ON SATAKE PARAMETERS FOR REPRESENTATIONS WITH PARAHORIC FIXED VECTORS. 1. Introduction

ON SATAKE PARAMETERS FOR REPRESENTATIONS WITH PARAHORIC FIXED VECTORS THOMAS J. HAINES Abstract. This article constructs the Satake parameter for any irreducible smooth J- spherical representation of a p-adic group, where J is any parahoric subgroup. The main novelty is for groups which are not quasi-split, and the construction should play a role in the geometric Satake isomorphism for such groups. 1. Introduction Let G be a conntected reductive group over a non-archimedean local field F. Let K G(F ) be a special maximal parahoric subgroup, cf. [HRa], [HRo]. Let π be an irreducible smooth representation of G(F ) with π K 0. We will show how to attach to π a natural element s(π) of the set [ĜI F Φ] ss /Int((ĜI F )). Here I F is the inertia subgroup of the Weil group W F, and Φ is the inverse of a Frobenius generator for Aut( F un /F ). The element s(π) will be characterized using the Satake isomorphism of [HRo] for the Hecke algebra H(G(F ), K). If G is unramified over F, then ĜI F = Ĝ and by results of [HRo], K is a special maximal compact subgroup of G(F ), and the Satake isomorphism in [HRo] coincides with the classical one described by Cartier [Car]. In this case π s(π) agrees with the usual Satake parameter for unramified groups which is described in [Bor]. In the general quasi-split case, the main complication is due to the disconnectedness of the reductive group ĜI F. When G is quasisplit over F and splits over a tamely ramified extension, then π s(π) has been constructed by Manesh Mishra [Mis], and also by X. Zhu [Zh, Prop. 6.4] (the latter for certain very special maximal parahoric subgroups). When G is not even quasi-split, we will use the theory of normalized transfer homomorphisms, developed in the Appendix to [H13], to define s(π). The main purpose of this article is to explain how to construct s(π) in general, and to pin down the set S(G) of elements in [ĜI F Φ] ss /Int((ĜI F ) ) which are of the form s(π) for some π. If G/F is quasi-split, then we will see that S(G) = [ĜI F Φ] ss /Int((ĜI F )); in general, S(G) will be an explicitly determined subset of (ĜI F Φ) ss /Int((ĜI F ) ), defined using the normalized transfer homomorphisms mentioned above. The same techniques show how to construct the s-parameter in the hypothetical Deligne- Langlands correspondence one could hope to prove for parahoric-spherical representations of general connected reductive groups. See 11. As above, parahoric should be understood in the sense of Bruhat-Tits [BT2], as the O F -points of a connected group scheme over O. 1

2 THOMAS J. HAINES So for example an Iwahori subgroup here is somewhat smaller than the naive notion that sometime appears in the literature under the same name, and therefore the Iwahori-Hecke algebra and its center are slightly larger (cf. [?] and [H13, Appendix]). The construction uses the Bernstein isomorphisms of [H13] describing the centers Z(G(F ), J) of the Hecke algebras H(G(F ), J). Here again, in the general case we make use of the normalized transfer homomorphisms, which also exist for such centers (see[h13]). Let G/F be any connected reductive group and let J be any parahoric subgroup. Let Π(G/F ) denote the set of isomorphism classes of irreducible smooth representations π of G(F ) on complex vector spaces, and for any compact open subgroup J G(F ) let Π(G/F, J) denote the subset consisting of those π with π J 0. Summary: We will define in a canonical way a subset S(G) [ĜI F Φ] ss /ĜI F and for every π Π(G/F, J) an element s(π) S(G), such that the following properties hold: (1) G/F is quasi-split if and only if S(G) = [ĜI F Φ] ss /ĜI F. (2) If K is a special maximal compact subgroup, then π s(π) gives the parametrization of K-spherical representations Π(G/F, K) S(G). (3) The parameter s(π) predicts part of the local Langlands parameter ϕ π that is conjecturally attached to π: ϕ π (Φ) = s(π) in [Ĝ Φ] ss/ĝ. (4) S(G) is explicitly described in terms of the normalized transfer homomorphisms of [H13], which also serve to relate parameters across inner twistings G G (where G is assumed quasi-split), in a manner compatible with a hypothetical generalized Jacquet-Langlands correspondence.. Acknowledgements. I thank Robert Kottwitz for a helpful conversation about the scope of the results in 4. 2. Notation Define L, G(L) 1, G(F ) 1, etc. Define δ P and i G P (σ), etc. Define n U etc. State that we use Kottwitz conventions on dual groups Ĝ and their Γ-actions, see [Ko84, 1]. Warning: the map T Ĝ is not generally Γ-equivariant, but this is the case when G is quasi-split... 3. First parametrization of K-spherical representations Fix a special maximal parahoric subgroup K G(F ). In G, choose any maximal F -split torus A whose associated apartment in the Bruhat-Tits building B(G ad, F ) contains the special vertex associated to K. Let M := C G (A) be the centralizer of A, a minimal F - Levi subgroup. Following [H13], we call the group of homomorphisms M(F )/M(F ) 1 C

