, L ∩ P is a maximal parahoric subgroup of L. The proof of Theorem 4.8 then depends on the following facts: (i) Any irreducible smooth representation of G which contains a cuspidal representation of a maximal parahoric subgroup must be supercuspidal, and induced from an open, compact mod centre subgroup of G. (See Proposition 4.1.) (ii) The intertwining algebra H (G, σ ) contains invertible elements which are supported on double cosets P dP where d is strongly (P , P) -positive. This is pointed out in Sections 2.4 and 3.3. (iii) There is an isomorphism of isotypic components V σ → (VU )σU , for any smooth representation (π, V) which contains the type σ . Here as usual VU denotes the (unnormalised) Jacquet module. This is pointed out in Lemma 3.6; to prove it one uses property (ii) above. We remark that results of this sort go back to Jacquet; see [Cs]. As a variation, let P̂ be the full centraliser in G of the facet associated to P , and σ̂ be an irreducible representation of Pˆ /U which contains σ . We show that (P̂ , σ̂ ) is an s-type for a singleton set s; see 4.7, 4.9. Lemma 3.9 is the vehicle we use to prove this; it is of independent interest. (See also the remark following Theorem 4.9.) Property (iii) above has another consequence. We have the algebras H (G, σ ), H (L, σU ), and their respective categories of unital modules. On the other hand we 165882.tex; 23/08/1999; 9:09; p.2 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 137 LEVEL ZERO G-TYPES have the categories SRσ (G), and SRσU (L), and the categorical equivalences mentioned above. Theorem 4.8 implies that the (unnormalised) induction functor and (the projection of) the Jacquet functor provide adjoint functors between SRσ (G), and SRσU (L). (See Theorem 4.10.) Theorem 4.12 says that these functors can be realised algebraically via (pullbacks of) a ring homomorphism from H(L, σU ) to H(G, σ ). This amounts to showing that one can apply Corollary 8.4 of [BK2]. Corrigenda (i) The group denoted by H in [M]3.15 and elsewhere in that paper, should be replaced by the group H = ker ν ′ in 1.6 below. (ii) Contrary to what is asserted in op. cit. 3.15 the group M′ ∩ P need not be special in M′ ; see 1.7 below. This does not affect the proofs. In particular, in op. cit. 4.14 the subgroup MJ need not be special. Notation and Conventions F : complete non-Archimedean field; o: ring of integers of F ; p: maximal ideal of o; Fq : residue field o/p (q = p n , where p is some prime number); G: connected reductive F -group; Z: maximal F -split torus in the centre of G; T: maximal F -split torus in G; ZG (X) (resp. NG (X)): centraliser (resp. normaliser) in G of X. In general if V is an algebraic F -variety we shall write V for the set V(F ); we make an exception for parabolic subgroups and their unipotent radicals: Remark. In this paper, the expression ‘parabolic subgroup’ will always mean ‘F -parabolic subgroup’. If P is such a group with unipotent radical U we shall write P(F ), U(F ) respectively for their F -rational points. If L is a Levi component for P, we shall write L = L(F ). We remark that all Levi decompositions will be assumed to be defined over F . In fact, we shall write P , Q, etc., for parahoric subgroups of G = G(F ). Other notation is explained as needed. 1. Preliminaries 1.1. We begin with a quick review of the relevant aspects of the theory of reductive groups. Thus let G denote a connected reductive group defined over F and let 8 be the set of relative roots with respect to some maximal F -split torus T; when 165882.tex; 23/08/1999; 9:09; p.3 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 138 LAWRENCE MORRIS necessary we shall write 1 for the set of simple roots corresponding to the choice of a minimal parabolic subgroup P0 . 1.2. Let P be a parabolic subgroup with a Levi decomposition P = L · U. THEOREM. (i) There is a unique parabolic subgroup P− containing L with Levi decomposition P− = L · U− with the property that U ∩ U− = {1}. (ii) Let P− be as in (i). There is an F -isomorphism of varieties U− × L × U → − P · P induced by the multiplication map; the image is a Zariski open subset in G. Proof. Except for the F -statements, (i) and (ii) are contained in [Bo] 14.21. If P is defined over F so is P− ([Bo] 20.5). The multiplication map is defined over F , so the rest of (ii) follows since the image of an F -morphism is an F -variety ([Bo] AG14.3). 2 DEFINITION 1.3. We shall call the group P− the opposite parabolic subgroup to P (with respect to L). PROPOSITION 1.4. If S is an F -split subtorus of T then ZG (S) is the Levi component of a parabolic subgroup of G. Proof. This is Proposition 20.4 of [Bo]. 2 1.5. We take T, 8, P0 , 1 just as above, and we write v W for the spherical Weyl group of the root system 8. Let 6 be the set of affine roots associated to a reduced root system v 6 in the same real vector space as the root system 8 with affine Weyl group W ′ . We assume that v 6 and 8 have the same Weyl group. This is equivalent to assuming that if α ∈ 8 then λ(α)α ∈ v 6 for a positive real number λ and that the map α → λ(α)α is onto. A typical element a of 6 can be written as Da + k where Da ∈ v 6 and k is an integer; we refer to Da as the gradient of a. There is also a homomorphism D: W ′ → v W . DEFINITION. An échelonnage E ⊂ 8 × 6 of 8 by 6 is a subset which satisfies the following properties: (E1) if (α, a) ∈ E then α is a positive multiple of Da; (E2) if w ∈ W ′ and (α, a) ∈ E then (Dw(α), wa) ∈ E; (E3) the projection maps from E to 8, 6 are onto. Remarks. (i) If (α, a) ∈ E we say that α, Da are associated. (ii) Let 8nd denote the set of non-divisible roots in 8. Then (E1) and (E3) imply that there is a bijection ρ: v 6 → 8nd such that α = µα ρ(α) with µα > 0. 1.6. Now we quickly review some aspects of Bruhat–Tits theory; as a general reference we suggest [T]. The group G = G(F ) is naturally furnished with the structure of a second countable locally compact Hausdorff totally disconnected group (= t.d. group, in brief). The work of Bruhat and Tits associates to (G, T) an échelonnage E ⊂ 8 × 6. We remind the reader that the ambient vector space 165882.tex; 23/08/1999; 9:09; p.4 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 LEVEL ZERO G-TYPES 139 V on which the roots in either 8 or v 6 act as functions is the real dual of the subspace of XF (T)⊗R generated by 8, where XF (T) denotes the lattice of rational characters. In turn, from this and a choice of simple affine roots in 6 one obtains a normal subgroup G′ in G, a compact open subgroup B in G′ and a subgroup N ′ = NG′ ′ (T ) in G′ , and a set of reflections S in W ′ such that (G′ , B, N ′ , S) is an affine Tits system with respect to the system 6. (For the definition of G′ see [BT2]5.2.11.) In particular there is a surjection ν ′ : N ′ → W ′ . We denote the kernel of ν ′ by H ; it is a compact normal subgroup of the group ZG (T)(F ). We note that N ′ ⊆ N = NG (T ), and the triple (G, B, N) is a generalised affine BN-pair in the sense of [M]3.2. The generalised affine Weyl group here is the quotient W = N/H ; we write νW : N → W for the natural projection. Any subgroup conjugate in G (or G′ ) to B is