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

The most degenerate unitary principal series representations pi(i lambda,delta) (lambda is an element of R, delta is an element of 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 pi(i lambda, delta)vertical bar(H) (branching law) with respect to all symmetric pairs (G, H). For N =2n with n >= 2, the restriction pi(i lambda, delta)vertical bar(H) remains irreducible for H = Sp(n, R) if lambda not equal 0 and splits into two irreducible representations if lambda = 0. The branching law of the restriction pi(i lambda, delta)vertical bar(H) is purely discrete for H = GL(n, C), consists only of continuous spectrum for H = GL(p, R) x GL(q, R) (p q = N), and contains both discrete and continuous spectra for H = 0(p, q) (p > q >= 1). Our emphasis is laid on geometric analysis, which arises from the restriction of 'small representations' to various subgroups. (C) 2010 Elsevier Inc. All rights reserved.