Infinite-Dimensional Representations of the Lie Algebra $mathfrak{g}mathfrak{l}left( {n,mathbb{C}} right)$ Related to Complex Analogs of the Gelfand–Tsetlin Patterns and General Hypergeometric Functions on the Lie Group GL $left( {n,mathbb{C}} right)$

摘要

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.