ON SATAKE PARAMETERS FOR REPRESENTATIONS WITH PARAHORIC FIXED VECTORS 3 the group X w (M) of weakly unramified characters on M(F ). The Kottwitz homomorphism [Ko97, 7] induces an isomorphism M(F )/M(F ) 1 = X (Z( M) I F Φ ), so that X w (M) = Z( M) I F Φ, a diagonalizable group over C. Given χ X w (M), the Iwasawa decomposition of [HRo, Cor. 9.1.2] allows us to define an element Φ K,χ i G P (χ)k by Φ K,χ (mnk) = δ 1/2 (m) χ(m) P for m M(F ), n N(F ), and k K (here N is the unipotent radical of an F -parabolic subgroup P having M as Levi factor). Then define the spherical function Γ χ (g) = Φ K,χ (kg) dk K where vol dk (K) = 1. Let π χ denote the smallest G-stable subspace of the right regular representation of G(F ) on C (G(F )) containing Γ χ. Then, as in [Car, 4.4], we see that π χ is irreducible, that π χ = πχ iff χ = w χ for some w W (G, A), and that every element of Π(G/F, K) is isomorphic to some π χ. Thus we have the following first parametrization of Π(G/F, K). Proposition 3.0.1. The map χ π χ sets up a 1-1 correspondence (3.0.1) (Z( M) I F ) Φ /W (G, A) Π(G/F, K). Moreover, if f H(G(F ), K), then π χ (f) acts on πχ K Satake isomorphism by the scalar S(f)(χ), where S is the (3.0.2) S : H(G(F ), K) C[Z( M) I F Φ /W (G, A)] of [HRo, Thm. 1.0.1]. Here the right hand side denotes the ring of regular functions on the affine variety Z( M) I F Φ /W (G, A). 4. Fixed-point subgroups under finite groups of automorphisms Steinberg [St] proved fundamental results on cyclic groups of automorphisms of a simply connected semisimple algebraic group. The aim here is to extend some of those results to finite groups of automorphisms of a reductive group. The following might be known, but we include a complete proof here due to the lack of a suitable reference. Notation: If a group J acts by automorphisms on an algebraic group P, we write P for the neutral component of P and often write P J, instead of (P J ). Proposition 4.0.1. Let H be a possibly disconnected reductive group over an algebraically closed field k. Assume that a finite group I acts by automorphisms on H and preserves a splitting (T, B, X), consisting of a Borel subgroup B, a maximal torus T in B, and a principal nilpotent element X = α (T,B) X α for some non-zero elements X α (Lie H ) α indexed by the B-positive simple roots (T, B) in X (T ). Let U be the unipotent radical of B, and let N = N(H, T ) be the normalizer of T in H. Then:

4 THOMAS J. HAINES (a) The algebraic group H I is reductive with identity component (H I ) = [(H ) I ] and with splitting (T I,, B I,, X I ), where B I, = T I, U I and where X I is a principal nilpotent in Lie(H I, ) constructed from X. (If char(k) 2, then X I = X.) (b) We have T I H I, = T I, and N I H I, = N(H I,, T I, ). (c) If W := W (H, T ) := N/T, then every element of W I has a representative in N I H I,, and thus N(H I,, T I, ) = W I. (d) The inclusion T H induces a bijection π 0 T I π 0 H I. Before beginning the proof, note that giving the data of X is equivalent to giving the data {x α } α (T,B) of root group homomorphisms x α : G a U α, where U α is the maximal connected unipotent subgroup of H normalized by T and with Lie(U α ) = (Lie H ) α. This is because the Lie functor gives an isomorphism Isom k Grp (G a, U α ) = Isom k (k, Lie(U α )). Proof. Quite generally (H I ) contains [(H ) I ] with finite index, and as both are connected algebraic groups, they coincide. Lemma 4.0.2. Let Ψ be a reduced root system in a real vector space V, with set of simple roots. Suppose I is a finite group of automorphisms of V which preserves Ψ and. Let ᾱ V denote the average of the I-orbit of α Ψ, and let Ψ I = {ᾱ α Ψ} and I = {ᾱ α }. Then (1) Ψ I is a possibly non-reduced root system in V I with set of simple roots I ; (2) W (Ψ I ) = W (Ψ) I, where W (Σ) denotes the Weyl group of a root system Σ. Proof. First assume I = τ. Let Ψ τ Ψ τ be defined by discarding those elements of Ψ τ which are smaller multiples of others. Then [St, 1.32, 1.33] shows that Ψ τ is a root system with Weyl group W (Ψ) τ. The only difference between Ψ τ and Ψ τ is that the latter could contain 1 2 α for α Ψ τ, and then only for a component of Ψ of type A 2n. Consideration of the root system for a quasi-split unitary group in 2n + 1 variables attached to a separable quadratic extension of a p-adic field (cf. [Tits, 1.15]) shows that adding such half-roots to a root system of form Ψ τ, still gives a root system; so Ψ τ is indeed a root system, with simple roots τ and with the same Weyl group W (Ψ) τ. Note that if τ is odd, then Ψ τ is again reduced (comp. [HN, Lemma 9.2]). Now decompose (V, Ψ) into a sum of simple systems (V j, Ψ j ). The action of I permutes these simple systems while the stabilizer of each component continues to act through its automorphism group. Therefore we may assume (V, Ψ) is simple. Using the classification, we may assume I acts through a faithful action of Z/2Z, Z/3Z, or S 3 on. In the last case, Ψ = D 4 and the I-orbits on Ψ + and on coincide with those of the subgroup Z/3Z S 3. Thus we may assume I is cyclic, and we may apply the preceding paragraph. We will apply Lemma 4.0.2 when Ψ = Ψ(H, T ) (resp. = (T, B)), the set of roots (resp. B-simple roots) for T in Lie(H ) in the vector space V = X (T ) R. The set of positive roots Ψ + is a disjoint union of subsets S α, where S α consists of all those positive roots whose projection to V I is proportional to that of α (comp. [St, Thm. 8.2(2 )]). We

