Descents of unipotent representations of finite unitary groups

Abstract

Inspired by the Gan-Gross-Prasad conjecture and the descent problem for classical groups, in this paper we study the descents of unipotent cuspidal representations of orthogonal and symplectic groups over finite fields.Date: August 29, 2019. 2010 Mathematics Subject Classification. Primary 20C33; Secondary 22E50.Let G(V ) be the identity component of the automorphism group of V and G(W ) ⊂ G(V ) the subgroup which acts as identity on W ⊥ . Let π and π ′ be irreducible representations of G(V ) and G(W ) respectively. The Gan-Gross-Prasad conjecture is concerned with the multiplicity Theorem 15.1], and will be explained in details shortly. According to whether dimV − dimW is odd or even, the above-Hom space is called the Bessel model or Fourier-Jacobi model. In the case of finite unitary groups, W. T. Gan, B. H. Gross and D. Prasad ([GGP2, Proposition 5.3]) showed that if π and π ′ are both cuspidal, then m(π, π ′ ) ≤ 1.We should mention that our formulation of multiplicities differs slightly from that in the Gan-Gross-Prasad conjecture [GGP1], up to taking the contragradient of π ′ . This is more suitable for the purpose of descents (c.f. [LW2]), which will be clear from the discussion below. On the other hand, in this paper we will restrict our attention to unipotent cuspidal representations of SO n (F q ) and Sp 2n (F q ), which are self-dual (see Proposition 5.5), and thus for π and π ′ unipotent cuspidal the above Hom-space coincides with Hom H(Fq) (π ⊗ π ′ , ν).Roughly speaking, for fixed G(V ) and its representation π, the descent problem seeks the smallest member G(W ) among a Witt tower which has an irreducible representation π ′ satisfying m(π, π ′ ) = 0, and all such π ′ give the first descent of π. To give the precise notion of descent, let us sketch the definition of the data (H, ν) following [GGP1] and [JZ1].We first consider the Bessel model. Let V n be an n-dimensional space over F q with a nondegenerate symmetric bilinear form (, ), which defines the special orthogonal group SO(V n ). We will consider various pairs of Hermitian spaces V n ⊃ V n−2ℓ−1 and the following partitions of n, (1.1) p ℓ = [2ℓ + 1, 1 n−2ℓ−1 ], 0 ≤ ℓ < n/2.Assume that V n has a decomposition V n = X + V n−2ℓ + X ∨

Lusztig correspondence and the Gan-Gross-Prasad problem

“…However, we can not get the complete answer in the Gan-Gross-Prasad problem directly in this way. In previous works [LW1,LW3], we have studied the Gan-Gross-Prasad problem of unipotent representations of finite unitary groups and in [LW2,Wang] for finite orthogonal groups and finite symplectic groups. In this paper, we will give a formula to reduce the Gan-Gross-Prasad problem of arbitrary representations to the unipotent representations, which is known in our previous work.…”

“…[GGP1,Theorem 16.1]) (c.f. [LW1,LW2] for details). If π and σ are complex irreducible representations of orthogonal groups, then the above Hom space is called the Bessel model.…”

“…For finite orthogonal groups, in [Wang], we reduce the Gan-Gross-Prasad problem of arbitrary representations to the uniform case. Then we can calculate it by Reeder's formula ( [R,LW1]). Although in general, explicit calculation with Reeder¡¯s formula is still quite involved, we can reduce this problem to a product of certain multiplicities of certain unipotent representations for finite unitary group and finite general linear groups, which has been calculated in [LW1].…”

“…Then we can calculate it by Reeder's formula ( [R,LW1]). Although in general, explicit calculation with Reeder¡¯s formula is still quite involved, we can reduce this problem to a product of certain multiplicities of certain unipotent representations for finite unitary group and finite general linear groups, which has been calculated in [LW1]. To do that, we first give a decomposition of Reeder¡¯s formula:…”

“…In Section 4, we recall the Lusztig correspondence. In Section 5, we recall the Reeder's formula in [R,LW1]. In Section 6, we prove Theorem 1.1 for unitary groups.…”