Generalized Whittaker functions for degenerate principal series of \(\mathrm{GL}(n, \mathbb{R})\)
The author studies generalized Whittaker functions of degenerate principle series of \(\mathrm{GL}(n, \mathbb{R})\). The first main result is a general characterization of the spaces of corresponding class one generalized Whittaker functions by an explicit system of differential operators. Using this result, the author presents detailed calculations in the case of the group \(\mathrm{GL}(4, \mathbb{R})\), considering degenerate principal series induced from different maximal parabolic subgroups. We emphasize that previously made results were mainly related to generalized principal series induced from a maximal parabolic subgroup \(P_{1, n}\) fixing the unit vector \(e_{n} = (0, \dotsc, 0, 1)\). In the considered case, the author examines the dimension of the spaces of class one generalized Whittaker functions, finds their bases in terms of hypergeometric functions in one and two variables and proves a corresponding multiplicity one theorem.
scientific article; zbMATH DE number 6126564
scientific article