ON SATAKE PARAMETERS FOR REPRESENTATIONS WITH PARAHORIC FIXED VECTORS 5 will always index S α by a minimal element α in this set (use the usual partial order on positive roots). For example (cf. loc. cit.), if I = τ, then S α comes in two types: (Type 1) S α is a τ-orbit {α, τα,... }, no two of which add up to a root; (Type 2) S α = {α, τα, β}, where β := α + τα is a root; this occurs only in type A 2n. For a root system of the form Ψ τ, we write (Ψ τ ) red (resp. (Ψ τ ) red )) for the root system we get by discarding vectors from Ψ τ which are shorter (resp. longer) multiples of others. For example, Ψ τ = (Ψ τ ) red. We now make the following temporary assumptions: (i) H I, is reductive with splitting (T I,, B I,, X I ), where X I denotes a principal nilpotent element of Lie(H I, ) constructed from X. (If char(k) 2, then X I = X.) (ii) Ψ(H I,, T I, ) can be identified with (Ψ I ) red if char(k) 2 and with (Ψ I ) red if char(k) = 2. (iii) If char(k) 2, the group U I is the product of the subgroups Uᾱ = Ga indexed by the various subsets S α, where Ad(T I, ) acts via ᾱ on Lie(Uᾱ) Lie(H I, ). In particular U I is connected. Furthermore, Uᾱ is contained in the product of the groups U α for α S α. If char(k) = 2, the same statements hold with Uᾱ replaced by U 2ᾱ. Remark 4.0.3. Property (iii) automatically implies another property: (iv) The map N I W (Ψ) I, n w n, has the property that for every subset S α, we have nu iᾱ n 1 = U wn(iᾱ), for i {1, 2}. Lifting step. Let sᾱ W (Ψ) I be the reflection corresponding to a simple root ᾱ for some α (T, B) (cf. Lemma 4.0.2). We wish to show it can be lifted to an element in N I H I,. Using (iii), we may choose u Uᾱ\{1} if char(k) 2 (resp. U 2ᾱ \{1} if char(k) = 2). Using the Bruhat decomposition for U in place of U, we can write uniquely u = u 1 nu 2, where u 2 U, n N, and u 1 U nun 1 ; since u is I-fixed, u 1, n, u 2 are too. The element n belongs to N I U I U iᾱ U I (i {1, 2}) and thus to N I H I, by (iii). The element w n W = W (Ψ) to which n projects is different from 1, is fixed by I, and is in the group generated by the reflections s τ(α), τ I. For the last statement, use (iii) and [Sp, 9.2.1] to show that n U ±α α U where α ranges over elements in S α (a Levi subset of Ψ + when α ), and then use the Bruhat decomposition again. Let Ψ I,+ denote the positive roots of Ψ I. If w n sᾱ then w n sends some root in Ψ I,+ \{ᾱ, 2ᾱ} to Ψ I,+. Then as an element of W (Ψ), w n makes negative some positive root outside S α, in violation of w n s τ(α), τ I. Thus w n = sᾱ, and n is the desired lift of sᾱ. By Lemma 4.0.2, W (Ψ) I = W (Ψ I ) and so any element w W (Ψ) I is a product of elements sᾱ as above; hence w can be lifted to N I H I,. This proves part of (c). Since N I clearly maps to W I = W (Ψ) I, the proof also shows that N I T I, U I, U I. As (H ) I = N I, U I by the Bruhat decomposition of H, we obtain (4.0.3) (H ) I = T I, U I, U I.

6 THOMAS J. HAINES From this we see [(H ) I ] contains the connected subgroup (T I ), U I, U I with finite index, and so (4.0.4) [(H ) I ] = (T I ), U I, U I. Hence (4.0.5) (H ) I = T I [(H ) I ]. We claim that (T I ) = T I [(H ) I ]. The inclusion is clear. As for the other, using (4.0.4) it is enough to show that T I U I, U I (T I ). But U I, U I lies in the image of [(H ) sc ] I [(H ) der ] I, and so we are reduced to the case where H is simply connected 1. In that case X (T ) has a Z-basis permuted by I, and so T I is already connected, and the result is obvious. It is clear that N I H I, N(H I,, T I, ). We claim that equality holds. The Bruhat decomposition for the reductive group H implies that an element of [(H ) I ] decomposes uniquely in the form unv where n N I [(H ) I ], v U I and u U I n U I. (Note n automatically belongs to [(H ) I ] since U I and U I n U I are connected, as follows from (iii, iv).) Since by (i,iii) [(H ) I ] is reductive with Borel subgroup B I, = T I, U I, the element also decomposes as u 1 n 1 v 1 where n 1 N(H I,, T I, ), v 1 U I, and u 1 U I n 1 U I. Comparing these decompositions, we see n = n 1, i.e., N(H I,, T I, ) = N I H I,. As N I H I, surjects onto W I, we deduce N I H I, /T I H I, W I. The above paragraphs show the left hand side is W (H I,, T I, ). At this point we have proved (a-d) assuming (i-iii) 2. Now we need to prove (i-iii). We first consider the case where I is generated by a single element τ. We use results of Steinberg [St], especially 8.2, 8.3. (Much of what we need also appears in [KS, 1.1].) We will adapt the proof of [St, Theorem 8.2]: it assumes H is semisimple and simply connected and only assumes τ fixes T and B, but the argument carries over when τ fixes a splitting because in loc. cit. step (5) we may take t = 1. Indeed, for each B-positive root α X (T ) let x α : G a H be the corresponding root homomorphism, and write (4.0.6) τx τα (y) = x τ (c τα y) for all y k and some constants c α k. Then the hypothesis that τ fixes X is equivalent to c α = 1, α (T, B). Since α(t) = c τα for α (T, B) by definition of t, we may choose t = 1. Further, [St] shows that U τ is connected by analyzing the conditions under which an element of the form x α (y α )x τα (y τα ) (indices ranging over S α ) belongs to U τ. To ease notation, write H 1 (resp. T 1, B 1 ) for H τ, (resp. T τ,, B τ, ). First suppose α (T, B) is such that S α is of Type 1. Consider the average ᾱ of the orbit {α, τα,... }. Since c α = 1 1 Since it fixes a splitting, the action of I on (H ) der can be lifted to give a compatible action on its simply-connected cover (H ) sc, for example by using the Isomorphism Theorem [Sp, 9.6.2]. 2 In fact only (i,iii) are needed in the argument.

