Beta functions of Bruhat–Tits buildings and deformation of $l^2$ on the set of $p$-adic lattices

For the space $\operatorname{Lat}_n$ 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 $\operatorname{Lat}_n$ is proposed. Conditions for the positive-definiteness of these kernels and an explicit Plancherel's formula are obtained.