Beta functions of Bruhat-Tits buildings and deformation of l{sup 2} on the set of p-adic lattices

Abstract

For the space Lat{sub n} of all lattices in an n-dimensional p-adic linear space an analogue of the matrix beta function is constructed; this beta function can degenerate to the Tamagawa zeta function. An analogue of Berezin kernels for Lat{sub n} is proposed. Conditions for the positive-definiteness of these kernels and an explicit Plancherel's formula are obtained.