Laplacian Operators and Radon Transforms on Grassmann Graphs

Abstract

Let Ω be a vector space over a finite field with q elements. Let G denote the general linear group of automorphisms of Ω and let us consider the left regular representation $\rho: {\rm G} \rightarrow {\cal B}(L_2({\rm X}))$ associated with 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 EndG(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.