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 ${1,z, \begin{pmatrix} 1& z\ 0 &z^i \end{pmatrix} (i>1)}$ 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 ${\lambda,\nu,\mu}$.