论文标题
有限符号组的波前集和下降方法
Wavefront sets and descent method for finite symplectic groups
论文作者
论文摘要
在\ cite {jz1}中，D。Jiang和L. Zhang提出了一个猜想，该猜想将波前集和在本地场中的下降方法相关联。最近，在\ cite {jlz}中，他们和D. Liu定义了某些不可约束的可允许的代表$π$的算术波前$ g（k）$的$ g（k）$定义在本地字段$ k $上，这是$ k $ k $ k $ k $ k $ rational-rational nilpotent nilpotent of liebert的子集的子集$ g（k）$ g（k）的lie the e y-G（k）lieparm of g（k），a y-parram of g（k） $π$。这些算术结构基于局部Langlands对应关系和局部Gan-Gross-Proasad猜想的合理性。他们还证明了算术波前集是$π$的不变性（它独立于选择Whittaker Datum \ Cite [theorem 1.1] {Jlz}）的选择，并提出了几种指出，以描述算术浪潮集之间的关系，分析波段集，分析波浪形集和Algebront Sets和Algebrain Wavefront Wavefront sets。 在本文中，我们研究了有限符号群体的不可减至表示的波前集，并描述了波前集，下降方法和有限的gan-gross-prasad问题之间的关系。可以明确计算gan-gross-prasad问题的有限字段{lw3，wang1，wang2}。它使我们能够在广义的Gelfand-Graev表示中计算出与某些Nilpotent Orbits相对应的多样性，这些表示对应于某些nilpotent Orbits，这些轨道是算术波前集的有限场类比。特别是，对于Cuspidal表示，我们给出了一定的多重性一个定理，并表明算术波前的有限场类比与G. lusztig和N. Kawanaka的意义相吻合。
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.
