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$$K$$-finite matrix elements of irreducible Harish-Chandra modules are hypergeometric functions. (English. Russian original) Zbl 1166.22009
Funct. Anal. Appl. 41, No. 4, 295-302 (2007); translation from Funkts. Anal. Prilozh. 41, No. 4, 60-69 (2007).
Let $$G$$ be a linear real semisimple Lie group, with maximal compact subgroup $$K$$. The main result of this paper concerns the $$K$$-finite matrix coefficients of minimal principal series representations of $$G$$.
Let $$P=MAN$$ be a minimal parabolic subgroup of $$G$$, $$\tau$$ an irreducible representation of the compact group $$M$$, and $$\chi$$ a character of the vector group $$A$$. Recall that such a character $$\chi=\chi_s$$ is given by a $$k$$-tuple $$s$$ of complex numbers, where $$k$$ is the dimension of $$A$$. Write $$\pi=Ind_{P}^{G}(\tau\otimes\chi)$$, a minimal principal series representation, and $$V$$ the space of $$K$$-finite vectors. Denote the dual module $$V^{\circ}$$. Let $$\sigma$$ and $$\theta$$ be $$K$$-types occurring in $$V$$ and $$V^{\circ}$$, respectively, and let $$v\in V_{\sigma}$$, $$w \in V^{\circ}_{\theta}$$, be elements of the corresponding $$K$$-isotypic subspaces. If $$\{,\}$$ denotes the natural pairing between $$V$$ and its dual, then we obtain a matrix coefficient $$\{\pi(g)v,w\}$$.
Main Theorem: (a) Suppose $$\pi$$ is irreducible. Then there exists a finite-dimensional representation $$\xi$$ of $$G$$ and a $$k$$-tuple $$t\in \mathbb C^k$$ such that $$\{\pi(g)v,w\}$$ can be written as a finite sum
$\{\pi(g)v,w\}=\sum_j h_j(g)\cdot p_jq_j\Psi_t(g).$ Here $$h_j$$ is a finite linear combination of matrix coefficients for $$\xi$$, $$p_j$$ and $$q_j$$ belong to the algebras of left and right $$G$$-invariant differential operators on $$G$$, respectively, and $$\Psi_t$$ is the spherical function on $$G$$ corresponding to $$t$$.
(b) If $$\pi$$ is reducible, we have a similar finite expression
$\{\pi(g)v,w\}=\lim_{\varepsilon\to 0}\sum_j h_j(\varepsilon,g)\cdot p_j(\varepsilon) q_j(\varepsilon)\Psi_{t+\varepsilon t'}(g)$ with $$h_j(\varepsilon,g)$$, $$p_j(\varepsilon)$$ and $$q_j(\varepsilon)$$ depending rationally on the parameter $$\varepsilon$$.
Since every irreducible admissible representation of $$G$$ can be realized as a constituent of a minimal principal series, we get an expression as in the second part of the theorem for every irreducible Harish-Chandra module of $$G$$.
Consequently, $$K$$-finite matrix coefficients of HC modules are hypergeometric functions.
As an immediate corollary, one obtains that each domain $$G \subset \Omega \subset G_{\mathbb C}$$ of holomorphy for all spherical functions is a domain of holomorphy for all $$K$$-finite matrix coefficients of irreducible Harish-Chandra modules of $$G$$.
MSC:
 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 33C80 Connections of hypergeometric functions with groups and algebras, and related topics 22E46 Semisimple Lie groups and their representations
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