Lusztig Correspondence and Howe Correspondence for Finite Reductive Dual Pairs

Abstract

Let (G, G ′ ) be a finite reductive dual pair of a symplectic group and an orthogonal group. The Howe correspondence establishes a correspondence between a subset of irreducible characters of G and a subset of irreducible characters of G ′ . The Lusztig correspondence is a bijection between the Lusztig series indexed by the conjugacy class of a semisimple element s in the connected component (G * ) 0 of the dual group of G and the set of irreducible unipotent characters of the centralizer of s in G * . In this paper, we prove the commutativity (up to a twist of the sign character) between these two correspondences under some restriction on the characteristic of the finite field.

Degree Differences in the Eta Correspondences

“…As in the proof of Lemma 3.13, the (n (+) − k (+) )-admissibility of ρ Λ (+) implies that both unipotent characters θ(ρ Λ (+) ), θ(ρ Λ (+) )sgn of O ǫ ′ ǫ n (+) (q) do not occur in the Θ-correspondence for the dual pair (Sp k ′(+) , O ǫ ′ ǫ n (+) ) for any k ′(+) < k (+) . Hence by [Pan19b] theorem 6.9 and remark 6.10, we see that both irreducible characters θ(ρ), θ(ρ)sgn of O ǫ n (q) do not occur in the Θ-correspondence for the dual pair (Sp k ′ , O ǫ n ) for any k ′ < k. We know that θ(ρ)χ O ǫ n ∈ E(G ′ ) −s ′ , and if we write…”

“…Write Υ(Λ) = µ1,µ2,...,µm 1 ν1,ν2,...,νm 2 and let d = |def(Λ)|. From [Pan19b] section 8, we can conclude the following.…”

“…[AM93] theorem 3.5). The following proposition on the Θ-correspondence of unipotent characters for a unitary dual pair is rephrased from [AMR96] théorème 5.15 (see also [Pan19b] proposition 5.13):…”

“…where G [λ] (s) is given in [AMR96] subsection 1.B (see also [Pan19b] subsection 2.2). One can see that…”

Lusztig correspondence and the Gan-Gross-Prasad problem

“…There are two types of periods in the Gan-Gross-Prasad conjecture: the Bessel periods and the Fourier-Jacobi periods and we can deduce the Fourier-Jacobi case from the Bessel case by the standard arguments of theta correspondence and see-saw dual pairs, which are used in the proof of local Gan-Gross-Prasad conjecture (see [GI, Ato]). A complete understanding of theta correspondence over finite fields is known in [AMR,P1,P2] and theta correspondence for the representations considering in this paper is quite simple. So in this paper, we only focus on the Bessel case.…”