called an Iwahori subgroup of G. The affine Tits system (G′ , B, N ′ , S) gives rise to a polysimplicial complex on which G and G′ act, preserving the simplicial structure. The geometric realisation of this complex is called the affine building associated to (G′ , B, N ′ , S); we denote it by I. In fact I is obtained by pasting together copies (called apartments) of an affine Euclidean space A whose underlying space of translations is V above. The points of A correspond to valuations of (G, ZG (T ), (Uα )α∈8 ). For more on this see [BT1]6.2. In particular, A embeds into I. The group N ′ acts on A as a group of affine automorphisms with kernel H . Furthermore, the affine root system 6 partitions A in the usual way into facets; it is this partition which gives rise to the underlying simplicial structure of I. Thus the facets of A are facets of I and any facet of I is a translate by an element of G′ of a facet of A. We remark that the choice of a different apartment amounts to choosing a different T; the resulting E is the same. The G′ -centralisers of facets in I are called parahoric subgroups; in particular the centralisers of chambers (facets of maximal dimension) in I are conjugates of B. Any parahoric subgroup is a compact open subgroup of G. See [BT1] 6.2, 6.5, and Section 2. Finally we have H = B ∩ N ′ . WARNING. The subgroup H that we employ here is not the H employed in [BT1, BT2]. The subgroup that we denote by H is denoted by H 0 in [BT2]4.6.3(4), or by Zo (O ♮ ) in ibid. 5.2.10. For many purposes the enlarged building I1 is a more convenient object; in particular it guarantees that the centralisers in G of facets will be compact open subgroups of G. It is defined as follows. Let V 1 denote the dual of XF (G) ⊗ R where as usual XF (G) is the group of rational characters of G. Then I1 = I × V 1 and the action of G on I (which we have not explicitly defined) is extended to one on I1 by defining θ: G × V 1 → V 1 by θ(g)(χ) = −ω(χ(g)), for all χ ∈ XF (G). We identify I with I × {0}, and we write G1 for the stabiliser in G of this set. A facet F in I corresponds to a facet F 1 = F × V 1 in I1 . We write P̂F for the centraliser in G of the facet F 1 ⊆ I1 . It is also the centraliser in G1 of F 1 , and it is the centraliser in G of the ‘facet’ F × {0} in I1 . We always have G′ ⊆ G1 ; if G is semisimple we have G = G1 , and I = I1 . We note that the group 165882.tex; 23/08/1999; 9:09; p.5 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 140 LAWRENCE MORRIS G′ ⊆ G = G(F ) is the subgroup of G generated by the connected centralisers (= parahoric subgroups) of facets of the enlarged building I1 . We have G/G′ = N/N ′ . We set  = N/N ′ . Let W = NG (T)(F )/H be the full ˜ where affine Weyl group associated to (G, T). It is a semidirect product W ′ ⋊  1 ˜ is the subgroup of elements stabilising some specified alcove in I ; in particular  ˜ maps isomorphically under the obvious projection map W → W/W ′ the group  to . The group N acts on A by affine transformations; this defines a map ν: N → Aff(A) (as in [BT1]), which factors through νW . Indeed the generalised affine Weyl group W is an extension of v W by the group D = ZG (T)(F )/H ; here D is an extension of the lattice ZG (T)(F )/ker νW by the finite Abelian group ker νW /H . 1.7. The choice of B amounts to choosing a set of simple affine roots 5 in 6, and one can attach a local Dynkin diagram to this in a way similar to the usual case of ordinary root systems. For example if G is split this diagram is just the usual completed Dynkin diagram. If F is a facet in A ⊂ I we take the set 6F of affine roots vanishing on F . The set of roots 8F ∈ 8 associated to this set is a not necessarily closed subroot system of 8: for example, if α ∈ 8F it need not be the case that 2α ∈ 8F ; we denote its closure by c 8F . We remark that it can happen that c 8F = 8 if F is a nonspecial vertex, even if 8 is reduced; if G is split this does not occur. In particular, let F be a facet in the closure of the chamber (alcove) corresponding to B. Then F also corresponds to a subset J = JF ⊆ 5 giving rise to a finite reflection group WJ and a subset of 5; the group WJ is generated by the fundamental reflections associated to the elements of J . Then WJ is the Weyl group for 8F (but not necessarily c 8F ), and the Dynkin diagram for 8F is obtained from the local Dynkin diagram by striking out all the nodes not corresponding to elements of J and all edges meeting such a node. Each of these objects only depend on F ; we sometimes write 8J instead. See [T] Section 1 and [BT1] 6.2,6.4. 1.8. The root system 8J has the following interpretation. Let P be the parahoric subgroup centralising the facet F . There is a short exact sequence 0 → U → P → M → 0 where U is an open compact pro- p subgroup of G and M is the group of Fq -rational points of a connected Fq -reductive group M. There is an obvious o-split torus scheme T whose generic fibre is T and whose reduction mod Fq gives a maximal Fq -split torus T in M. The root system for M with respect to T is then just 8J . See [T] 3.5.1. 1.9. The structure of P can be described more precisely as follows. First, for any element α of 8nd let a(α, F ) be the smallest affine root which is nonnegative on F and which corresponds to α by the map in 1.5: i.e. ρ(Da(α, F )) = α. For each affine root a with ρ(Da) = α there is a compact open subgroup Ua of Uα = Uα (F ). Let U + (F ) be the group generated by all the Ua(α,F ) for α ∈ 8+ nd = 8nd ∩ 8+ and define U − (F ) in a similar way. Here 8+ denotes the set of positive roots with respect to 1. Finally let N ′ (F ) be the subgroup in N ′ which fixes F pointwise. 165882.tex; 23/08/1999; 12:03; p.6 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 141 LEVEL ZERO G-TYPES THEOREM. (i) The product map α∈8+ Ua(α,F ) → U + (F ) is bijective for every nd ordering of the factors, and similarly for U − (F ). (ii) PF = U − (F ) · U + (F ) · N ′ (F ). Q (iii) If F is a chamber, the product map α∈8nd Ua(α,F ) × H → B is a bijection for every ordering of the product. ∗ (iv) Let U be as in 1.8. For each Ua(α,FQ) as above let Ua(α,F ) = Ua(α,F ) ∩ U and ∗ ∗ let H = H ∩ U . Then the product map α∈8nd Ua(α,F ) × H ∗ → U is a bijection for every ordering of the product. Proof. Statements (i) and (ii) are proved in 6.4.9 and 7.1.8 of [BT1]. Statement (iii) is proved in 6.4.48 of op. cit. Statement (iv) also follows from that result, on using the concave function fF∗ of 6.4.23 of op. cit. 2 Q 1.10. Let P = PJ be as in 1.8. There are then the subroot systems 8J ⊂ c 8J ⊂ 8. Since c 8J is closed there is a connected reductive F -subgroup M ⊂ G containing T and which has the relative root system c 8J with respect to T. Indeed this group is generated by those root groups Uα with α ∈ c 8J , and by T. (One may have M = G when P is maximal but not special.) We let M′ ⊆ M(F ) be the subgroup generated by the Uα (F ) with α ∈ c 8J , and by H . PROPOSITION. (i) If P = PF for a facet F in the apartment A with respect to T then P ∩ M′ is a parahoric subgroup of M, and there is a short exact sequence 0 → U ∩ M′ → PF ∩ M′ → M → 0. (ii) Similarly, if P̂ = P̂F for a facet F in the apartment A with respect to T then P̂ ∩ M′ centralises a facet for M, and there is a short exact sequence 0 → U ∩ M′ → PˆF ∩ M′ → M̂ → 0. Proof. Let M′0 be the group generated by the Uα with α ∈ c 8J . Then M′ = H · M′0 . Taking the valuated root system (ϕα )α∈8 that gives rise to the affine Tits system (G′ , B, N ′ , S) and applying [BT1] 7.6.3 (see also 1.12 below) to the groups G1 = M′ and G01 = M′0 we see that we obtain a valuated root system on M′ . Now observe that the group T1 of loc. cit. is just the group H · (ZG (T)(F ) ∩ M′0 ). In particular this enables us to apply Corollary 7.6.5 of op. cit., which implies (i), and the proof of (ii) is similar. 