ON SATAKE PARAMETERS FOR REPRESENTATIONS WITH PARAHORIC FIXED VECTORS 7 for all α (T, B), [St, Thm. 8.2 (2)] shows that we may construct a root homomorphism with values in U H 1 by the formula x H 1 ᾱ (4.0.7) x H 1 ᾱ (y) = x α (y) x τα (y). Now suppose α (T, B) is such that S α is of Type 2: S α = {α, τ(α), β}, where β := α+τα is a root of H. Then β/2 = ᾱ. Following loc. cit. Thm. 8.2, we may normalize the homomorphism x β such that [x α (y), x τα (y )] = x β (yy ), where [a, b] := a 1 b 1 ab. We stress that x β depends on the choice of ordering (α, τα) of the set {α, τα}. Now assume char(k) 2. Then since c α = c τα = 1, according to [St, Thm.8.2(2)] we may define x H 1 ᾱ : G a H 1 by (4.0.8) x H 1 ᾱ (y) = x α (y) x τα (y) x β ( y 2 /2). If char(k) = 2, then we define x H 1 2ᾱ : G a H 1 by (4.0.9) x H 1 2ᾱ (y) = x β(y). Define U iᾱ to be the image of x H 1 iᾱ, for i {1, 2}. Then loc. cit. 8.2 shows that U τ is the product of the subgroups U iᾱ corresponding to the various S α s. Thus property (iii) holds for I = τ. Now we consider (i). The argument of the Lifting step above used only Lemma 4.0.2 and property (iii), and so can be used here to show that N τ H τ, W τ. Let R H τ, be the unipotent radical. By [St, Cor. 7.4], R is contained in a τ-stable Borel subgroup of H, which we may assume to be B; hence R U τ. But by the surjectivity of N τ H τ, W τ and by (iii, iv), only the trivial subgroup of U τ can be normalized by N τ H τ,. Hence R = 1 and H τ, is reductive. Since U τ is connected it follows that B τ, = T τ, U τ. Also, H τ, /B τ, is proper, so B τ, is a parabolic subgroup of H τ,. Thus B τ, is a Borel subgroup of H τ,, being a connected solvable parabolic subgroup of a reductive group. It follows that T τ, is a maximal torus of H τ,. Finally, we need to construct the splitting X τ. If char(k) 2, then the definition of x H 1 ᾱ above shows that the simple roots for Ψ(H τ,, T τ, ) are the averages ᾱ of the τ-orbits of the α (T, B). For a simple root α (T τ,, B τ, ), let X α := X α. α (T,B) ᾱ=α One can check taking differentials of (4.0.7) and (4.0.8) that X α Lie(H τ, ) α, and so X = α X α gives the desired splitting. If char(k) = 2, we have to be more careful: Lie(H τ, ) can be smaller than Lie(H ) τ, and in fact when S α is Type 2, X α + X τα will not belong to Lie(H τ, ). Nevertheless, we can define X τ to be the splitting corresponding to the collection of root-group homomorphisms (4.0.10) {x H 1 ᾱ (y)} {x H 1 2ᾱ (y)}

8 THOMAS J. HAINES where in the first (resp. second) union is over subsets S α of Type 1 (resp. Type 2) attached to α (T, B). Along the way, we have verified (ii) when I = τ. Note again that when τ is odd Ψ τ is reduced. Thus we have proved (i-iii) hold when I = τ. Now suppose I = S 3, which will arise in the same way as in the proof of Lemma 4.0.2. Write I = τ 1, τ 2, where τ 1 generates the normal subgroup of order 3, and τ 2 is of order 2. Then by applying the above argument first with τ = τ 1 and then with τ = τ 2 (note that τ 2 fixes the splitting X τ 1 = X), we see that (i-iii) also hold in this case. Now consider the most general case, where I is arbitrary. Let Z denote the center of H. We have a short exact sequence (4.0.11) 1 Z I (H ) I (H ad )I H 1 (I, Z). As Z has finite I -torsion, we see H 1 (I, Z) is finite and thus [(H ) I ] surjects onto [(H ad )I ]. So we have an exact sequence for H (4.0.12) 1 Z I H I, H I, (H ad )I, 1. This exact sequence shows that H I, is reductive if (Had )I, is reductive. Similarly, (i-iii) for H follow formally from (i-iii) for Had. Thus, we are reduced to assuming H = Had. Then H is a product of simple groups, which are permuted by I and which each carry an action by the stabilizer subgroup of I. We may therefore assume H is simple, and the classification shows we may assume I = Z/2Z, Z/3Z, or S 3. Each of these cases was handled above, and we conclude that (i-iii) indeed hold for adjoint groups. Remark 4.0.4. Some of Proposition 4.0.1 appears in [Ri, Lemma A.1], but with the unnecessary assumption that the order I is prime to char(k). The latter assumption is indeed necessary to prove that H I is reductive, when I is finite but is not assumed to fix any pair (T, B) (see [PY, Thm. 2.1, Rem. 3.5], on which [Ri] relies). Lemma 4.0.5. Assume H, B, T, X, and I are as in Proposition 4.0.1. Let Z denote the center of H. (i) Suppose (H ad )I is connected. Then the natural map Z I π 0 (H ) I is surjective. (ii) Let T ad denote the image of T in (H ) ad. If (T ad ) I is connected, then Z I (H ) I, = (H ) I. Proof. For part (i), the key point is that H I, surjects onto (Had )I by (4.0.12). Part (ii) follows immediately from (i) and Proposition 4.0.1(d). Lemma 4.0.6. Suppose H, B, T are as in Proposition 4.0.1, and assume I = τ fixes B, T but not necessarily X. Also assume char(k) = 0 and that H is connected. In the group H τ, the set (H τ) ss of semisimple elements in the coset H τ is the set of H-conjugates of T τ.

