Self-dual representations of SL(2,F): an approach using the Iwahori–Hecke algebra

Abstract Let F be a non-Archimedean local field and Let be an irreducible smooth Iwahori-spherical representation of G. It is easy to see that such representations are always self-dual. The space V of π admits a non-degenerate G-invariant bilinear form which is unique up to scaling. It can be shown that the form is either symmetric or skew-symmetric and we set accordingly. In this article, we use the Bernstein–Lusztig presentation of the Iwahori–Hecke algebra of G and show that


Introduction
Let G be a group and ðp; VÞ be an irreducible complex representation of G. Suppose that p ' p Ú (here p Ú is the dual or contragredient representation). Using Schur's lemma, we can show that there exists a non-degenerate G-invariant bilinear form on V which is unique up to scalars, and consequently is either symmetric or skew-symmetric. Accordingly, we set e p ð Þ ¼ 1 if the form is symmetric; À1 if the form is skewÀsymmetric; & which we call the sign of p.
The sign eðpÞ is well understood for connected compact Lie groups and certain classes of finite groups of Lie type. If G is a connected compact Lie group, it is known that the sign can be computed using the dominant weight attached to the representation p (see [3] pg. 261-264). For certain finite classical groups, computing the sign involves difficult conjugacy class computations. In [6], Prasad introduced a nice idea to compute the sign for a certain class of representations of finite groups of Lie type. He has used this idea to determine the sign for many classical groups of Lie type. In recent times, there has been a significant interest in studying these signs in the setting of reductive p-adic groups. In [7], Prasad extended the results of [6] to the case of reductive padic groups and computed the sign of certain classical groups. The disadvantage of his method is that it works only for representations admitting a Whittaker model. In [8], Roche and Spallone discuss the relation between twisted sign (see Section 1 in [8]) and the ordinary sign and describe a way of studying the ordinary sign using the twisted sign. In an earlier work [2], we used the ideas of Roche and Spallone [8] to study the sign for non-generic Iwahori-spherical representations of SLðn; FÞ (for arbitrary n and where characteristic F is zero). The key idea in this work was to reduce the problem to computing the twisted sign of a certain generic representation of a Levi subgroup of G and use Prasad's method to compute the sign.
In this article, we reprove a specific case of the main result of [2] using the Bernstein-Lusztig presentation of the Iwahori-Hecke algebra (explained in Section 4). The advantage of using this presentation is that we don't have to restrict ourselves to any special classes of representations to study the sign. Also we don't have to impose any restrictions on the characteristic of the field F. In future, we hope to study the problem for SLðn; FÞ using similar techniques.
Before we proceed further, we note that in the case when G ¼ SLð2; FÞ; any irreducible smooth Iwahori-spherical representation is always self-dual. We refer the reader to Theorem 2.2 in [1], for a proof of this.
For completeness, we state our main result below.

Preliminaries on signs
In this section, we briefly discuss the notion of signs associated to self-dual representations. Let F be a non-Archimedean local field and G be the group of F-points of a connected reductive algebraic group. Let ðp; VÞ be a smooth irreducible representation of G. We write ðp Ú ; V Ú Þ for the smooth dual or contragredient of ðp; VÞ and h ; i for the canonical non-degenerate Ginvariant pairing on V Â V Ú (given by evaluation). Let t : ðp; VÞ ! ðp Ú ; V Ú Þ be an isomorphism. The map t can be used to define a bilinear form on V as follows It is easy to see that ð ; Þ is a non-degenerate G-invariant form on V, i.e. it satisfies, Let ð ; Þ Ã be a new bilinear form on V defined by This form is again non-degenerate and G-invariant. It follows from Schur's Lemma that for some non-zero scalar c. A simple computation shows that c 2 f61g: Indeed, We set c ¼ eðpÞ: It clearly depends only on the equivalence class of p. To summarize, the form ð ; Þ is symmetric or skew-symmetric and the sign eðpÞ determines its type.

The Hecke algebra
Let G be a locally profinite group and C 1 c ðGÞ be the space of all functions f : G ! C which are locally constant and compactly supported. Let l be a Haar measure on G. For f 1 ; f 2 2 C 1 c ðGÞ; define The algebra HðGÞ ¼ ðC 1 c ðGÞ; ÃÞ is an associative C-algebra and is called the Hecke algebra of G. For K a compact open subgroup of G, we write HðG; KÞ for the subalgebra of HðGÞ of K biinvariant functions. To be more precise, The above action gives V the structure of a smooth HðGÞ-module.
Proposition 3.2. Let ðp; VÞ be an irreducible smooth representation of G and V K be the subspace of K-fixed vectors in V. The space V K is either zero or a simple module over HðG; KÞ. The process V ! V K induces a bijection between the equivalence classes of irreducible smooth representations ðp; VÞ of G such that V K 6 ¼ 0, and isomorphism classes of simple HðG; KÞ-modules.
Proof. We refer the reader to Section 4 in [4] for a proof of the above proposition.