2 1.11. There is a bijective correspondence between parahoric subgroups contained in P and (Fq -rational points of) parabolic subgroups of the group M. This correspondence is realised by Q 7 → U \Q. This is part (i) of Proposition 5.1.32 of [BT2]. 1.12. We conclude this section by comparing parahoric subgroups of a Levi component L = L(F ) (as in 1.2) with parahoric subgroups of G. Let L be a Levi component of G defined by some T subset 2 of the set of simple roots 1 as in 1.3. Thus L = ZG (S) where S = α∈2 ker α. The set 2 is a basis for a closed subroot system 8L ; indeed this last is the root system for (L, T). Let L′ , L1 be 165882.tex; 23/08/1999; 9:09; p.7 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 142 LAWRENCE MORRIS the analogues for L of G′ , G1 , and let L0 be the subgroup of L generated by the root groups Uα (F ), α ∈ 8L . Then L′ = H · L0 and L1 = (ZG (T ) ∩ L1 )L0 . For the time being let L1 be any subgroup of L which is generated by L0 and a subgroup ZG (T )1 ⊆ ZG (T ) which contains ZG (T ) ∩ L0 . According to [BT1]7.6.3 if ϕ = (ϕα )α∈8 is a valuation for (G, ZG (T ), (Uα )α∈8 ) then ϕL1 = (ϕα )α∈8L is a valuation for (L1 , ZG (T )1 , (Uα )α∈8L ). We write V (L), A(L), I(L), . . . , etc. to denote the correspondingTobjects for L that have been defined previously for G. We also let V 1 (L) = α∈2 ker α, where now the intersection is taken in the vector space V of 1.6, and we define V1 (L) = V /V 1 (L); this last space can be identified with V (L). In particular there is a natural map v π : V → V (L). If we suppose that a v W -invariant inner product has been chosen on V with orthogonal projection p: V → V 1 (L) then V (L) can be identified with ker p. As before, we can form the buildings I(L), I(L1 ); as complexes these are the same, with the action of L extending that of L1 . We then have the following facts. (i) Let π be the map A → A(L) defined by ϕ + v → ϕL1 +v π(v). Proposition 7.6.4 in [BT1]) says that (a) there is a unique L1 -equivariant map π̃: L1 · A → I(L1 ) extending π ; the inverse image of an apartment, half-apartment, wall, is an apartment, halfapartment, wall in I; (b) there is a unique action V 1 (L) × L1 · A → L1 · A extending the action V 1 (L) × A → A; this action factors through π̃ and the quotient map defines a bijection (L1 · A)/V 1 (L) → I(L1 ). Note that L1 · A has the structure of a polysimplicial complex, inherited from that of I. The definition of affine roots for I(L1 ) implies that if F is a facet in L1 · A then π̃ (F ) lies in a unique facet, but this image is not necessarily a facet. (ii) Let  ⊆ L1 · A ⊆ I; write P̂ for the pointwise centraliser of . Then [BT1]7.6.5 says in particular that (a) P̂ ∩ L1 ⊆ Pˆπ̃ () (the pointwise centraliser in L1 of π̃()), and (b) if the subgroup ZG (T )1 is contained in ker(p ◦ ν) where ν: NG (T ) → Aff(A) then P̂ ∩ L1 = P̂π̃( ). (iii) Now choose a point ϕ ∈ A and consider the affine subspace ϕ +ker p. We can then form I′ = L0 · (ϕ + ker p) ⊆ I since L0 ⊆ G. According to [BT2]4.2.17, (a) the restriction of π̃ in (i)(a) provides an L0 -equivariant isometry π ′ : I′ → I(L) extending the map ϕ + ker p → A(L); (b) the inverse j of π ′ provides a bijection (y, v) 7 → j (y) + v from I(L) × V 1 (L) to L · A; (c) there is a homomorphism θ(L1 ): L1 → V 1 (L) such that for any ℓ ∈ L1 , y ∈ I(L), v ∈ V 1 (L), ℓ ·(j (y) + v) = j (ℓ ·y) + v +θ(L1)(ℓ), and θ(L1 )|ZG (T ) = p ◦ ν; θ(L1 )|L0 = 0. 165882.tex; 23/08/1999; 9:09; p.8 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 143 LEVEL ZERO G-TYPES The affine subspace ϕ + ker p inherits a polysimplicial decomposition from A. We note that the isometries π ′ , j take facets to facets. (iv) Taking L1 = L in (iii) and applying the definitions of I1 , I(L)1 one deduces op. cit. 4.2.18 that I(L)1 can be isometrically identified with the smallest subset of I1 stable by L and containing the ‘enlarged’ apartment A × V 1 . Under this identification the map θL for the enlarged building I(L)1 which corresponds to the map θ in 1.6 for G, is given by θ(L) + θ; thus L1 = ker θ(L) ∩ ker θ. LEMMA 1.13. Let F be a facet in L · A, P̂ = PˆF and P ⊆ P̂ the corresponding parahoric subgroup. Then (i) P̂ ∩ L1 = P̂ ∩ L is the centraliser of a facet in I(L)1 ; (ii) P ∩ L′ = P ∩ L is a parahoric subgroup in L. Proof. From 1.12(i) we see that π̃ (F ) lies in a facet FL in I(L) and FL identifies with a facet j (FL ) in the complex I′ of 1.12(iii). Applying 1.12(ii) to the group L′ we see that if P = PF is a parahoric subgroup in G′ then P ∩ L′ is the parahoric subgroup in L′ for the facet j (FL). Now, P ∩ L ⊆ ker θ(L) ∩ ker θ by 1.12(iii)(c), hence P ∩ L ⊆ P̂j (FL )×{0}×{0} . But this last group only differs from its connected component by elements of (ZG (T ) ∩ L1 ) − H ⊆ ZG (T ) − H and these cannot lie in P in any case. This proves (ii). Applying 1.12(i)–(iii) in a similar way to the group L1 we see that if P̂ ⊂ G1 fixes pointwise a facet in L · A then Pˆ ∩ L1 is the full centraliser in L1 of a facet in I(L1 ) = I(L) , hence the centraliser in L1 of a facet in I(L1 ) = I(L). Thus it is the centraliser in L of a facet in I(L)1 . 2 2. Parabolics and Parahorics 2.1. We now fix a facet F ⊆ A and let P = PF be the associated parahoric subgroup, with corresponding short exact sequence 0 → U → P → M → 0 as in 1.8, and associated root system 8J = 8F . As in 1.6 we write Pˆ for the full centraliser in G1 of F . We remark that P is the group of integral points of the connected component of a smooth o-group scheme P̂ such that P̂ (o) = P̂ . There is an exact sequence 0 → U → Pˆ → M̂ → 0, where M̂ is the group of rational points of a reductive Fq -group M̂, and the group denoted M in 1.8 is the identity component of M̂. Recall the group M in 1.10; it has a centre containing an F -split component S. The centraliser of S is a (connected) reductive F -group L. Note that S is the F -split component of ZL : we have L ⊃ M so the F -split component of ZL centralises M and contains S, hence it must be S. Moreover, S = Z(G) if and only if P is maximal. THEOREM. The group L is the Levi component of a parabolic subgroup P = L·U; set L = L(F ). Furthermore, the following properties hold. 165882.tex; 23/08/1999; 9:09; p.9 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 144 LAWRENCE MORRIS (i) P̂ ∩ L = Q̂ is the centraliser in L of a vertex of I(L)1 , and Q = P ∩ L is a maximal parahoric subgroup of L, which is contained in Q̂. There are short exact sequences 0 → U ∩ L → Q → M → 0, 0 → U ∩ L → Q̂ → M̂ → 