ON SATAKE PARAMETERS FOR REPRESENTATIONS WITH PARAHORIC FIXED VECTORS 9 Proof. Conjugates of elements in T τ, are semi-simple since τ has finite order and char(k) = 0. Conversely, suppose hτ is semi-simple. Since H is connected, [St, 7.3] ensures that some τ-conjugate of h lies in B; hence we may assume h B. Now by loc. cit. Theorem 7.5 applied to the (disconnected) group B τ, Int(hτ) fixes a maximal torus T B. Write T = bt b 1 for some b B. We obtain b 1 hτ(b) N B = T, thus hτ is H-conjugate to T τ. 5. The parameter space Assume G is a quasi-split connected reductive group over F, and fix an F -rational maximal torus/borel subgroup T B thereof. Let Ĝ be the complex dual group of G. By definition, it carries an action by the absolute Galois group Γ over F, which factors through a finite quotient and fixes a splitting of the form ( B, T, X) (cf. [Ko84, 1.5]), where we may assume T is the complex dual torus for T. Note that since G is quasi-split with Γ-fixed pair (B, T ), the Γ-action on T inherited from Ĝ agrees with that derived from the Γ-action on X (T ) = X ( T ). We shall use this below, applied to the torus T sc. The group ĜI F is reductive by Proposition 4.0.1 and carries an action by τ = Φ which fixes the splitting ( T I F,, B I F,, X) (see the proof of Proposition 4.0.1). Write I for I F in what follows. By Lemma 4.0.6 applied with H = ĜI,, we have a surjection T I, [ĜI, Φ] ss /ĜI,. Let Ẑ = Z(Ĝ). Since the torus T sc in G sc is I-induced, its dual torus T ad is also I-induced, and so ( T ad ) I is connected. Then from Lemma 4.0.5 (ii) with H = Ĝ, we see ẐI ĜI, = ĜI. Now multiplying on the left by ẐI, the above surjection gives rise to a surjection (5.0.13) ( T I ) Φ [ĜI Φ] ss /ĜI. Let N = N(Ĝ, T ) and Ŵ = W (Ĝ, T ). By 4.0.1(c), Ŵ I = W (ĜI,, T I, ), and (Ŵ I ) Φ = W (ĜΓ,, T Γ, ). Further, we see that ( N I ) Φ = N Γ surjects onto (Ŵ I ) Φ = Ŵ Γ. We thus have a well-defined surjective map (5.0.14) ( T I ) Φ /Ŵ Γ [ĜI Φ] ss /ĜI. Proposition 5.0.1. The map (5.0.14) is bijective. Proof. It remains to prove the injectivity, which is similar to [Mis, Prop. 11]. Suppose there exist s, t T I and zg 0 Z I ĜI, with (zg 0 ) 1 sφ(zg 0 ) = t. Write U for the unipotent radical of B. Via the Bruhat decompsition write g 0 = u 0 n 0 v 0 where n 0 N I ĜI,, v 0 U I, and u 0 U I n 0 U I. We have sφ(u 0 )Φ(zn 0 )Φ(v 0 ) = u 0 (zn 0 )v 0 t, and thus (sφ(u 0 )s 1 ) sφ(zn 0 ) Φ(v 0 ) = u 0 (zn 0 t) (t 1 v 0 t).

10 THOMAS J. HAINES Uniqueness of the decomposition yields sφ(zn 0 ) = zn 0 t. The image of zn 0 N I in Ŵ I is therefore Φ-fixed, so lifts (cf. Prop. 4.0.1) to some element n 1 N Γ ; write zn 0 = t 1 n 1 for some t 1 T I. The resulting equation n 1 1 (t 1 1 sφ(t 1))n 1 = t shows that s and t have the same image in ( T I ) Φ /Ŵ Γ. Corollary 5.0.2. The set [ĜI F canonically isomorphic to ( T I F ) Φ /Ŵ Γ. has the structure of an affine algebraic variety If G is not quasi-split over F, consider an inner twisting ψ : G G where G /F is quasisplit. Then ψ induces an Γ-equivariant isomorphism Ĝ Ĝ, and thus an isomorphism [Ĝ I F Φ] ss /Ĝ I F [ĜI F. and the right hand side inherits the structure of an affine algebraic variety from the left hand side. 6. Second parametrization of K-spherical representations Now again assume G is quasi-split over F. Let A be a maximal F -split torus in G, and suppose T = Cent G (A); let W = W (G, T ) and recall that since G is quasi-split, W Γ is the relative Weyl group W (G, A). There is a Γ-equivariant isomorphism W = Ŵ. Putting this together with Proposition 5.0.1 yields the following result. Proposition 6.0.1. Assume G is quasi-split over F. There is a natural bijection (6.0.15) ( T I F ) Φ /W (G, A) [ĜI F. Theorem 6.0.2. Assume G is quasi-split over F and K is any special maximal compact subgroup of G(F ). We have the following parametrization of Π(G/F, K) Π(G/F, K) (3.0.1) ( T I F ) Φ /W (G, A) (6.0.15) [ĜI F. We define the Satake parameter s(π) for π Π(G/F, K) to be the image of π under this map. Then s(π) may be characterized as follows: it is the unique element of the affine variety [ĜI F such that tr(f π) = S(f)(s(π)) for all f H(G(F ), K), where we view the Satake transform S(f) as a regular function on [ĜI F via (6.0.15).

