Generating function for the spherical functions on a Gelfand pair of exceptional type
Let \(X= G/K\) be a non-compact Riemannian symmetric space of rank 1, where \(G\) is a connected simple Lie group and \(K\) its maximal compact subgroup. If \(G= KAN\) is an Iwasawa decomposition, then \(\dim A= 1\). It is known that \(G\) is one of \(\text{SO}_0(1, n)\), \(\text{SU}(1, n)\) and \(\text{SP}(1,n)\) or the exceptional group \(F_{4(- 20)}\). Let \(M\) be the centralizer of \(A\) in \(K\), then the Martin boundary \(K/M\) of \(X\) is generally not a symmetric space, but \((K, M)\) is a Gelfand pair. In the case of real hyperbolic spaces, on the boundary \(K/M= \text{SO}(n)/\text{SO}(n- 1)\) we have the classical generating function expansion: \[ (1- 2tx+ t^2)^{- (n- 2)/2}= \sum^\infty_{p= 0} C^{(n- 2)/2}_p(x) t^p, \] where \(C^{(n- 2)/2}_p\) is the Gegenbauer polynomial. This paper investigates a similar problem in the exceptional case. More precisely, the Martin boundary \(K/M\) of the Cayley hyperbolic plane is isomorphic to \(S^{15}\) by the identification \(kM\mapsto kF^1_2= F^u_2+ F^v_3\) (following the notation of \textit{R. Takahashi} [Analyse harmonique sur les groupes de Lie. II. Lect. Notes Math. 739, 511-567 (1979; Zbl 0447.43008)]), \(u,v\in \text{\textbf{0}}\), \(|u|^2+ |v|^2= 1\). Let \[ \varphi_{p, q}(kF^1_2)= c_{pq}\cdot C^3_p(\text{Re}(u)/|u|)|u|^p {}_2F_1(- q, p+ q+ 7; p+ 4; |u|^2) \] be the spherical function of \(K/M\). Then the author proves: if \(k\in K\), \(w\in \text{\textbf{0}}\), \(|w|= 1\) and \(0\leq r< 1\), then \[ \int_{\text{SO}(7)} [1- 2r\text{ Re}(\alpha(u) w)+ r^2]^{- 7} d\alpha= \sum^\infty_{p, q= 0} \gamma_{pq} \varphi_{pq}(kF^1_2) C^3_p(\text{Re}(w))r^{p+ 2q}. \] This formula is considered to give a generating function for the function \(\varphi_{pq}\). This interpretation of the generating functions is similar to the classical cases (cf. the author [Proc. Jap. Acad. Ser. A 68, 140-142 (1992; Zbl 0790.43014); J. Math. Kyoto Univ. 33, 1125-1142 (1993; Zbl 0801.43001)]).
scientific article; zbMATH DE number 645700