0. (ii) Let P be a parabolic subgroup containing L with Levi decomposition P = L · U. There is a homeomorphism in the p-adic topology U ∩ U− × Pˆ ∩ L × U ∩ U → P̂ , and there is a similar decomposition for the group P . Proof. The first assertion follows from 1.4 and the first exact sequence follows from the observation that Q contains the subgroup P ∩ M′ and this group projects onto M as in 1.10. Now let P = L · U be an F - parabolic for which L is a Levi component. (Note that if P is maximal then P = G trivially satisfies (i), (ii) and (iii).) Applying 1.13 we see that Pˆ ∩ L is the centraliser of a facet in the enlarged building for L, and Q = P ∩ L is the corresponding parahoric subgroup. The remark above implies that M is the connected reductive subgroup of L associated to Q as in 1.10. The definition of affine roots for I(L) and the identifications of 1.12 imply immediately that π̃ (F ) is a point; in fact it is not difficult to see by unravelling the definitions in 1.12–13 that it must be a vertex. Alternatively, if Q were not maximal in L we could repeat the argument above in L itself to produce a proper Levi component K within L with the same properties (with respect to L and Q). Since M is the connected reductive subgroup of L associated to Q we have K = ZL (S) = L. It follows that Q is a maximal parahoric subgroup in L as claimed. For the last part of (i) observe that P ∩ L contains P ∩ M′ which projects onto M as in 1.10; similarly Pˆ ∩ L contains P̂ ∩ M′ which projects onto M̂. To prove (ii) recall from 1.2 that given any parabolic subgroup P = L · U with opposite parabolic subgroup P− = L · U− there is an isomorphism of varieties induced by multiplication: U × L × U− → P · P− and the image is an open set in G. In particular, if P is defined over F we can take F -valued points to get a homeomorphism in the p-adic topology on G. Now consider the restriction P̂ ∩ U × Pˆ ∩ L × P̂ ∩ U− → Pˆ ∩ (P · P− )(F ). To finish we need only show that the image is all of P̂ . Let x ∈ P̂ ; by (i) we can find l ∈ Pˆ ∩ L with y = l −1 x ∈ U . If I is any Iwahori subgroup contained in P then U ⊂ I ; this follows immediately from 1.11. Invoking [BT1] 6.4.9, 6.4.48 we see that (ii) is true if we replace Pˆ by I , hence it is true if we replace Pˆ by U . (See 1.9.) Write y = u1 mu2 with m ∈ U ∩ L, u1 ∈ U ∩ U(F ), u2 ∈ U ∩ U− (F ). Thus x = lu1 mu2 . Since P ∩ L normalises U ∩ U we can rewrite this as x = vlmu2 with v ∈ U ∩ U. The argument for P is the same. 2 Remark 2.2. Although (P ∩ L)0 is maximal in L, it is not usually special in L (as easy examples show), even if P ∩ M(F ) is special in M(F ). (Observe that P ∩ M(F ) = P ∩ M′ because M′ is to M(F ) as G′ is to G.) 165882.tex; 23/08/1999; 9:09; p.10 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 145 LEVEL ZERO G-TYPES 2.3. We assume that L is standard with respect to the basis 1 of 1.3. Thus L = L2 for some 2 ⊆ 1, and we write S for its split centre. Write L = L(F ) as usual; observe that T ⊆ L. The generalised affine Weyl group WL = WL,aff for L is an extension of D (see 1.6) by W2 = v Wρ(2) ⊆ v W . Let X∗ (S) denote the group of rational cocharacters of S. Recall that there is a homomorphism HS : S → X∗ (S) ⊗ R defined by HS (s)(χ) = −ordF (χ(s)) if χ ∈ X(S) = XF (S). Let DS = im(HS ). 2.4. Now let P , L, Q = P ∩ L be the particular subgroups of Section 2.1. The F split torus S acts on U by conjugation; from this one obtains a set of weights which we denote by 8(P, S)+ . The elements of this set can be obtained by considering the nontrivial restrictions to S of the roots in 8+ ; if we write 1(P, S) for the set of nontrivial restrictions of the elements of the basis 1 then each element of 8(P, S)+ can be expressed as a linear combination of elements of 1(P, S) with nonnegative coefficients. (As usual, we are assuming L is standard.) Since the elements of 8(P, S) are rational characters for S, obtained by restriction from the elements of 8 we can write S + = {s ∈ S(F ) = S|HS (s)(α) > 0}, α ∈ 1(P, S). We define S ++ by replacing inequality by strict inequality in the definition above. LEMMA. (i) Let s ∈ S + . Then s(U(F ) ∩ U )s −1 ⊆ U(F ) ∩ U ; s −1 (U− (F ) ∩ U )s ⊆ U− (F ) ∩ U. (ii) If s ∈ S ++ then (a) For any pair of compact open subgroups H1 and H2 of U(F ) there is a nonnegative integer n such that s n H1 s −n ⊆ H2 . (b) For any pair of compact open subgroups K1 and K2 of U− (F ) there is a nonpositive integer n such that s n K1 s −n ⊆ K2 . Proof. In (i) we shall only prove the second assertion. We suppose that the parabolic subgroup P corresponds to the subset 2 ⊆ P 1. The group U− is directly spanned by root groups Uγ where γ ∈ 8nd and γ = α∈1 mα α with at least one −1 α∈ / 2 with mα < 0. It suffices P to show that P s Uγ ,r s ⊆ Uγ ,r if Uγ ,r ⊆ Uγ (F ) is a valuation group. Write γ = α ∈/ 2 mα α+ β∈2 mβ β. If s ∈ S then [BT2]5.1.22(2) ; the assertion for s ∈ S + follows implies that s −1 Uγ ,r s = Uγ ,r−Pα∈2 / (HS (s),α)mα immediately. For (ii) it is enough to show (c.f. [BK2] 6.14) that if s ∈ S ++ then \ [ s n (U(F ) ∩ U )s −n = {1}, s n (U(F ) ∩ U )s −n = U(F ), n> 0 n6 0 or again that sUγ ,r s −1 ( Uγ ,r , s −1 Uγ ,r s ) Uγ ,r , if Uγ ,r ⊆ Uγ (F ) ⊂ U(F ) is a valuation group. This follows from the argument for (i). 2 165882.tex; 23/08/1999; 9:09; p.11 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 146 LAWRENCE MORRIS 2.5. In the language of [BK2]6.16, Lemma 2.4 says that the elements of S which lie in S ++ are strongly (P, P )-positive. 3. Invertible Elements in H(G, σ ) 3.1. We retain the notation of the previous sections. In particular, P = PJ and we have the short exact sequence of 1.8: 0 → U → P → M → 0. Let (σ, V ) be an irreducible cuspidal representation of the group M = M(Fq ) with contragredient representation (σ̌ , V ∨ ) This inflates to a representation of the group P , and we can form the compactly induced representation c-IndG P (σ̌ ). The intertwining algebra G EndG (c-IndP (σ̌ )) is isomorphic to the algebra H (G, σ ) = H(σ ) of σ̌ -spherical functions on G with compact support where the multiplication in the latter is given by the standard convolution product (see [M] Section 4, or [BK2] Section 2.6). In [M] this algebra was analysed and described by generators and relations. Roughly speaking it is an affine Hecke algebra twisted by a group algebra (with a 2-cocycle). Indeed, let SJ = {w ∈ W | wJ = J } and put NJ = P ∩ N; then SJ is a complement in NW (WJ ) to the finite group WJ and, moreover, one has NW (WJ ) NN (P ∩ M′ ) ≃ ≃ SJ . NJ WJ For this see [M] 4.16, 6.1. It then follows that W (J, σ ) = W (σ ) = {w ∈ SJ | wσ ≃ σ } is well defined. (Note that σ can be viewed as a representation on P ∩ M′ .) PROPOSITION. There is a (canonically defined) affine Coxeter subgroup R(σ ) ⊂ W (σ ) together with a (canonically defined) complement C(σ ) W (σ ) = R(σ ) ⋊ C(σ ). Moreover, there is a canonical choice for a set of simple roots in the affine root system associated to R(σ ), once a set of set of positive roots has been chosen in 6. This is proved in [M] 7.3. Henceforth we suppose that a set of positive affine roots for the affine system 6 has been chosen, as well as the matching affine basis in the root system associated with R(σ ). 