ON SATAKE PARAMETERS FOR REPRESENTATIONS WITH PARAHORIC FIXED VECTORS 11 7. Construction of parameters: quasi-split case Continue with the assumptions of 6. Let J G(F ) be any parahoric subgroup, and let π Π(G/F, J). By [H13, 11.5], there exists a weakly unramified character χ ( T I F ) Φ /W (G, A) such that π is an irreducible subquotient of the normalized induction i G B (χ). We have the Bernstein isomorphism of [H13, 11.9.1]3 (7.0.16) S : Z(G(F ), J) C[( T I F ) Φ /W (G, A)]. By loc. cit. 11.8, z Z(G(F ), J) acts on i G B (χ)j by the scalar S(z)(χ). Definition 7.0.1. Define s(π) [ĜI F to be the image of χ ( T I F ) Φ /W (G, A) under the bijection (6.0.15). Then we have the following characterization of s(π): an element z Z(G(F ), J) acts on π J by the scalar S(z)(s(π)). 8. Review of transfer homomorphisms In order to define Satake parameters for general groups, we need to recall the normalized transfer homomorphisms introduced in [H13, 11]. Let G be a quasi-split group over F. Let F s denote a separable closure of F, and set Γ = Gal(F s /F ). Recall that an inner form of G is a pair (G, Ψ) consisting of a connected reductive F -group G and a Γ-stable G ad (F s )-orbit Ψ of F s -isomorphisms ψ : G G. The set of isomorphism classes of pairs (G, Ψ) corresponds bijectively to H 1 (F, G ad ). Choose any maximal F -split tori A G and A G. Let M = Cent G (A) and T = Cent G (A ), a maximal F -torus in G. Now choose an F -parabolic subgroup P G having M as Levi factor, and an F -rational Borel subgroup B G having T as Levi factor. Then there exists a unique parabolic subgroup P G such that P B and P is G (F s )-conjugate to ψ(p ) for every ψ Ψ. Let M be the unique Levi factor of P containing T. Then define Ψ M = {ψ Ψ ψ(p ) = P, ψ(m) = M }. The set Ψ M is a nonempty Γ-stable M ad (F s )-orbit of F s -isomorphisms M M, and so (M, Ψ M ) is an inner form of M. Choose any ψ 0 Ψ M. Then since ψ 0 A is F -rational, ψ 0 (A) is an F -split torus in Z(M ) and hence ψ 0 (A) A. As ψ 0 induces a Γ-equivariant homomorphism ˆψ 0 : Z( M) Z( M ) T, we get a morphism of affine algebraic varieties t A,A : (Z( M) I F ) ΦF /W (G, A) ( T I F ) Φ F /W (G, A ) ˆm ˆψ 0 ( ˆm) where ( ) designates the Galois action on G. (For Weyl group equivariance, see [HRo, 12.2].) The morphism t A,A is independent of the choices of P and B. 3 We use the letter S for this map, because when J = K is it just the Satake isomorphism (3.0.2).

12 THOMAS J. HAINES We now recall the definition of a normalized version of t A,A, for which we need to refine the choice of ψ 0 Ψ M somewhat. Following [H13, Lemma 11.2.4], given the choice of P M and B T used to define Ψ M, choose any F un -rational ψ 0 Ψ M and define a morphism of affine algebraic varieties (8.0.17) t A,A : (Z( M) I F ) ΦF /W (G, A) ( T I F ) Φ F /W (G, A ) ˆm δ 1/2 B ˆψ 0 (δ 1/2 P ˆm). This makes sense as δ P (resp. δ B ) is a weakly unramified character of M(F ) (resp. T (F )), and so can be regarded as an element of (Z( M) I F ) ΦF (resp. ( T I F ) Φ F ), by [H13, (3.3.2)]. Lemma 8.0.1. The morphism t A,A is well-defined and independent of the choice of P, B, and F un -rational ψ 0 Ψ M used in its construction. Proof. The independence statement and the implicit Weyl-group invariance is proved in [H13, 11.12.4]. Lemma 8.0.2. The map (8.0.17) is a closed immersion. Proof. We first prove that the map (Z( M) I F ) ΦF ( T I F ) Φ F given by ˆm δ 1/2 B ˆψ0 (δ 1/2 P ˆm) is a closed immersion. For this it is clearly enough to show that the unnormalized map ˆm ˆψ 0 ( ˆm) is a closed immersion. But this follows from the surjectivity of the corresponding map (8.0.18) t A,A : X ( T ) Φ F I F X (Z( M)) Φ F I F, which was proved in [H13, Remark 11.12.2]. In fact this is done by interpreting (8.0.18), via the Kottwitz isomorphism, as the map (8.0.19) T (F )/T (F ) 1 M (F )/M ψ 1 0 (F ) 1 M(F )/M(F ) 1. We therefore have a surjective normalized variant (8.0.20) t A,A : T (F )/T (F ) 1 M (F )/M ψ 1 0 (F ) 1 M(F )/M(F ) 1. Now recall that in [H13, 11.12.3] is constructed a bijective map (8.0.21) W (G, A) ψ 0 W (G, A )/W (M, A ) defined as follows. Let S be the F un -split component of T. Choose a maximal F un -split torus S G which is defined over F and which contains A, and set T = Cent G (S). Choose ψ 0 Ψ M such that ψ 0 is defined over F un and has ψ 0 (S) = S, hence also ψ 0 (T ) = T. Now suppose w W (G, A). We may choose a representative n N G (S)(L) Φ F (cf. [HRo]). There exists m n N M (S )(L) such that ψ 0 (n)m n N G (A )(F ). Then define ψ 0 (w) to be the image of ψ 0 (n)m n in W (G, A )/W (M, A ).

