Generalized Whittaker Functions for Degenerate Principal Series of GL(4, $\mathbb{R}$)

Abstract

For degenerate principal series representations of $\mathrm{GL}(n,\mathbb{R})$, we show that the spaces of corresponding class one generalized Whittaker functions are characterized by explicit systems of differential operators. By using this characterization, we give detailed calculations on $\mathrm{GL}(4,\mathbb{R})$. We examine the dimensions of the spaces of generalized Whittaker functions and give their generators in terms of hypergeometric functions of one and two variables. We show that generalized Whittaker functions have multiplicity one by using the theory of hypergeometric functions.