On the Hecke Module of $\text{GL}_n(k[[z]])\backslash \text{GL}_n(k((z)))/\text{GL}_n(k((z^2)))$

TL;DR

This paper characterizes the structure of a specific Hecke module related to double cosets in general linear groups over local fields, providing explicit formulas for its coefficients under certain conditions.

Contribution

It offers a classification of double cosets in $ ext{GL}_n(k[[z]])ackslash ext{GL}_n(k((z))) / ext{GL}_n(k((z^2)))$ and derives a closed formula for the associated Hecke module coefficients.

Findings

Unique block diagonal matrix representatives for cosets.

Explicit formulas for Hecke module coefficients.

Conditions under which the formulas apply.

Abstract

Every double coset in $\text{GL}_m(k[[z]])\backslash \text{GL}_m(k((z)))/\text{GL}_m(k((z^2)))$ is uniquely represented by a block diagonal matrix with diagonal blocks in $\{1,z, \begin{pmatrix} 1& z\\ 0 &z^i \end{pmatrix} (i>1)\}$ if $char(k) \neq 2$ and $k$ is a finite field. These cosets form a (spherical) Hecke module $\mathcal{H}(G,H,K)$ over the (spherical) Hecke algebra $\mathcal{H}(G,K)$ of double cosets in $K\backslash G/H$, where $K=\text{GL}_m(k[[z]])$ and $H=\text{GL}_m(k((z^2)))$ and $G=\text{GL}_m(k((z)))$. Similarly to Hall polynomial $h_{\lambda,\nu}^{\mu}$ from the Hecke algebra $\mathcal{H}(G,K)$, coefficients $h_{\lambda,\nu}^{\mu}$ arise from the Hecke module. We will provide a closed formula for $h_{\lambda,\nu}^\mu$, under some restrictions over ${\lambda,\nu,\mu}$.