ON SATAKE PARAMETERS FOR REPRESENTATIONS WITH PARAHORIC FIXED VECTORS 13 Now the desired surjectivity of (8.0.22) C[T (F )/T (F ) 1 ] W (G,A ) t A A C[M(F )/M(F ) 1 ] W (G,A) follows immediately using the surjectivity of (8.0.20) and the isomorphism (8.0.21), because W (M, A ) and N M (S )(L) act trivially on M (F )/M (F ) 1. Recall the definition of the normalized transfer homomorphism on the level of Bernstein centers. Definition 8.0.3. ([H13, 11.12.1]) Let J G(F ) and J G (F ) be any parahoric subgroups and choose compatible maximal F -split tori A resp. A to be in good position 4 with respect to J resp. J. Then we define the normalized transfer homomorphism t : Z(G (F ), J ) Z(G(F ), J) to be the unique homomorphism making the following diagram commute Z(G (F ), J ) t Z(G(F ), J) C[X ( T ) Φ F IF ] W (G,A ) S t A,A C[X (Z( M)) Φ F I F ] W (G,A). S We use S to denote the Bernstein isomorphisms described in [H13, 11.10.1]. As explained in loc. cit., t is independent of the choices for A, A, and is a completely canonical homomorphism. Corollary 8.0.4. The normalized transfer homomorphism t : Z(G (F ), J ) Z(G(F ), J) is surjective. We now present an alternative way to characterize the maps t, taken from [H13, 11.12.6]. Proposition 8.0.5. Choose A, A, ψ 0 Ψ M as in Definition 8.0.3. For each subtorus A L A, let L = Cent G (A L ), A = ψ 0 (A), and L = ψ 0 (L), so that ψ 0 restricts to an inner twising L L of F -Levi subgroups of G resp. G. Set J L = J L(F ). Then the family of normalized transfer homomorphisms t : Z(L (F ), J L ) Z(L(F ), J L) is the unique family with the following properties: (a) The t are compatible with contant term homomorphisms, in the sense that the following diagrams commute for all L: Z(G (F ), J ) t Z(G(F ), J) c G L c G L Z(L (F ), J L ) t Z(L(F ), J L ). 4 This means that in the Bruhat-Tits building B(Gad, F ), the facet corresponding to J is contained in the apartment corresponding to A.

14 THOMAS J. HAINES (b) For L = M and z Z(M (F ), J M ), the function t(z) is given by integrating z over the fibers of the Kottwitz homomorphism κ M (F ). (Note M ad is anisotropic over F.) 9. Construction of parameters: general case Suppose G is any connected reductive group over F, and J G(F ) is a parahoric subgroup. Choose G, P, B and an F un -rational ψ 0 Ψ M as above. Let π Π(G/F, J). By [H13, 11.5], there exists a weakly unramified character χ (Z( M) I F ) Φ /W (G, A) such that π is an irreducible subquotient of the normalized induction i G P (χ). We have the Bernstein isomorphism of [H13, 11.9.1] S : Z(G(F ), J) C[(Z( M) I F ) Φ /W (G, A)]. By loc. cit. 11.8, z Z(G(F ), J) acts on i G P (χ)j by the scalar S(z)(χ). Definition 9.0.1. Define s(π) [ĜI F to be the image of χ ( T I F ) Φ /W (G, A) under the map (Z( M) I F ) Φ /W (G, A) t A,A ( T ) I F Φ /W (G, A ) (6.0.15) [Ĝ I F Φ ] ss /Ĝ I F ˆψ 0 [ĜI F. Let S(G) be the image of this map, which is a closed subvariety of [ĜI F by Lemma 8.0.2. Then we have the following characterization of s(π): for all z Z(G (F ), J ) we have tr( t(z ) π) = dim(π J ) S(z 1 )( ˆψ 0 (s(π))). Since the left hand side is well-defined (in particular independent of the choice of F un - rational ψ 0 ) and since t A,A and (6.0.15) are likewise independent of the choice of ψ 0, this equality has the side benefit of showing the the isomorphism ˆψ 0 in the definition above, is also independent of the choice of F un -rational ψ 0. 10. A transfer map Π(G, K) Π(G, K ) Let K G(F ) and K G (F ) be special maximal parahoric subgroups. define an operation We shall Π(G/F, K) Π(G /F, K ) π π which is dual to t : H(G (F ), K ) H(G(F ), K). Using ψ 0 as in the definition of t above, we identify L G L G. Given π we have s(π) S(G) [Ĝ I F Φ ] ss /Ĝ I F = S(G ). Definition 10.0.1. We define π Π(G /F, K ) to be the unique isomorphism class with s(π ) = s(π).

