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

The most degenerate unitary principal series representations πiλ,δ (λ ∈ R, δ ∈ 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 πiλ,δ|H (branching law) with respect to all symmetric pairs (G,H). For N = 2n with n 2, the restriction πiλ,δ|H remains irreducible for H = Sp(n,R) if λ = 0 and splits into two irreducible representations if λ = 0. The branching law of the restriction πiλ,δ|H is purely discrete for H = GL(n,C), consists only of continuous spectrum for H = GL(p,R) × GL(q,R) (p + q = N), and contains both discrete and continuous spectra for H = O(p, q) (p >q 1). Our emphasis is laid on geometric analysis, which arises from the restriction of ‘small representations’ to various subgroups. © 2010 Elsevier Inc. All rights reserved.