Infinite-dimensional representations of the Lie algebra \(\mathfrak{gl}(n,\mathbb C)\) related to complex analogs of the Gelfand-Tsetlin patterns and general hypergeometric functions on the Lie group \(\text{GL}(n,\mathbb C)\).
In 1950 Gel'fand-Tsetlin bases for irreducible finite-dimensional representations of the Lie algebra \(\mathfrak{gl}(n,\mathbb C)\) were introduced by \textit{I. M. Gel'fand} and \textit{M. L. Tsetlin} [Dokl. Akad. Nauk SSSR, N. Ser. 71, 825--828 (1950; Zbl 0037.15301)] and in 1965 their explicit matrix elements were studied by \textit{I. M. Gel'fand} and \textit{M. I. Graev} [Izv. Akad. Nauk SSSR, Ser. Mat. 29, 1329--1356 (1965; Zbl 0139.30701)]. Here it is shown that this procedure still works when complex Gel'fand-Tsetlin \(n\)-patterns are considered. The author studies the infinite-dimensional representations of \(\mathfrak{gl}(n,\mathbb C)\) constructed in this way together with the local representation of the group \(\text{GL}(n,\mathbb C)\) it yields by exponentiation.
representations of Lie groups and Lie algebras
scientific article; zbMATH DE number 2093656