On Gelfand pairs and degenerate Gelfand-Graev modules of General Linear groups of degree two over principal ideal local rings of finite length

TL;DR

This paper proves that the pair (G(R), B(R)) forms a strong Gelfand pair for degree two general linear groups over certain local rings and fully decomposes degenerate Gelfand-Graev modules for rings of small length.

Contribution

It establishes the strong Gelfand pair property for (G(R), B(R)) and provides a complete decomposition of DGG modules for rings of length up to four.

Findings

(G(R), B(R)) is a strong Gelfand pair.

Complete decomposition of DGG modules for rings of length ≤ 4.

Characterization of multiplicities in DGG modules independent of residue field size.

Abstract

Let $R$ be a principal ideal local ring of finite length with a finite residue field of odd characteristic. Denote by $G(R)$ the general linear group of degree two over $R$, and by $B(R)$ the Borel subgroup of $G(R)$ consisting of upper triangular matrices. In this article, we prove that the pair $(G(R), B(R))$ is a strong Gelfand pair. We also investigate the decomposition of the degenerate Gelfand-Graev (DGG) modules of $G(R)$. It is known that the non-degenerate Gelfand Graev module (also called non-degenerate Whittaker model) of $G(R)$ is multiplicity-free. We characterize the DGG-modules where the multiplicities are independent of the cardinality of the residue field. We provide a complete decomposition of all DGG modules of $G(R)$ for $R$ of length at most four.