On the Hecke Module of $GL_n(k[[z]]) GL_n(k((z)))/GL_n(k((z^2)))$

Every double coset in GLₘ(k[[z]]) GLₘ(k((z)))/GLₘ(k((z²))) is uniquely represented by a block diagonal matrix with diagonal blocks in \1,z, pmatrix 1& z\\ 0 &zⁱ pmatrix (i>1)\ if char(k) ≠ 2 and k is a finite field. These cosets form a (spherical) Hecke module H(G,H,K) over the (spherical) Hecke algebra H(G,K) of double cosets in K G/H, where K=GLₘ(k[[z]]) and H=GLₘ(k((z²))) and G=GLₘ(k((z))). Similarly to Hall polynomial hλ,νμ from the Hecke algebra H(G,K), coefficients hλ,νμ arise from the Hecke module. We will provide a closed formula for hλ,ν^μ, under some restrictions over λ,ν,μ.