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KLYACHKO MODELS of P-ADIC SPECIAL LINEAR GROUPS 1. Introduction Let F Be a Field, Let Um(F) Denote the Group of M-By-M Unipotent

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KLYACHKO MODELS of P-ADIC SPECIAL LINEAR GROUPS 1. Introduction Let F Be a Field, Let Um(F) Denote the Group of M-By-M Unipotent PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY S 0002-9939(2010)10640-8 Article electronically published on November 24, 2010 KLYACHKO MODELS OF p-ADIC SPECIAL LINEAR GROUPS JOSHUA M. LANSKY AND C. RYAN VINROOT (Communicated by Wen-Ching Winnie Li) Abstract. We study Klyachko models of SL(n, F ), where F is a non- Archimedean local field. In particular, using results of Klyachko models for GL(n, F ) due to Heumos, Rallis, Offen and Sayag, we give statements of exis- tence, uniqueness, and disjointness of Klyachko models for admissible represen- tations of SL(n, F ), where the uniqueness and disjointness are up to specified conjugacy of the inducing character, and the existence is for unitarizable rep- resentations in the case F has characteristic 0. We apply these results to relate the size of an L-packet containing a given representation of SL(n, F )to the type of its Klyachko model, and we describe when a self-dual unitarizable representation of SL(n, F ) is orthogonal and when it is symplectic. 1. Introduction Let F be a field, let Um(F ) denote the group of m-by-m unipotent upper tri- angular matrices over F , and let Mm,l(F )bethesetofm-by-l matrices over F (not necessarily invertible). For each integer k satisfying 0 ≤ 2k ≤ n, define the subgroup Gk of GL(n, F )by: NX (1.1) G = N ∈ U − ,S ∈ Sp(2k, F),X ∈ M − (F ) . k S n 2k n 2k,2k Here we define, for k>0, the symplectic group Sp(2k, F) to be the stabilizer of the − form corresponding to the skew-symmetric matrix J = 1k . Fix a nontrivial k 1k + additive character θ : F → C,andforeachk, define a character ψk on Gk as follows: (1.2) n− 2k−1 NX If g ∈ G ,g= , and N =(a ), then define ψ (g)=θ a . k S ij k i,i+1 i=1 In other words, ψk is only nontrivial on the unipotent factor of Gk.Whenn = 2m,thenψm is just the trivial character on the subgroup Gm = Sp(2m, F ), and when k =0,ψk is a nondegenerate character of the unipotent subgroup Un(F )of GL(n, F ). Suppose that F = Fq is a finite field, let G =GL(n, Fq), and for each k,0≤ 2k ≤ n, define the induced representation T =IndG (ψ ). Klyachko [8] claimed that k Gk k Received by the editors September 3, 2009 and, in revised form, June 10, 2010. 2000 Mathematics Subject Classification. Primary 22E50. Key words and phrases. Klyachko model, p-adic special linear group, multiplicity one, L- packets, self-dual representations. c 2010 American Mathematical Society 1 2 JOSHUA M. LANSKY AND C. RYAN VINROOT for any complex irreducible representation (π, V )ofG,dimC HomG(π, Tk) ≤ 1for every k, and there exists a unique k,0≤ 2k ≤ n, such that dimC HomG(π, Tk)=1. After Klyachko’s original paper, Inglis and Saxl [7] gave the first complete proof to Klyachko’s claim. We call an embedding of the representation (π, V ) in the induced representation Tk a Klyachko model of the representation π. Klyachko’s original result states that every irreducible representation of GL(n, Fq) has a unique Klyachko model, and in particular, all of the induced representations Tk are multiplicity-free, and Tk and Tl have no isomorphic subrepresentations when k = l. Now consider the case that F is a non-Archimedean local field, with G = GL(n, F ). For each k,0≤ 2k ≤ n, define the representation Tk by T =IndG (ψ ), k Gk k where Ind denotes the ordinary (nonnormalized) induced representation for a locally compact totally disconnected group. In this case, we have the following result on Klyachko models of representations of GL(n, F ). Theorem 1.1 (Heumos and Rallis, Offen and Sayag). Let G =GL(n, F ),where F is a non-Archimedean local field. Let (π, V ) be any irreducible admissible repre- sentation of G. We have the following: n/2 (1) dimC HomH (π, Tk) ≤ 1. k=0 (2) If F has characteristic 0 and (π, V ) is unitarizable, then there exists a unique k such that dimC HomG(π, Tk)=1. Heumos and Rallis [6] proved that, if n =2m, then for any π,dimC HomH (π, Tm) ≤ 1; that is, any irreducible admissible representation has a unique symplectic model if one exists. They also proved that, in this case, the set