3.2. The definition of W (σ ) implies the existence of a 2-cocycle µ: W (σ )×W (σ ) → C× , which is nontrivial only on C(σ ) × C(σ ) . (See [M] 6.2, 7.11.) THEOREM. The algebra H (σ ) is generated by elements Tw , w ∈ W (σ ) subject to the following relations. Let w ∈ W (σ ), t ∈ C(σ ) and let v be a reflection in R(σ ) corresponding to a simple root a (chosen as above in 3.1). (i) Tw Tt = µ(w, t)Twt ; (ii) Tt Tw = µ(t, w)Tt w ; 165882.tex; 23/08/1999; 9:09; p.12 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 147 LEVEL ZERO G-TYPES (iii) (iv) Tv Tw =  if w −1 a > 0; Tvw , pa Tvw + (pa − 1)Tw , if not; Tw Tv =  Twv , if wa > 0; pa Twv + (pa − 1)Tw , if not. Here pa 6 = 1 is a nonnegative power of p (the residue characteristic), and the element Tw is supported on one double coset of the form P ẇP where ẇ is an element in N(T ) such that νW (ẇ) = w. This is Theorem 7.12 in [M]. We remark that R(σ ) can be trivial. 3.3. Now consider the translations T (J ) in W (σ ) provided by the group of rational points of the split centre of M′ . They always provide a lattice in W (σ ) of rank at least as large as the lattice of all translations in the group denoted RJ in [M]7.3. (See also [M]2.6-2.7.) Further, their definition and that of the 2-cocycle µ, ensure that µ restricted to T (J ) is always trivial. (See remark (b) following loc. cit.). If we take w = v in 3.2(iii) and (iv) we see that Tv is invertible when v is a fundamental reflection in the ‘quotient’ affine root system. Then, by writing an arbitrary w ∈ R(σ ) as a minimal product of such reflections, we see that Tw is invertible for any such w, again using 3.2(iii) and (iv). In general we can express w = rc where r ∈ R(σ ) and c ∈ C(σ ); since Tc is invertible by 3.2(i), it follows that Tw is invertible by 3.2(i) or (ii) once again. In particular we have the following result. LEMMA. The elements Td , d ∈ T (J ), are invertible. 3.4. Let H (G) = {f : G → C | f locally constant, compact support}. This is algebra with multiplication defined by convolution f ∗ h(x) = Ran associative −1 f (xg )h(g) dg. G With σ as above define eσ ∈ H (G) by eσ (x) =  (1/vol P ) dim(σ ) trace (x −1 ), if x ∈ P ; 0, if not. This is an idempotent in H (G); we then have the algebra eσ ∗ H(G) ∗ eσ which has as an identity the element eσ . From Proposition 4.2.4 of [BK1] there is a canonical isomorphism ϒ: H (σ ) ⊗C EndC (V ) → eσ ∗ H(G) ∗ eσ . It is realised in one direction in the following manner. We identify the left side with H (σ )⊗C V ⊗C V ∨ where we denote the dual of V by V ∨ . Then ϒ(8⊗v⊗v̌) is the function φ(g) = dim(σ )hv, 8(g)v̌i, where h , i denotes the canonical pairing on V ×V ∨ . The isomorphism ϒ implies that the algebras H (σ ), eσ ∗H (G)∗eσ are Morita equivalent, hence their module categories are equivalent.This is realised as follows. If M is an H(σ )-module then M ⊗C V is the corresponding eσ ∗ H(G) ∗ eσ (≃ H (σ ) ⊗C EndC (V ))-module. Conversely, if N is an eσ ∗ H (G) ∗ eσ -module, 165882.tex; 23/08/1999; 9:09; p.13 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 148 LAWRENCE MORRIS we view V ∨ as a right EndC (V )-module and form V ∨ ⊗EndC (V ) N. We then get a H (σ )-module via the right factor since there is an embedding H(σ ) → eσ ∗ H(G) ∗ eσ . For more details we refer the reader to [BK1] Ch.4. We shall denote the equivalence between the module categories by ϒ ∗ . 3.5. Now we take P as in Section 2 with respect to P . We denote by σU the representation of L ∩ P on V by restriction of σ : it is also the inflation of σ on M = M(Fq ) (notation of 1.8) hence is irreducible. Let (π, V) be a smooth representation of G. We denote by V σ the σ -isotypic Rpart of V. Recall that there is a representation of H (G) on V defined by π(f )v = G f (x)π(x)v dx. Given (σ, V ) as above, and (π, V) a smooth representation of G we define Vσ = HomP (V , V) ≃ HomG (c − IndG P (σ ), V), the isomorphism following from Frobenius reciprocity for compact induction. RThe algebra H (σ̌ ) acts on the left −1 on c − IndG P (σ ) via convolution φ ∗ f (x) = G φ(y)f (y x) dy, if φ ∈ H(σ̌ ), G and f ∈ c − IndP (σ ). On the other hand there is a canonical anti-isomorphism of algebras with identity provided by the map φ 7 → φ̌ where φ̌(x) = (φ(x −1 ))ˇ. This means that Vσ is canonically a left H (σ )-module. There is an obvious evaluation map Vσ ⊗ V → V σ ; in terms of the canonical isomorphism ϒ of 3.4 one deduces that there is a natural isomorphism of eσ ∗ H(G) ∗ eσ -modules V σ ≃ ϒ ∗ (Vσ ⊗ V ) provided by this evaluation map. See [BK2]2.13 for more details on this. 3.6. From Theorem 2.1 we have (i) (P ∩ U(F )) · (P ∩ L) · (P ∩ U− (F )) = P ; (ii) σ is trivial on P ∩ U(F ), P ∩ U− (F ), since it factors through L ∩ P . In the terminology of [BK2]6.1, (i) and (ii) say that the pair (P , σ ) is decomposed with respect to (L, P). Indeed 2.1 says that it is decomposed with respect to (L, P′ ) where P′ is any parabolic which contains L as Levi component. Let s ∈ S. Recall from Section 2, that s lies in the split centre of L by construction. We have already seen that the elements Tν(s) (ν as in 1.12(ii)(b)) are invertible, hence any non zero element of H (G, σ ) which is supported on P sP is invertible. Lemma 2.4 says that an abundance of such s are strongly (P, P ) positive. The above observations tell us that Theorem 7.9 of [BK2] is applicable in this situation. We immediately deduce the following lemma. LEMMA. Let (π, V) be a smooth representation of G. Write (πU , VU ) for the Jacquet module of (π, V) with respect to P. Then there is a canonical isomorphism V σ → (VU )σU . This isomorphism can be described as follows. Let r: V → VU denote the quotient map. We then obtain a map q: HomP (V , V) = Vσ → HomQ (V , VU ) = (VU )σU by composing with r; here Q = P ∩ L as in Section 2. The map q then induces the isomorphism in Lemma 3.6. 165882.tex; 23/08/1999; 9:09; p.14 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 LEVEL ZERO G-TYPES 149 Remark. If P is not maximal then the Levi component L of 2.1 is proper. Suppose that (π, V) is irreducible admissible containing (σ, V ). Then 4.1 implies that the Jacquet module VU cannot be zero. In particular, (π, V) cannot be supercuspidal. This gives an alternative proof of [M1]3.5. We point that each of these proofs requires some knowledge of the structure of the Hecke algebra. 