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 ∨

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We define Eisenstein series on rank 2 hyperbolic Kac-Moody groups over R, induced from quasi-characters. We prove convergence of the constant term and hence the almost everywhere convergence of the Eisenstein series. We define and calculate the degenerate Fourier coefficients. We also consider Eisenstein series induced from cusp forms and show that these are entire functions.

Inspired by the Gan-Gross-Prasad conjecture and the descent problem for classical groups, in this paper we study the descents of unipotent representations of unitary groups over finite fields. We give the first descents of unipotent representations explicitly, which are unipotent as well. Our results include both the Bessel case and Fourier-Jacobi case, which are related via theta correspondence.

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