The spherical transform of a Schwartz function on the Heisenberg group
Let \(K\subset U(n)\) be a compact Lie group acting on the \((2n+1)\)-dimensional Heisenberg group \(H_n=V\times\mathbb{R}\). We say that \((H_n, K)\) is a Gelfand pair if the convolution algebra \(L^1_K(H_n)\) of integrable \(K\)-invariant functions on \(H_n\) is commutative. In this case, the Gelfand space \(\Delta=\Delta(K,H_n)\) is equipped with the Plancherel measure, and the spherical transform \(^\wedge\): \(L^2_K(H_n)\to L^2_K(\Delta)\) is an isometry. The main result of this paper provides a characterization of the set \({\mathcal S}_K(H_n)^\wedge=\{\widehat f| f\in{\mathcal S}_K(H_n)\}\) of spherical transforms of \(K\)-invariant Schwartz functions on \(H_n\). The authors show that a function \(F\) on \(\Delta\) belongs to \({\mathcal S}_K(H_n)^\wedge\) if and only if the functions obtained from \(F\) via application of certain derivatives and difference operators satisfy decay conditions. In order to describe these conditions let \(\mathbb{C}[V]=\sum_{\alpha\in\Lambda}P_\alpha\subset{\mathcal F}\) be the decomposition of \(\mathbb{C}[V]\) into \(K\)-irreducible subspaces \(P_\alpha\). Here \({\mathcal F}\) is the Fock space on which the Heisenberg group acts via the irreducible representation \(\pi\). For every \(\alpha\in\Lambda\), \(\lambda\in \mathbb{R}^*\), let \(\Phi_{\alpha,\lambda}\) denote the corresponding \(K\)-spherical function \[ \Phi_{\alpha,\lambda}(z,t)={1\over d_\alpha} \sum^{d_\alpha}_{j=1} \biggl \langle\pi\bigl(\sqrt{|\lambda|}z, \lambda t\bigr) v_j,v_j\biggr \rangle, \] where the \(v_j\)'s denote an orthogonal basis of \(P_\alpha\). These \(\Phi_{\alpha,\lambda}\) form a dense open subset of \(\Delta\). The other elements \(\Phi_w\), \(w\in V/K\), of \(\Delta\) are coming from the one-dimensional representations of \(H_n\). Hence \(\Delta=\Lambda\times\mathbb{R}^*\cup V/K\). The \(K\)-spherical transform \(^\wedge\) for \(f\in L^1_K(H_n)\) is the function \[ \widehat f(\Phi)=\int_{H_n}f(z,t)\overline{\Phi(z,t)}dzdt,\;\Phi\in\Delta. \] A continuous function \(F\) on \(\Delta\) is said to be rapidly decreasing if the restriction of \(F\) to \(V/K\) is in \({\mathcal S}_K(V)\), if the map \(\lambda\mapsto F(\alpha, \lambda)\) is smooth on \(\mathbb{R}^*\) for every \(\alpha\in\Lambda\) and if for each \(m,N\geq 0\) there exists a constant \(C_{m,N}\) for which \(|\partial_\lambda^mF(\alpha,\lambda)|\leq{C_{m,N}\over|\lambda|^{m+N}(2|\alpha|+n)^N}\) for all \((\alpha,\lambda)\in\Lambda\times\mathbb{R}^*\). The difference operators \({\mathcal D}^+\) and \({\mathcal D}^-\) are defined by \[ {\mathcal D}^+(F(\alpha,\lambda))=\sum_{|\beta|=| a|+1}{d_\beta\over d_\alpha}{\beta\brack\alpha}F(\beta,\lambda)-\bigl(|\alpha|+n\bigr)F(\alpha,n) \] and \[ {\mathcal D}^-(F(\alpha,\lambda))=|\alpha| F(\alpha,n)-\sum_{|\beta|=| a|-1}{d_\beta\over d_\alpha}{\beta\brack\alpha}F(\beta,\lambda). \] Finally let \(M^{+\choose -}F(\alpha,\lambda)=(\partial_\lambda-\left({1\over\lambda}\right){\mathcal D}^{+\choose -})F(\alpha,\lambda)\), \({+\choose -}\lambda>0\), and similarly for \(M^-\). Finally, the Schwartz space \({\mathcal S}(\Delta)\) is the set of all functions \(F:\Delta\to\mathbb{C}\) for which \((M^+)^l(M^-)^mF\) is rapidly decreasing for all \(m,l\geq 0\). The authors show that the spherical transform is an isomorphism of \({\mathcal S}_K(H_n)\) onto \({\mathcal S}(\Delta)\). The authors also consider spherical series expansions for \(K\)-invariant Schwartz-functions on \(H_n\) modulo its center.
scientific article; zbMATH DE number 1216766
scientific article