One algebra of double cosets for a general linear group over a finite field

Let ${\mathbb{F}}_q$ be finite field with $q$ elements. Let $\alpha ⩽n$ be positive integers. Consider the general linear group $\mathrm{G}\mathrm{L}(\alpha +n,{\mathbb{F}}_q)$ and its subgroup $H(n)$, which fixes the first $\alpha$ basis elements in ${\mathbb{F}}_q^{\alpha +n}$. Denote ${\mathcal{A}}_n$ by the convolution algebra of $H(n)$-biinvariant functions on $\mathrm{G}\mathrm{L}(\alpha +n,{\mathbb{F}}_q)$. We describe algebras ${\mathcal{A}}_n$ in terms of generators and relations and show that the family ${\mathcal{A}}_n$ admits a natural interpolation to arbitrary complex $n$ (the field ${\mathbb{F}}_q$ and $\alpha$ are fixed).