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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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