Generalized Whittaker functions for degenerate principal series of $GL(4,\R)$

TL;DR

This paper characterizes generalized Whittaker models for degenerate principal series of $GL(4, )$ using differential operators, computes their dimensions, bases, and proves multiplicity one results with hypergeometric functions.

Contribution

It introduces a new characterization of generalized Whittaker models as kernels of differential operators and applies this to $GL(4, )$, providing explicit bases and multiplicity results.

Findings

Dimensions of the generalized Whittaker models are computed.

Explicit bases are given in terms of hypergeometric functions.

Multiplicity one of the models is established.

Abstract

We give a characterization of a generalized Whittaker model of a degenerate principal series representation of $GL(n,\R)$ as the kernel of some differential operators. By this characterization, we investigate some examples on $GL(4,\R)$. We obtain the dimensions of the generalized Whittaker models and give their basis in terms of hypergeometric functions of one and two variables. We show the multiplicity one of the generalized Whittaker models by using the theory of hypergeometric functions.