Wavefront sets and descent method for finite symplectic groups

TL;DR

This paper investigates wavefront sets of irreducible representations of finite symplectic groups, linking them to descent methods and finite Gan-Gross-Prasad problems, and establishes their coincidence with Lusztig and Kawanaka wavefront sets.

Contribution

It provides explicit calculations of wavefront sets for finite symplectic groups and demonstrates their equivalence with Lusztig and Kawanaka wavefront sets, extending the understanding of representation invariants.

Findings

Calculated multiplicities of irreducible representations in generalized Gelfand-Graev representations.

Established that finite field analogues of arithmetic wavefront sets match Lusztig and Kawanaka wavefront sets.

Proved a multiplicity one theorem for cuspidal representations.

Abstract

In \cite{JZ1}, D. Jiang and L. Zhang proposed a conjecture which related the wavefront sets and the descent method in the local fields case. Recently, in \cite{JLZ}, they and D. Liu define the arithmetic wavefront set of certain irreducible admissible representation $\pi$ of a classical group $G(k)$ defined over local field $k$, which is a subset of $k$-rational nilpotent orbits of the Lie algebra of $G(k)$, by the arithmetic structures of the enhanced L-parameter of $\pi$. These arithmetic structures are based on the rationality of the local Langlands correspondence and the local Gan-Gross-Prasad conjecture. They also prove that the arithmetic wavefront set is an invariant of $\pi$ (it is independent of the choice of the Whittaker datum \cite[Theorem 1.1]{JLZ}), and propose several conjectures to describe the relationship between arithmetic wavefront sets, analytic wavefront sets and algebraic wavefront sets. In this paper we study wavefront sets of irreducible representations for finite symplectic groups and describe the relationship between wavefront sets, descent method and finite Gan-Gross-Prasad problem. The finite fields case of Gan-Gross-Prasad problem can be calculated explicitly \cite{LW3,Wang1,Wang2}. It allows us to calculate the multiplicity of an irreducible representation in the generalised Gelfand-Graev representation corresponding to certain nilpotent orbits which are finite fields analogies of the arithmetic wavefront sets. In particular, for cuspidal representations, we give certain multiplicity one theorem and show that the finite fields analogy of arithmetic wavefront sets coincides with the wavefront sets in the sense of G. Lusztig and N. Kawanaka.