ON SATAKE PARAMETERS FOR REPRESENTATIONS WITH PARAHORIC FIXED VECTORS 15 Clearly π is characterized by the equalities for all f H(G (F ), K ) tr( t(f ) π) = S( t(f ))(s(π)) = S(f )(s(π )) = tr(f π ). The middle equality follows from the diagram in Definition 8.0.3 (taking J = K and J = K ). We remark that the character identity characterizes π in terms of π because f t(f ) is surjective (Cor. 8.0.4), and hence π π is injective. 11. Relation with local Langlands correspondence 11.1. Construction of s-parameter in Deligne-Langlands correspondence. The Satake parameter s(π) should give us part of the local Langlands parameter associated to π Π(G/F, J). Conjecture 11.1.1. Let W F := W F C be the Weil-Deligne group. If π Π(G/F, J) has local Langlands parameter ϕ π : W F L G, then (11.1.1) ϕ π (Φ) = s(π) as elements in [Ĝ Φ] ss/ĝ. Note that we still denote by s(π) its image under the natural map [ĜI F [Ĝ Φ] ss/ĝ. Remark 11.1.2. Put another way, the conjecture predicts the s-parameter in the Deligne- Langlands triple (s, u, ρ) which is hypothetically attached to a representation π with Iwahorifixed vectors (note that the works of Kazhdan-Lusztig [KL], Lusztig [L1, L2] construct the entire triple unconditionally, but not for the most general p-adic groups groups). Remark 11.1.3. This is similar to [Mis, Thm. 2], which discusses the case where G is quasi-split and split over a tamely ramified extension of F. When G/F is quasi-split, (11.1.1) is predicted by the compatibility of LLC with normalized parabolic induction (see [H13, 5.2]), as follows. Write G, T, B, etc. in place of G, T, B etc. from 8. By [H13, 11.5], there is a weakly unramified character χ on T (F ) such that π is a subquotient of i G B (χ). By LLC+ (loc. cit. 5.2), we expect ϕ π = ϕ χ, the latter taking values in L T L G. Now, the local Langlands correspondence for tori implies that ϕ χ exists unconditionally, and has ϕ χ (Φ) = χ Φ, where χ is viewed as an element of ( T I F ) ΦF. But clearly s(π) = χ Φ as well. Thus, the conjecture gives an essentially new prediction only when G is not quasi-split. In fact, it shows that the normalized transfer homomorphisms, used to define s(π) in the general case, are really telling us what ϕ π (Φ) should be.

16 THOMAS J. HAINES 11.2. Compatibility with generalized Jacquet-Langlands correspondence. Now return to the usual notation, where G is general and G is its quasi-split inner form. Let us identify L G with L G 1 using the isomorphism ˆψ 0 : L G L G. Given π Π(G/F, J), we may choose any π Π(G /F, J ) such that s(π ) = s(π). Note that if J = K, then π is unique, but in general it will not be. Since s(π) = s(π ) by construction of π, we expect ϕ π (Φ) = ϕ π (Φ). Since π and π are J-(resp. J )-spherical, ϕ π and ϕ π should be trivial on I F, and so we expect ϕ π WF = ϕ π WF. This is compatible with what a generalized Jacquet-Langlands correspondence would entail, on the level of infinitesimal characters (cf. [H13, 5]). Namely, π should give rise to the composition W F ϕ π L G ˆψ 1 0 L G which we call ϕ, which in turn should give rise to an L-packet Π ϕ for the group G. The map π Π ϕ would be part of a generalized Jacquet-Langlands correspondence. However, usually we would not expect π Π ϕ. For example, if D/F is a quaternion algebra over its center F, and if G = D, J = O D, G = GL 2, J = GL 2 (O F ), and π = 1 D (the trivial representation of D on C), then π = 1 GL2 (F ), while Π ϕ = JL(π) is the Steinberg representation of GL 2 (F ). On the other hand, if we restrict to W F, we get an agreement of infinitesimal characters ϕ WF = ϕ π WF = ϕ π WF. Thus, while π might not belong to the L-packet Π ϕ, it will at least belong to the infinitesimal class Π ϕ WF containing Π ϕ. References [Bor] A. Borel, Automorphic L-functions, In: Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math., vol. 33, part 2, Amer. Math. Soc., Providence, RI, 1979, pp. 27 61. [BT2] F. Bruhat, J. Tits, Groupes réductifs sur un corps local. II, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 5-184. [Car] P. Cartier, Representations of p-adic groups: a survey In: Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math., vol. 33, part 1, Amer. Math. Soc., Providence, RI, 1979, pp. 111 155. [H13] T. Haines, The stable Bernstein center and test functions for Shimura varieties. To appear in the proceedings for the London Mathematical Society - EPSRC Durham Symposium on Automorphic Forms and Galois Representations, Durham, July 18-28, 2011. arxiv:1304.6293. [HN] T. Haines, B. C. Ngô, Alcoves associated to special fibers of local models, Amer. J. Math. 124 (2002), 1125-1152. [HRa] T. Haines, M. Rapoport, On parahoric subgroups, Advances in Math. 219 (2008), 188-198; appendix to: G. Pappas, M. Rapoport, Twisted loop groups and their affine flag varieties, Advances in Math. 219 (2008), 118-198. [HRo] T. Haines, S. Rostami, The Satake isomorphism for special maximal parahoric Hecke algebras, Representation Theory 14 (2010), 264-284. [KL] D. Kazhdan, G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153215. [Ko84] R. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51, no.3, (1984), 611-650.

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