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].

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].

Reviewer: Angela Pasquale (Clausthal-Zellerfeld)

##### MSC:

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) |