Generating functions and integral representations for the spherical functions on some classical Gelfand pairs
Let \(G\) be \(SO_ 0(1,n)\), the connected component of the identity element in the real case \(\mathbb{F}=\mathbb{R}\), \(SU(1,n)\) in the complex case \(\mathbb{F}=\mathbb{C}\) and \(U(1,n;\mathbb{H})=Sp(1,n)\) in the quaternion case \(\mathbb{F}=\mathbb{H}\). The group \(G\) acts on the unit ball \(B(\mathbb{F}^ n)\) and the unit sphere \(S(\mathbb{F}^ n)\). The isotropy group \(K\) of the point \(O\in B(\mathbb{F}^ n)\) in \(G\) is just a maximal compact subgroup and \(G/K=B(\mathbb{F}^ n)\), \(K\cong SO(n)\) in the real case, \(U(n)\) in the complex case and \(Sp(1)\times Sp(n)\) in the quaternion case. Let \(G=KAN\) be the Iwasawa decomposition and \(M\) be the centralizer of \(A\) in \(K\). Then \(M\) is the isotropy group of the point \(e_ 1={^ t}(1,0,\dots,0)\in S(\mathbb{F}^ n)\) in \(K\), \(M\cong SO(n-1)\) in the real case, \(U(n-1)\) in the complex case and \(Sp(1)\times Sp(n-1)\) in the quaternion case, and \(K/M\cong S(\mathbb{F}^ n)\). In the real case, \(SO(n-1)\cong S(\mathbb{R}^ n)\), it is well known that each zonal spherical function \(\varphi\) depends only on the first coordinate \(\eta_ 1\), \((\eta={^ t}(\eta_ 1,\dots,\eta_ n)\in S(\mathbb{R}^ n))\), \(\varphi(\eta)=\varphi(\eta_ 1)\) and there is a unique nonnegative integer \(p\) such that \(\varphi(\eta_ 1)=C^{(n-2)/2}_ p(\eta_ 1)/C_ p^{(n-2)/2}(1)\), where \(C^{(n-2)/2}_ p\) is the Gegenbauer polynomial, for which there is the well known generating function \[ (1-2tz+t^ 2)^{-(n-2)/2}=\sum_{p=0}^ \infty C_ p^{(n-2)/2}(z) t^ p,\quad -1\leq z\leq 1,\quad -1<t<1. \] In the paper under review the author finds generating functions for the zonal spherical functions in the complex and the quaternion cases. He also gives integral presentations for the zonal functions in these situations. In the complex case, \(L^ 2(\mathbb{S}(\mathbb{C}^ n))=\bigoplus_{p,q=0}^ \infty H_{p,q}^{(n)}\), where \(H^{(n)}_{p,q}\) denotes the space of restrictions to \(\mathbb{S}(\mathbb{C}^ n)\) of harmonic polynomials \(f(\xi,\overline{\xi})\) on \(\mathbb{C}^ n\) which are homogeneous of degree \(p\) in \(\xi\) and degree \(q\) in \(\overline{\xi}\), each zonal spherical function \(\varphi\) belongs to one of these subspaces, say \(\varphi=\varphi^{(n)}_{p,q}\). The corresponding generating function for \(\varphi_{p,q}^{(n)}\) was found following Theorem 1.1: If \(w,z\in {\mathbb{C}}\), \(| w|<1\), \(| z|\leq 1\) then \[ (1- 2\text{Re}(wz)+| w|^ 2)^{1-n}=\sum_{p,q=0}^ \infty a_{p,q}^{(n)} Q_{p,q}^{(n)}(z) w^ p\overline{w}^ q, \] where \(Q_{p,q}^{(n)}(\eta_ 1)=\varphi_{p,q}^{(n)}(\eta),\) \(\eta=(\eta_ 1,\dots,\eta_ n)\in\mathbb{S}(\mathbb{C})^ n)\) and \(a_{p,q}^{(n)}={\Gamma(n+p-1)\over\Gamma(n-1)\Gamma(p+1)}{\Gamma(n+q- 1)\over\Gamma(n-1)\Gamma(q+1)}\). The series on the right hand side converges absolutely and uniformly for \(| z|\leq 1\) and \(| w|\leq\rho\) for each \(\rho < 1\). The function \(Q_{p,q}^{(n)}\) has also integral presentation and Rodrigues' formula \[ a_{p,q}^{(n)}Q_{p,q}^{(n)}(z)={(n-1)_ p\over 2\pi ip!}(| z|^ 2-1)^{2-n}\int_{|\zeta-z|=r}{\zeta^ p(\zeta\overline{z}-1)^{n+q-2}\over (\zeta-z)^{q+1}}d\zeta = \] \[ = {(n-1)_ p\over p!q!}(| z|^ 2-1)^{2-n}{\partial^ q\over\partial z^ q}[z^ p(| z|^ 2-1)^{n+q-2}], \] where \(| z| < 1\). If one puts \(w=re^{i\theta}\), then one has \[ (1- 2\text{Re}(e^{i\theta}z)+r^ 2)^{1-n} =\sum_{p,q}^ \infty a_{p,q}^{(n)}Q_{p,q}^{(n)} e^{i(p-q)\theta}r^{p+q}. \] This means that the zonal functions \(\varphi_{p,q}^{(n)}\) appear as the coefficients in the expansion of the left hand side by the powers of \(r\) and the spherical functions of \(U(1)\cong\mathbb{S}(\mathbb{R}^ 2)\). This interpretation for generating functions is adapted to the quaternion case.
scientific article; zbMATH DE number 584586