In this article, we prove a generalisation of Bochner-Godement theorem. Our result deals with Olshanski spherical pairs $(G,K)$ defined as inductive limits of increasing sequences of Gelfand pairs ${(G\left(n\right),K\left(n\right))}_{n\ge 1}$. By using the integral representation theory of G. Choquet on convex cones, we establish a Bochner type representation of any element $\varphi$ of the set ${𝒫}^{♮}\left(G\right)$ of $K$-biinvariant continuous functions of positive type on $G$.

We compute the heat kernel on the classical and nonisotropic Heisenberg groups, and on the free step two nilpotent groups $N_{n,2}$, by an elementary method, in particular without using Laguerre calculus.

Let G be a real Lie group and H a lattice or, more generally, a closed subgroup of finite covolume in G. We show that the unitary representation ${\lambda }_{G/H}$ of G on L²(G/H) has a spectral gap, that is, the restriction of ${\lambda }_{G/H}$ to the orthogonal complement of the constants in L²(G/H) does not have almost invariant vectors. This answers a question of G. Margulis. We give an application to the spectral geometry of locally symmetric Riemannian spaces of infinite volume.

We prove a spectral Paley-Wiener theorem for the Heisenberg group by means of a support theorem for the twisted spherical means on ${ℂ}^n.$ If $f\left(z\right)e^{\frac{1}{4}{\left|z\right|}^2}$ is a Schwartz class function we show that $f$ is supported in a ball of radius $B$ in ${ℂ}^n$ if and only if $f\times {\mu }_r\left(z\right)=0$ for $r\>B+\left|z\right|$ for all $z\in {ℂ}^n.$ This is an analogue of Helgason’s support theorem on Euclidean and hyperbolic spaces. When $n=1$ we show that the two conditions $f\times {\mu }_r\left(z\right)={\mu }_r\times f\left(z\right)=0$ for $r\>B+\left|z\right|$ imply a support theorem for a large class of functions with exponential growth. Surprisingly enough,this latter...

We give a simple proof of a result of R. Rochberg and M. H. Taibleson that various maximal operators on a homogeneous tree, including the Hardy-Littlewood and spherical maximal operators, are of weak type (1,1). This result extends to corresponding maximal operators on a transitive group of isometries of the tree, and in particular for (nonabelian finitely generated) free groups.

The group SU(1,d) acts naturally on the Hilbert space $L²\left(Bd{\mu }_{\alpha }\right)(\alpha >-1)$, where B is the unit ball of ${ℂ}^d$ and $d{\mu }_{\alpha }$ the weighted measure ${(1-|z\left|²\right)}^{\alpha }dm\left(z\right)$. It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic...