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

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 $\{1,z,\left(\begin{array}{cc}1& z\\ 0& z^i\end{array}\right)(i>1)\}$ if $char(k)\ne 2$ 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 $\lambda ,\nu ,\mu$.