有限辛群的波前集和下降法
Wavefront sets and descent method for finite symplectic groups
摘要：
在\cite{JZ1}中，D. Jiang和L. Zhang提出了一个关于波前集和局部域情形下的下降法的猜想。最近，在\cite{JLZ}中，他们和D. Liu利用$π$的增强L-参数的算术结构，定义了局部域$k$上经典群$G(k)$的某种不可约容许表示$π$的算术波前集，该局部域是$G(k)$的李代数的$k$ -有理幂零轨道的子集.这些算术结构是基于局部Langlands对应和局部Gan-Gross-Prasad猜想的合理性。他们还证明了算术波前集是$π$的不变量（它与Whittaker基准\cite的选择无关[定理1.1]{JLZ}），并提出了几种描述算术波前集、解析波前集和波前代数集之间关系的方法。本文研究了有限辛群不可约表示的波前集，描述了波前集、下降法和有限Gan-Gross-Prasad问题之间的关系。有限域情形的Gan-Gross-Prasad问题可以用\cite{LW3,Wang1,Wang2}显式计算。它使我们能够计算的多重性的不可约表示在广义Gelfand-Graev表示对应于某些幂零轨道，这是有限域类比的算术波前集。特别地，对于尖点表示，我们给出了一定的重数1定理，并证明了算术波前集的有限域模拟与G. Lusztig和N.川中
英文摘要：
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 $π$ 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 $π$. 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 $π$ (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.
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