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Hypergeometric functions on reductive groups. (English) Zbl 0987.33008
Saito, M.-H. (ed.) et al., Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30-July 4, 1997, and in Kyoto, Japan, July 7-11, 1997. Singapore: World Scientific. 236-281 (1998).
The theory of \(A\)-hypergeometric functions is a generalization of the classical theory of hypergeometric functions. When \(A \subset\mathbb Z^m\) is a finite set of characters of the torus \((\mathbb C^*)^m\), the \(A\)-hypergeometric functions have been introduced by Gel’fand, Kapranov and Zelevinsky in the late 1980s. See the survey article by I. M. Gel’fand, M. M. Kapranov and A. V. Zelevinskij [in: M. Kashiwara and T. Miwa (eds.). Special functions, Proc. Hayashibara Forum, Okayama/Jap. 1990, ICM-90 104-121 (1991; Zbl 0785.33008)]. The paper under review defines the \(A\)-hypergeometric functions in the general case in which \(A\) is a finite set of irreducible representations of a reductive group \(H\).
Let \(M_A\) denote the space of functions on \(H\) which are linear combinations of matrix coefficients of the elements in \(A\). Then \(M_A\) is a finite-dimensional subspace of \(\mathbb C[H]\) which is both left and right \(H\)-invariant. Suppose that \(H\) can be regarded as a subset of the dual \(M_A^*\) of \(M_A\) by means of the direct sum of the representations in \(A\). Then two spherical varieties, \(Y_A\) and \(X_A\), are associated with \(A\). Under certain homogeneity assumptions, \(Y_A\) is a conical variety, defined as the Zariski closure of \(H\) in \(M_A^*\), and \(X_A\) is its projectivization. \(Y_A\) has a natural structure of reductive algebraic semigroup. Let \(\mathbf h\) denote the Lie algebra of \(H\). For every \(h \in \mathbf h\), let \(L_h\) and \(R_h\) respectively denote the infinitesimal generators of the left and right actions of \(H\) on \(M_A\). Finally, let the elements \(f\) of the homogeneous ideal \(I_A\) of \(Y_A\) be identified with differential operators \(P_f\) on \(M_A\). Then the \(A\)-hypergeometric system corresponding to the character \(\chi:{\mathbf h}\rightarrow \mathbb C\) is the system of differential equations on \(M_A\) \[ \begin{aligned} L_h\Phi&=R_h \Phi=\chi(h)\Phi, \quad h \in {\mathbf h}, \\ P_f \Phi&=0, \quad f \in I_A. \end{aligned} \] Its holomorphic solutions (defined on some open subset of \(M_A\)) are called the \(A\)-hypergeometric functions. As in both the classical and the toric cases, also in the general case treated in this paper, the \(A\)-hypergeometric functions can be represented by means of power series and Euler integrals.
The first four sections of the paper present a concise and nicely written exposition of the preliminaries needed for the \(A\)-hypergeometric functions: monomials and polynomials on groups and the related algebraic geometry, formal power series and distributions on groups, the Fourier transform of the matrix coefficients of irreducible representations of \(GL_n(\mathbb C)\) and the matrix gamma function. The \(A\)-hypergeometric functions are introduced in Section 5. In the final section, some generalizations of the Gauss, Pochhammer and Appell functions are presented as applications.
For the entire collection see [Zbl 0949.00022].

33C80 Connections of hypergeometric functions with groups and algebras, and related topics
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
20G05 Representation theory for linear algebraic groups
22E30 Analysis on real and complex Lie groups
32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)