Geometric analysis on small unitary representations of GL(N,R)

Abstract

The most degenerate unitary principal series representations π_{iλ,δ} (with λ \in R, δ\in Z/2Z) of G = GL(N,R) attain the minimum of the Gelfand-Kirillov dimension among all irreducible unitary representations of G. This article gives an explicit formula of the irreducible decomposition of the restriction π_{iλ,δ}|_H (branching law) with respect to all symmetric pairs (G,H). For N=2n with n \geq 2, the restriction π_{iλ,δ}|_H remains irreducible for H=Sp(n,R) if λ\neq0 and splits into two irreducible representations if λ=0. The branching law of the restriction π_{iλ,δ}|_H is purely discrete for H = GL(n,C), consists only of continuous spectrum for H = GL(p,R) \times GL(q,R) (p+q=N), and contains both discrete and continuous spectra for H=O(p,q) (p>q\geq1). Our emphasis is laid on geometric analysis, which arises from the restriction of `small representations' to various subgroups.