An Olshanski spherical pair \((G,K)\) is the inductive limit of an increasing sequence of Gelfand pairs \((G(n), K(n))\).
How can a spherical function for \((G, K)\) be obtained as limit of spherical functions for \((G(n), K(n))\)?
For a locally compact group \(G\) and a compact subgroup \(K\), \(L^1(K\backslash G/K)\) is the convolution algebra of \(K\)-biinvariant integrable functions on \(G\). Assume that \((G,K)\) is a Gelfand pair, i.e., the algebra \(L^1(K\backslash G/K)\) is commutative. A spherical function for the Gelfand pair \((G, K)\) is a continuous \(K\)-biinvariant function \(\varphi\) on \(G\) with \(\varphi(e)=1\), and \[ \int_{K} \varphi(xky) dk= \varphi(x)\varphi(y), \;\;\;\;\;\;x,y \in G. \] Let \(\Sigma\) denote the Gelfand spectrum of the Gelfand pair \((G, K)\) and \(\varphi(\sigma, x)\) the spherical function associated to \(\sigma \in \Sigma\).
The author considers the semi-direct product \(G(n)=K(n) \ltimes H(n)\), where \(H(n)=W(n) \times \mathbb{R}\) is a Heisenberg group, where \(W(n)\) is a complex Euclidean vector space, Sym(\(n, \mathbb{C}\)), \(M(n, \mathbb{C)}\) or Skew(\(2n, \mathbb{C}\)), and \(K(n)\) is a group of automorphisms of \(H(n)\).
{ Theorem. } (a) Let \(\sigma^{(n)} \in \Sigma_{n}\) be a sequence for the Gelfand pair \((G(n), K(n))\). Assume that \(\sigma^{(n)}\) is convergent and \[ \lim_{n \rightarrow \infty}\sigma^{(n)}= \sigma \in \Sigma. \] Then \[ \lim_{n \rightarrow \infty}\varphi_{n}(\sigma^{(n)},x)=\varphi(\sigma,x) \] uniformly on compact sets in \(H\).
(b) If \(\sigma^{(n)} \in \Sigma_{n}\) is a sequence for the Gelfand pair \((G(n), K(n))\) such that \[ \lim_{n \rightarrow \infty}\varphi_{n}(\sigma^{(n)},x)=\varphi(x) \] uniformly on compact sets in \(H\), where \(\varphi\) is a continuous function on \(H\), then \(\sigma^{(n)}\) is convergent, \(\lim_{n \rightarrow \infty}\sigma^{(n)}= \sigma \in \Sigma\), and \(\varphi(x)=\varphi(\sigma, x)\) on \(H\).