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

Abstract

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