3.7. The fact that the elements Td are invertible has a further consequence. Note that in addition to H(G, σ ) there is also the intertwining algebra H (L, σU ) for the pair (Q, σU ). Let ϕ ∈ H (L, σU ) have support QℓQ for some ℓ ∈ L. Because (P , σ ) is decomposed relative to (L, P ) there is a unique element T ϕ = 8 in H (G, σ ) with support in P ℓP ; see [BK2]6.3. Let H + (L, σU ) ⊂ H (L, σU ) denote the collection of functions whose support is contained in a union of double cosets of the form QℓQ where ℓ is positive relative to (P , P). Corollary 6.12, and Theorem 7.2 of op. cit. then tell us in particular the following. THEOREM. (i) H + (L, σU ) is a subalgebra of H (L, σU ) with the same identity element. (ii) The map T induces an injective homomorphism of algebras with identity T : H + (L, σU ) → H(G, σ ). (iii) The map T in (ii) extends uniquely to an injective homomorphism of algebras with identity t: H (L, σU ) → H (G, σ ). We remark that the proof of (i) and (ii) does not require the existence of an invertible element Td , but that of (iii) does. 3.8. We now have accumulated the following results concerning the pair (P , σ ) and its relation with any parabolic subgroup P containing the Levi component L: (i) the pair (P , σ ) is decomposed with respect to (L, P); (ii) the representation σU is smooth irreducible for the (maximal) parahoric subgroup Q = P ∩ L in L; (iii) there is a strongly (P, P )-positive element s ∈ S ⊂ Z(L)(F ) such that P sP supports an invertible element of H (σ ). In the language of [BK2]8.1 the pair (P , σ ) is a cover for the pair (Q, σU ). 3.9. The following lemma will be used in 3.10 below; it is of independent interest. We start with a Levi component L in the group G. Suppose that Jˆ ⊃ J are compact open subgroups in G. Now let τ̂ be a smooth irreducible representation of Jˆ whose restriction τ̂ | J contains τ . LEMMA. Suppose (i) (J, τ ) is a cover for (JL , τL ) in the sense of [BK2]8.1. (ii) if P is any parabolic subgroup containing L with Levi decomposition P = L·U and opposite parabolic P− = L·U− then Jˆ = (J ∩U− (F ))(Jˆ ∩L)(J ∩U(F )). 165882.tex; 23/08/1999; 9:09; p.15 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 150 LAWRENCE MORRIS (iii) (Jˆ ∩ L)/ker(τ̂ |(Jˆ ∩ L)) ∼ = Jˆ/ker τ̂ . ˆ Then (J , τ̂ ) is a cover for the pair (Jˆ ∩ L, τ̂ | (Jˆ ∩ L)). Proof. Assumption (ii) guarantees an Iwahori decomposition for Jˆ with respect to (L, P), and assumption (iii) ensures that (Jˆ, τ̂ ) is decomposed with respect to (L, P) for any P containing L as Levi component. Thus our pair (Jˆ, τ̂ ) satisfies condition (i) of loc. cit., and condition (ii) is trivially satisfied by construction. We must verify condition (iii). ˆ Now define τ∗ = IndJJ (τ ); then τ̂ occurs in τ∗ . Just as before we can define the algebras H (G, τ̂ ), H (G, τ∗ ). According to [BK1]4.1.3, there is a canonical isomorphism of algebras Ŵ: H (G, τ ) → H (G, τ∗ ) with the property that if φ ∈ H (G, τ ) has support J xJ then Ŵ(φ) has support Jˆx Jˆ, and if 8 ∈ H(G, τ∗ ) has support Jˆx Jˆ then Ŵ −1 (8) has support J xJ . On the other hand, the algebra H (G, τ̂ ) can be identified (non canonically) with a subalgebra of H (G, τ∗ ). To see this it is enough to replace the representations in the algebras in question by their contragredients, since taking contragredients commutes with induction. Denoting contragredients by ‘ ∨ ’ we see that H (G, (τ̂ )ˇ ) can be identified with some τ∗ spherical functions which transform via τ̂ . Indeed, let V∗ denote the space of τ∗ ; then V∗ = ⊕ni=1 Vi where Vi runs through the not necessarily distinct irreducible constituents of τ∗ . We can then identify τ̂ with (at least) one of these, and the assertion follows from this. Moreover we see that the identity in H (G, (τ∗ )ˇ ) can be written as a sum of the identities of the algebras EndC (Vi ) corresponding to the irreducible constituents Vi counted according to multiplicity. We conclude that indeed H(G, τ̂ ) can be identified with a subalgebra of H (G, τ ); furthermore the identity of H(G, τ̂ ) occurs as a nonzero direct summand of the identity of H (G, τ ). Let s be an element of the split centre S of L. It fixes L pointwise under conjugation, hence does the same to any subgroup of L; in particular it fixes pointwise the subgroups JˆL = Jˆ ∩ L, JL = P ∩ L. It follows that s fixes τ∗ , τ̂ , τ (not merely up to isomorphism); hence there are nonzero spherical functions φs∗ , φ̂s , φs in H (G, τ∗ ), H(G, τ̂ ), H(G, τ ) respectively. Furthermore the isomorphism H(G, τ ) ≃ H (G, τ∗ ) identifies φs with a non zero multiple of φs∗ . Since (J, τ ) is a cover for (JL , τL ) condition (iii) of Definition 8.1 in [BK2] says that there is an s such that φs is invertible. It follows that φs∗ is invertible in H (G, τ∗ ). Now φs∗ is a direct sum of operators φs(1), φs(2) , . . . , φs(r) corresponding to the irreducible constituents of τ∗ , since s acts trivially on each constituent. Since φs∗ is invertible so is each φs(1) , φs(2) , . . . , φs(r) . But φ̂s is a non zero multiple of one of these, hence it is invertible in the subalgebra H (G, τ̂ ). It follows that Condition 8.1(iii) holds for the pair (Jˆ, τ̂ ) as well. 2 VARIANT 3.10. We resume the notation and conventions of 1.6, 1.12 and 2.1. In particular if F is a facet in I we write P = PF for the corresponding parahoric subgroup and we write P̂ = P̂F ⊆ G1 for the full centraliser of F . We then write M̂ = P̂ /U ; it is the group of Fq -rational points of a disconnected Fq -reductive 165882.tex; 23/08/1999; 9:09; p.16 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 151 LEVEL ZERO G-TYPES group whose connected component is the group M of 1.8. We suppose that we are given an irreducible representation σ̂ of M̂ which contains σ ; as usual we view it also as a representation of P̂ . We also write Q̂ = P̂ ∩ L. We now let σ̂U be the restriction to Q̂ of σ̂ . It is immediate from 2.1 that the hypotheses (ii) and (iii) of 3.9 hold for the pair (P̂ , σ̂ ), and we have already seen in 3.8 that hypothesis (i) holds. We immediately deduce the following. COROLLARY. The pair (P̂ , σ̂ ) is a cover for the pair (Q̂, σ̂U ). 4. G-types 4.1. We continue with the notation of Section 3. We begin by recalling a result from [M1]; see also the remark following 3.7. Namely, let L be a connected reductive F -group with L = L(F ); let Q be a maximal parahoric subgroup of L with short exact sequence 0 → U → Q → M → 0 and suppose that (σ, V ) is an irreducible cuspidal representation of M. We regard (σ, V ) as a representation of Q by inflation. PROPOSITION. Let (τ, V) be an irreducible smooth representation of L containing (σ, V ). Then (τ, V) is supercuspidal, and there is an irreducible smooth representation (ρ, W ) of Q+ = NL (Q) containing (σ, V ) such that (τ, V) = c − IndLQ+ (ρ). Proof. This is proved in [M1] Sections 1–2. 2 4.2. Next we recall some ideas and results from [BK2] Sections 3–4. First, we consider pairs (L, ρ) where L is a (rational) Levi subgroup, L = L(F ), and ρ is an irreducible supercuspidal representation of L. As usual if g ∈ G we write g ρ for the (supercuspidal) representation on gLg −1 defined by g ρ(ℓ) = ρ(g −1 ℓg). Finally, we let Xu (G) denote the group of unramified quasicharacters of the (rational points of the) reductive group G: the elements of Xu (G) are finite products of quasicharacters of the form g 7 → |φ(g)|s for some s ∈ C and some φ ∈ XF (G), where XF (G) denotes the rational character group of G. DEFINITION. The pairs (L, ρ), (L′ , ρ ′ ) are inertially equivalent if there is a g ∈ G and ξ ∈ Xu (L′ ) such that L′ = gLg −1 and g ρ ≃ ρ ′ ⊗ ξ . We denote the equivalence class containing (L, ρ) by [L, ρ]. We write B(G) for the set of equivalence classes arising from the relation in the definition above. 