Geometric analysis on small unitary representations of [formula omitted

Abstract

The most degenerate unitary principal series representations π i λ , δ ( λ ∈ R , δ ∈ Z / 2 Z ) 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 i λ , δ | H ( branching law) with respect to all symmetric pairs ( G , H ) . For N = 2 n with n ⩾ 2 , the restriction π i λ , 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 π i λ , 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.