2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70682Let $Ω$ be a vector space over a finite field with q elements. Let G denote the general linear group of endomorphisms of $Ω$ and let us consider the left regular representation $ρ: G \to B(L_2(X))$ associated to the natural action of G on the set X of linear subspaces of $Ω$. In this paper we study a natural basis B of the algebra $End_{G}(L_2(X))$ of intertwining maps on $L_2(X)$. By using a Laplacian operator on Grassmann graphs, we identify the kernels in B as solutions of a basic hypergeometric difference equation. This provides two expressions for these kernels. One in terms of the q-Hahn polynomials and the other by means of a Rodrigues type formula. Finally, we obtain a useful product formula for the mappings in B. We give two different proofs. One uses the theory of classical hypergeometric polynomials and the other is supported by a characterization of spherical functions in finite symmetric spaces. Both proofs require the use of certain associated Radon transforms.32 pagesCombinatoricsRepresentation Theory05A30; 05E30; 20G40text