4.3. If P is a parabolic subgroup with Levi decomposition P = L · U we let δP denote the associated modulus quasicharacter; it provides an unramified quasicharG acter of L. We write IndG P to denote unnormalised induction from P to G and ιP to −1/2 G G denote normalised induction. These are related by ιP (τ ) = IndP (τ ⊗ δP ). The G left adjoint for ιG P is denoted by rP ; it is simply the unnormalised Jacquet functor 165882.tex; 23/08/1999; 9:09; p.17 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 152 LAWRENCE MORRIS 1/2 (of 3.6) tensored by δP . If (π, V) is an irreducible smooth representation of G, there is always a parabolic subgroup P with Levi decomposition P = L · U such that π is equivalent to a subquotient of ιG P (ρ) for some irreducible supercuspidal representation of L; see [Cs]. The resulting inertial class I(π ) = [L, ρ] is determined uniquely by π , and is called the inertial support of π . Note that since δP is an unramified quasicharacter of L, the remarks above imply that the inertial class could have been defined by replacing ι by Ind. Let SR(G) denote the category of smooth representations of G. If S ⊂ B(G) we write SRS (G) for the full subcategory of SR(G) whose objects are those objects (π, V) of SR(G) for which every irreducible subquotient has inertial support in S. If S = {s} we shall simply write SRs (G) rather than SRS (G). According to Proposition 2.10 of [BD] the category SR(G) is the direct product of the categories SRs (G) as s runs through B(G). This means that (a) for each smooth representation V, and for each s ∈ B(G) there is a unique G-subspace V s which is an object in SRs (G), maximal with respect to this property, and V is the direct sum of the V s as s runs through B(G); (b) if V, W are objects in SR(G) then HomG (V, W ) is the direct product of the various HomG (V s , W s ). DEFINITION. Let S be a subset of B(G). An S-type in G is a pair (K, σ ) where K is a compact open subgroup of G, and σ is an irreducible smooth representation of K with the following property: an irreducible smooth representation (π, V) of G contains σ if and only if I(π ) ∈ S. If S = {s} is a singleton, we shall abuse notation and write ‘s-type’. 4.4. Definition 4.3 has significant consequences, some of which we shall list below. In what follows, (K, σ ) always denotes an S-type. If (π, V) is a smooth representation we shall write V[σ ] for the G-module generated by the σ -isotypic vectors. Recall that one can form eσ ∗ V which provides an eσ ∗ H(G) ∗ eσ module. Composing this with the Morita equivalence of 3.4 then provides a functor Mρ : SRσ (G) → H (σ )-Mod. We then have the following result. THEOREM ([BK2] Theorem 4.3). (i) There is a uniquely determined G-space U such that V = V[σ ] ⊕ U. (ii) If V = V[σ ] then any irreducible G-subquotient of V contains σ . (iii) The functor Mρ above provides an equivalence of categories SRσ (G) → H (σ ) − Mod. (iv) SRS (G) = SRσ (G) 4.5. In [BK2] the authors provide many examples of s-types drawn from their work on linear and special linear groups. The prototype of all s-types is the pair (B, 1) where B is the centraliser of an alcove in the ‘enlarged’ building for G and 1 is the trivial representation of B. The full centraliser is typically larger than the 165882.tex; 23/08/1999; 9:09; p.18 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 LEVEL ZERO G-TYPES 153 corresponding Iwahori subgroup (connected centraliser). The admissible form of 4.4(iv) in this case is due to Borel [B]; see [BK2] for a simple proof of the more general situation, based on ideas in [MW]. THEOREM. Let (σ, V ) be an irreducible cuspidal representation as above and suppose that P is a maximal parahoric subgroup. Then (σ, V ) is an S-type, for a finite set S. Proof. Let (π, V) be an irreducible smooth representation containing (σ, V ). From 4.1 we can write π = c − IndG P + (ρ) where ρ is an irreducible smooth representation for P + which contains σ . Let χ denote the central quasicharacter for π , and let π ′ be another such representation which also contains σ and which also has ′ central quasicharacter χ. We suppose that π ′ = c − IndG P + (ρ ). The representations ′ ρ , ρ are determined on ZU , where U is the prounipotent radical of P hence we can write ρ ′ = ρ ⊗ τ where τ is an irreducible representation of the finite group P + /ZU . In particular if we fix a central quasicharacter there are only finitely many choices for the representation ρ and hence there are only finitely many such π containing σ with prescribed central quasicharacter. Now suppose that we consider π and π ′ as above but with possibly different central quasicharacters χ, χ ′ . We have χ| Z ∩ P = χ ′ | Z ∩ P in any case. Let Zc be the kernel of the map HZ defined in 2.3 for the Levi component G. From 2.3 there is an exact sequence 0 → Zc → Z → 3 → 0 where 3 is a lattice of finite rank and rank3 = split rank Z. On the other hand if H is the group in 1.6, then Zc ⊂ H ⊂ P for any parahoric subgroup P centralising a facet in A, since Z ⊂ T. In particular χ −1 · χ ′ is trivial on Zc hence comes from a quasicharacter on 3. Now 3 is a lattice of the same rank as the dual of the rational character group XF (G) of G. Indeed XF (G) is a subgroup of finite index in XF (Z) as one sees from the isogeny Z × Gder → G. Practically by definition, any quasicharacter of 3 is a (finite) product of ones of the form z(mod Zc ) 7 → |ψ(z)|s for some s ∈ C and some ψ ∈ XF (Z). It follows immediately that any quasicharacter of Z which is trivial on Zc is the restriction of an unramified quasicharacter of G (i.e. one which is a product of ones of the form g 7 → |φ(g)|s for some s ∈ C and some φ ∈ XF (G)). In particular χ −1 ·χ ′ is such a quasicharacter. Thus replacing π by π ⊗φ for a suitable unramified quasicharacter φ of G we see that π ⊗ φ and π ′ have the same central quasicharacter and we are in the situation of the previous paragraph. 2 Remark 4.6. One can easily produce examples (σ, V ) for which the set S is not a singleton, by considering the case where σ is unipotent cuspidal. In fact, many of the cases considered in [M1] provide such examples. VARIANT 4.7. By modifying the pair (P , σ ) slightly the set S can be reduced to a singleton. Indeed we know from 4.1 that any irreducible smooth representation (π, V) containing (σ, V ) has the form π = c − IndG P + (ρ), where ρ is an irreducible smooth representation for P + which contains σ . Since P is maximal it fixes an 165882.tex; 23/08/1999; 9:09; p.19 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 154 LAWRENCE MORRIS ‘enlarged’ vertex v × V 1 in I1 , and P + is the stabiliser in G of v × V 1 . It follows that G1 ∩ P + = P̂ is the centraliser in G1 of v × V 1 . Let σ̂ be any irreducible component of ρ | Pˆ . The group P̂ is open compact in G; in fact it is the maximal compact subgroup of P + . THEOREM. (Pˆ , σ̂ ) is a [G, π ]-type. Proof. To say that π ′ , π are inertially equivalent means that π ′ ≃ π ⊗ χ where + χ is an unramified quasicharacter of G. But then π ′ ≃ c − IndG P + (ρ ⊗ (χ|P )). Since χ is trivial on G1 hence Pˆ , it follows that π ′ contains σ̂ . On the other hand ′ ′ if π ′ contains σ̂ then π ′ = c − IndG P + (ρ ) where ρ is an irreducible constituent + of c − I ndP̂P (σ̂ ). Now P + /P̂ can be identified with a subgroup of