of admissible rep- resentations of GL(n, F ) which have symplectic models is disjoint with the set of representations which have Whittaker models. Finally, Heumos and Rallis proved statements (1) and (2) of Theorem 1.1 for n ≤ 4 and conjectured that these state- ments hold for all n. Theorem 1.1 was proved completely by Offen and Sayag in a series of papers [9, 10, 11]. Now notice that the groups Gk are also subgroups of the special linear group SL(n, F ). In this paper, we study Klyachko models of the group SL(n, F )whenF is a non-Archimedean local field. Since there is more than one orbit of nondegenerate characters of the unipotent subgroup of SL(n, F ), we must consider conjugates of the characters ψk in (1.2) in these models. Our main result, Theorem 2.1, is the analogue of Theorem 1.1 for the special linear group. The main difference in the result is in the statement of uniqueness and disjointness of Theorem 2.1, where we can only obtain uniqueness and disjointness of Klyachko models up to conjugation of the character ψk by an element of a certain group. We give two applications of Theorem 2.1. In the first, Corollary 2.1, we relate the type of the Klyachko model of a representation of SL(n, F )tothesizeofthe L-packet containing that representation. In the second, Corollary 3.1, we describe when a self-dual unitarizable representation of SL(n, F ) is orthogonal and when it is symplectic. KLYACHKO MODELS OF p-ADIC SPECIAL LINEAR GROUPS 3 2. Klyachko models of special linear groups From now on, we let F be a non-Archimedean local field, let G =GL(n, F ), let H =SL(n, F ), and let Gk be as in (1.1) for each k such that 0 ≤ 2k ≤ n.Notethat ∼ ∼ × G = H D,whereD = F is the group of matrices of the form diag(x, 1,...,1) for x ∈ F ×. We will often identify G/H with D and hence with F ×.NotethatH contains each Gk,andD normalizes Gk whenever 0 ≤ 2k<n. Let (ρ, W ) be an irreducible admissible representation of H. By [14, Prop. 2.2], there is an H-embedding of (ρ, W ) as a direct summand of some irreducible ad- missible representation (π, V )ofG. From results in [2, 14], we know that if ρ is unitarizable, then (π, V ) can also be taken to be unitarizable. Given any g ∈ G, define gρ to be the representation of H on W given by gρ(h)= − ∼ ρ(g 1hg). Denote by G(ρ) the subgroup {g ∈ G|gρ = ρ} of G. By [14, Cor. 2.3], G(ρ) is an open normal subgroup of finite index in G.Sinceρ is stable under conjugation by G(ρ), we may extend ρ to a representation of the group G(ρ). We let D(ρ)=D ∩ G(ρ). g If g ∈ G normalizes Gk,andψ is a character of Gk,denoteby ψ the character − × ∼ α → ψ(g 1αg). In this case, if g has image x ∈ F under the map G → G/H = ∼ × D = F , we have (in the notation of (1.2)) n− 2k−1 g −1 ψk(g)=θ x a1,2 + ai,i+1 when 0 ≤ 2k<n− 1. i=2 If n is odd and 2k = n − 1, then ψk is trivial, hence unaffected by conjugation. If n is even and 2k = n,thenψk is again trivial, although Gk = Sp(2k, F) is no longer γ −1 normalized by nontrivial elements of D. In this case, given γ ∈ D, Gk = γGkγ is the symplectic group defined by the skew-symmetric matrix γJkγ. So, in general, γ γ γ ψk is a character of Gk,andGk = Gk in all cases except when 2k = n.Wefirst prove a result which relates representations of H, G(ρ), and G. Lemma 2.1. Let (ρ, W ) be an irreducible admissible representation of H and let (π, V ) be an irreducible admissible representation of G that contains (ρ, W ) upon restriction. Consider (ρ, W ) as an irreducible representation of G(ρ).Then G ∼ IndG(ρ)(ρ) = π. Proof. Since the restriction of (π, V )toG(ρ)contains(ρ, W ), it follows that G IndG(ρ)(ρ) is contained in G G ⊗ ⊗ G IndG(ρ)(π)=IndG(ρ)(π 1)=π IndG(ρ)(1), G which is completely reducible. Therefore, IndG(ρ)(ρ) is completely reducible, and the number of irreducible components therein is given by G dimC EndG(IndG(ρ)(ρ)) . An application of Mackey’s theorem [1, Exer. 4.5.5] shows that this dimension is 1. The following lemma relates models for representations of G with those of H. Lemma 2.2. Let (ρ, W ) be an irreducible admissible representation of H and let (π, V ) be an irreducible admissible representation of G that contains (ρ, W ) upon restriction. Suppose that for some γ ∈ D and some k with 0 ≤ 2k ≤ n, (ρ, W ) H γ G γ G embeds in Indγ ( ψ ).Then(π, V ) embeds in T =Indγ ( ψ )=Ind (ψ ). Gk k k Gk k Gk k 4 JOSHUA M.
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