Beta functions of Bruhat-Tits buildings and deformation of on the set of -adic lattices

Abstract

For the space of all lattices in an -dimensional -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 is proposed. Conditions for the positive-definiteness of these kernels and an explicit Plancherel's formula are obtained.