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

Archita Gupta, Pooja Singla

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.

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

Paper Structure

This paper contains 14 sections, 36 theorems, 82 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of regular representations of ${\mathbf{G}_{\ell}}$ for $\ell$ even
  4. Construction of regular representations of ${\mathbf{G}_{\ell}}$ for $\ell$ odd
  5. Alternative construction of $\mathbf{ss}$ representations of ${\mathbf{G}_{\ell}}$
  6. Alternative construction of $\mathbf{sns}$ representations of ${\mathbf{G}_{\ell}}$ for odd $\ell$
  7. Proof of \ref{['thm:multiplicity-bounds']} and \ref{['thm:intermediate-steps-multiplicity-bounds']}
  8. Split semisimple case
  9. Split nonsemisimple Case
  10. Proof of \ref{['thm:borel-strong-Gelfand-pair']}
  11. Decomposition of degenerate whittaker models of lg for lg
  12. $\mathbf{GL}_2(\mathfrak{o}_\ell)\, \mathrm{for}\, \ell=2$
  13. $\mathbf{GL}_2(\mathfrak{o}_\ell)\, \mathrm{for}\, \ell=3$
  14. $\mathbf{GL}_2(\mathfrak{o}_\ell)\, \mathrm{for}\, \ell=4$

Key Result

Theorem 1.1

The pair $(\mathbf{GL}_2({\mathfrak{o}}_\ell),{\mathbf{B}({\mathfrak{o}}_\ell)})$ is a strong Gelfand pair for every $\ell \geq 1$.

Theorems & Definitions (64)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 54 more