On a Lie group NA that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, the heat semigroup ${\mu }_t$ generated by a second order subelliptic left-invariant operator ${\sum }_{j=0}^m{Y}_j+Y$ is considered. Under natural conditions there is a $\mu {̌}_t$-invariant measure m on N, i.e. $\mu {̌}_t*m=m$. Precise asymptotics of m at infinity is given for a large class of operators with Y₀,...,Yₘ generating the Lie algebra of S.

The asymptotics of spherical functions for large dimensions are related to spherical functions for Olshanski spherical pairs. In this paper we consider inductive limits of Gelfand pairs associated to the Heisenberg group. The group K = U(n) × U(p) acts multiplicity free on 𝓟(V), the space of polynomials on V = M(n,p;ℂ), the space of n × p complex matrices. The group K acts also on the Heisenberg group H = V × ℝ. By a result of Carcano, the pair (G,K) with G = K ⋉ H is a Gelfand pair. The main results...