The Bernstein-Lusztig presentation for the Iwahori-Hecke algebra
In this section, we briefly explain the Bernstein-Lusztig presentation for the Iwahori-Hecke algebra of SLð2; FÞ: We refer the reader to [5] (see Chapter 3, Section 1), for more details about the presentation in a very general setup. Throughout, we let G ¼ SLð2; FÞ; where F is a non-Archimedean local field. We write o for the ring of integers in F, p for the unique maximal ideal in o with generatorand k F for the finite residue field of cardinality q. Let I be the subgroup of G consisting of matrices of the form I is called the Iwahori subgroup of G. We normalize the Haar measure l such that lðIÞ ¼ 1: We write H ¼ HðG; IÞ for the Iwahori-Hecke algebra of G. We let T denote the subgroup of diagonal matrices in G, and let T o ¼ T \ I: We write W ¼ N G ðTÞ=T for the (finite) Weyl group, W ¼ N G ðTÞ=T o for the (infinite) affine Weyl group. Let It can be shown thatW ¼ hs 0 ; s 1 j s 2 0 ¼ 1; s 2 1 ¼ 1i: We let R ¼ C½q 1=2 ; q À1=2 : For L & G; we write v L for the characteristic function of L. Let h ¼ q À1 v IxI : It can be shown that h is an invertible element in H and A ¼ Span R fh n j n 2 Zg is an abelian subalgebra of H: For w 2W ; we let N w ¼ q À1=2 v IwI : Proposition 4.1. Let s ¼ s 1 and B ¼ fh n ; N s h n j n 2 Zg: Then B is an R-basis for H and H is generated as an algebra subject to the following relations: a. ðN s Àq 1=2 ÞðN s þ q À1=2 Þ ¼ 0: b. hN s ÀN s h À1 ¼ bðh þ 1Þ:

Reformulation using the Iwahori-Hecke algebra
In this section, we reformulate the sign of the representation in terms of the sign of a simple module over H: To be more precise, we show that eðpÞ is the same as eðMÞ where M is a simple module over H: Throughout, we let ðp; VÞ to be an irreducible smooth self-dual representation of G with nontrivial vectors fixed under the Iwahori subgroup. We write M ¼ V I for the subspace of vectors in V fixed under I, VðIÞ ¼ Span C fpðkÞvÀv j v 2 V; k 2 Ig: It can be shown that V ¼ V I VðIÞ and dim C ðMÞ jWj; where W is the finite Weyl group. Consider the action of H on V given in It is easy to see that the above action makes M Ú a module over H: Since p ' p Ú ; using Proposition 3.2, it follows that M ' M Ú as simple H-modules. LetT 2 Hom H ðM; M Ú Þ be an isomorphism. As before, we define a bilinear form ðð ; ÞÞ on M as follows. For m 1 ; m 2 2 M; we set Clearly, the above bilinear form is non-degenerate and is H-invariant in the following sense.
Let ðð ; ÞÞ Ã be a new bilinear form on M defined by This form is again non-degenerate and H-invariant. It follows from Schur's Lemma that for some non-zero scalar c. As earlier, it is easy to see that c 2 f61g: We set c ¼ eðMÞ and call it the sign of M.
It is easy to see that that ð ; Þj MÂM is non-degenerate and H-invariant and hence it follows that eðpÞ ¼ eðMÞ: We record it in the following lemma.
Lemma 5.2. eðpÞ ¼ eðMÞ: Proof. Let w 2 M and suppose that ðw; vÞ ¼ 0; 8v 2 M: For x 2 VðIÞ; clearly we have ðw; xÞ ¼ 0: It is enough to check this when x ¼ aÀpðkÞa; for a 2 V; k 2 I: Indeed, we have From this it follows that w ¼ 0 and ð ; Þj MÂM is non-degenerate. It is a trivial computation to check that ð ; Þj MÂM satisfies invariance property of Lemma 5.1. The result follows. w

Main theorem
In this section, we prove the main result of this article. For the sake of clarity, we recall some notation we need. We let G ¼ SLð2; FÞ and ðp; VÞ an irreducible smooth Iwahori-spherical representation of G. We write M ¼ V I for the subspace of V of vectors fixed under the Iwahori subgroup I in G and ðp; MÞ for the corresponding irreducible representation of the Iwahori-Hecke algebra H: Throughout we let A to be the abelian subalgebra of H as before. Since N s satisfies the quadratic relation N 2 s ÀbN s À1 ¼ 0; b ¼ q 1=2 Àq À1=2 it follows that the minimal polynomial for N s (as an operator on M) is the polynomial xÀq 1=2 or x þ q À1=2 or ðxÀq 1=2 Þðx þ q À1=2 Þ: We consider all these cases separately. We let M 1 ¼ KerðN s Àq 1=2 Þ and M 2 ¼ KerðN s þ q À1=2 Þ: Proof. Consider the restriction pj A of p. It is easy to see that the restriction pj A is irreducible. Indeed, let W be a non-zero subspace of M invariant under A and w 6 ¼ 0 2 W: Since It follows that W is invariant under H: Since M is an irreducible representation of H; we have W ¼ M. Therefore pj A is irreducible and dim C ðMÞ ¼ 1: