Infinite-Dimensional Representations of the Lie Algebra \mathfrakgl(n,C) Related to Complex Analogs of the Gelfand–Tsetlin Patterns and General Hypergeometric Functions on the Lie Group GL(n,C)

Abstract

Complex analogs of the Gelfand–Tsetlin patterns are introduced. Infinite-dimensional representations of \(\mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right)\) in the vector spaces spanned on these patterns are constructed. Exponentials of these representations are described. These exponentials are operators T(x), x∈GL(n,C), defined only in neighborhoods of the identity element of GL(n,C). A system of differential-difference equations for matrix elements of operators T(x) is constructed. Explicit formulas for matrix elements are obtained for the case x∈Z±, where Z+ and Z− are the triangular unipotent subgroups. Representations of \(\mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right)\) are also constructed; bases of these representations consist of Gelfand–Tsetlin patterns having infinitely many rows.