\(L^2\) spectral decomposition on the Heisenberg group associated to the action of \(\text{U}(p,q)\)
Let \(H_n = \mathbb C^n \times {\mathbb R}\) with group law \((z,t)(z',t')=(z+z',t+t'-{1 \over 2} \Im B(z,z'))\), where \(B(z,w) = \sum_{j=1}^p z_j\overline{w_j}-\sum_{j=p+1}^n z_j\overline{w_j}\). Suppose \(p+q=n\) and let \(U(p,q)\) denote the set of \(g \in GL_n(\mathbb C)\) for which \(B(gz, gw)=B(z,w)\). Now \(U(p,q)\) acts on \(H_n\) by \(g\cdot (z,t) = (gz,t)\), where \(g\in U(p,q)\) and \((z,t) \in H_n\). Let \(\{ X_1, \dots , X_n, Y_1, \dots ,Y_n,T\}\) be the standard basis of the Lie algebra of \(H_n\) and let \[ L=\sum_{j=1}^p (X_j^2+Y_j^2)-\sum_{j=p+1}^n (X_{j}^2+Y_{j}^2). \] In this paper the authors obtain, using the Plancherel inversion formula, the joint spectral decomposition of \(L^2(H_n)\) with respect to \(L\) and \(T\) \[ f=\sum_{k\in {\mathbb Z}} \int_{-\infty}^{\infty} f \ast S_{\lambda, k} |\lambda|^n\, d\lambda,\quad f\in S(H_n), \] where each \(S_{\lambda, k}\) is a tempered distribution that is \(U(p,q)\)-invariant. By studying the confluent hypergeometric equation the authors are able to explicitly compute the distributions \(S_{\lambda,k}\). The authors then proceed to compute \( \int_{-\infty}^{\infty} f \ast S_{\lambda, k} |\lambda|^n\, d\lambda.\)
scientific article; zbMATH DE number 1537805