the lattice G/G1 , and it contains the group denoted 3 in the proof of 4.5 because P + contains Z. It follows that P + /P̂ is a sublattice of G/G1 of the same rank, hence any quasicharacter of it extends to a quasicharacter of G/G1 . Now observe that if ρ, ρ ′ both contain σ̂ then they are determined on Z ∩ P̂ by the central character of σ̂ ; since Z is a split F -torus this means that the representation σ̂ can be extended to Z P̂ (by an unramified quasicharacter of Z). Clifford–Mackey theory then implies that ρ ′ = ρ ⊗ χ for some quasicharacter χ of P + P̂ , and then that π ′ = π ⊗ χ ′ for some extension χ ′ to G of χ. 2 Remark. Note that this result says that each irreducible constituent σ̂ of ρ|Pˆ is an s-type for the same singleton s. 4.8. We now combine 3.8, 4.5, and [BK2] Theorem 8.3, to deduce the following result. THEOREM. If (σ, V ) is an irreducible cuspidal representation as above where P is not necessarily maximal, then (σ, V ) is an S-type for a finite set S. Proof. Let L be as in 2.1. Applying Theorem 4.5 to L and the pair (Q, σU ) we see that (Q, σU ) is an SL -type for some finite set SL . Here SL consists of a finite set of inertial equivalence classes with respect to L of the form [L, τ ] where τ is an irreducible supercuspidal representation of L. On the other hand 3.8 says that (P , σ ) is a G-cover for the pair (Q, σU ). Theorem 8.3 of [BK2] then says that in this situation (P , σ ) is an SG -type where SG is the finite set formed from the inertial equivalence classes with respect to G of the elements in SL . Briefly, the argument goes as follows. First, let (π, V) be an irreducible smooth representation of G containing (σ, V ). There is always an irreducible supercuspidal representation τ of L containing σU such that π is isomorphic to a G-subspace of IndG P (τ ). Indeed 3.6 implies that the unnormalised Jacquet module (πU , VU ) contains σU . Since σU is an SL -type Proposition 2.10 of [BD] (described in 4.3 above) and part (iv) of Theorem 4.4 imply that some irreducible quotient τ has I(τ ) ∈ SL . Since δP is unramified the same is true on replacing the unnormalised Jacquet module by the normalised version. Frobenius reciprocity then implies that I(π ) ∈ SG . 165882.tex; 23/08/1999; 9:09; p.20 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 LEVEL ZERO G-TYPES 155 To go the other way, let SG be as in the preceding paragraph, and suppose that I(π ) ∈ SG . This means that π occurs as a subrepresentation of ιG P (ρ) for some (L, ρ) with [L, ρ] in S, and by construction ρ contains σU . One may now apply 3.6 to see that π contains σ . 2 VARIANT 4.9. Again, by replacing the pair (P , σ ) by the pair (P̂ , σ̂ ) where P̂ is the full centraliser of the appropriate facet and σ̂ is an irreducible smooth representation of P̂ which contains σ as in 3.10, we can deduce the following. THEOREM. (Pˆ , σ̂ ) is an s-type for a singleton set s. Proof. We know from Variant 3.10 that (P̂ , σ̂ ) is a G-cover for the pair (Q̂, σ̂U ). Here the (P̂ , σ̂ ) is with respect to G, while (Q̂, σ̂U ) is with respect to L. The result then follows immediately from Variant 4.7 and Theorem 8.3 of [BK2]. 2 Remark. The technique above can be codified into a general principle. We revert to the notation of 3.9, and assume we have pairs (Jˆ, τ̂ ), (J, τ ) satisfying the conditions in Lemma 3.9. Assume further that (Jˆ ∩ L, τ̂ | (Jˆ ∩ L)) is an s-type for a singleton s. The argument above implies that (Jˆ, τ̂ ) is an sG -type for a singleton sG . 4.10. Now we recall some results in [BD]. First, let s = [L, τ ] ∈ B(G) and let (L, τ ) be a representative for it. Then (L, τ ) determines a class sL ∈ B(L). If we change the representative then it must have the form (g L, g τ ⊗ χ) for some g ∈ G and χ ∈ Xu (g L). If we write L′ for g L and sL′ for the resulting class in B(L′ ) then conjugation by g provides an equivalence of categories SRsL (L) ≃ SRsL′ (L). Second, if we interpret [BD] 2.8 in the language above (c.f. [BK2] 2.3,6.1) we obtain the following statements. THEOREM. (i) Let (π, V) be an object of SRs (G). Then (πU , VU ) is an object of the subcategory 5t SRt (L) of SR(L) where t runs through the NG (L) orbit of sL . (ii) The representation (π, V) is an object of SRs (G) if and only if there are parabolic subgroups P of G each of which has Levi component L, and smooth ` representations τL ∈ SRsL (L) and a G-injection π → P IndG (τ P L ). 4.11. The unnormalised Jacquet functor provides a functor rU : SRS (G) → SR(L). Composing this with the projection functor pSL : SR(L) → SRSL (L) guaranteed by Theorem 4.4(i) we obtain a functor rU : SRS (G) → SRSL (L), since this last category is also the category SRσU (L) by 4.4(iv). Going the other way, 4.10(ii) implies that the unnormalised induction functor Ind takes the category SRsL (L) to the category SRs (G). Here s is the class determined by sL as in the proof of Theorem 4.8. It follows that Ind takes SRSL (L) to the category SRS (G). If τ is an object in SRsL (L) we then have HomG (π, IndG P (τ ) ≃ HomL (rU (π ), S L τ ) ≃ HomL (p rU (π ), τ ). 165882.tex; 23/08/1999; 9:09; p.21 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 156 LAWRENCE MORRIS In other words, we have the following result. PROPOSITION. The unnormalised Jacquet functor rU provides a functor rU : SRS (G) → SRSL (L). It has a right adjoint functor provided by the unnormalised induction functor Ind. Remark. If we used normalised induction here we would have to (un)twist the −1/2 Jacquet functor by δP . 4.12. If f : A → B is a homomorphism of associative rings, and M is a B-module we write f ∗ (M) for the A-module M induced by f . If N is an A-module we write f∗ (N) for the B-module HomA (B, N). Theorem 4.8 guarantees equivalences of categories SRσ (G) → H (G, σ ) − Mod, SRσU (L) → H(L, σU ) − Mod. Furthermore, Proposition 4.11 implies that unnormalised induction provides a functor SRσU (L) → SRσ (G), and that the Jacquet functor rU provides a functor rU : SRσ (G) → SRσU (L). Recall the injective algebra homomorphism tP : H (L, σU ) → H (G, σ ) of 3.7. Applying Corollary 8.4 of [BK2] to this we immediately obtain the following result. THEOREM. Each of the following diagrams is commutative: SRσ (G) rU ❄ SRσU (L) SRσ (G) Mσ✲ H (G, σ ) − Mod tP∗ ❄ ✲ H (L, σU ) − Mod; MσU Mσ✲ H (G, σ ) − Mod ✻ Ind SRσU (L) tP∗ ❄ ✲ H (L, σU ) − Mod. MσU Acknowledgements It is a pleasure to thank Colin Bushnell and Philip Kutzko for sharing their ideas and results at various opportune times, and for several detailed comments on the paper itself; in particular Colin Bushnell pointed out a blunder in an earlier version. It is also a pleasure to thank the referee for a number of detailed and helpful comments. Some of the results obtained here are proved in [MP] Section 6 by 165882.tex; 23/08/1999; 9:09; p.22 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614 LEVEL ZERO G-TYPES 157 different methods. 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Soc., Providence, RI, 1979, pp. 29–69. 165882.tex; 23/08/1999; 9:09; p.23 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 19 Jun 2020 at 17:45:33, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1001019027614