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The most degenerate unitary principal series representations{\pi}_{i{\lambda},{\delta}} (with {\lambda} \in R, \delta \in Z/2Z) of G =GL(N,R) attain the minimum of the Gelfand-Kirillov dimension among allirreducible unitary representations of G. This article gives an explicitformula of the irreducible decomposition of the restriction{\pi}_{i{\lambda},{\delta}}|_H (branching law) with respect to all symmetricpairs (G,H). For N=2n with n \geq 2, the restriction{\pi}_{i{\lambda},{\delta}}|_H remains irreducible for H=Sp(n,R) if{\lambda}\neq0 and splits into two irreducible representations if {\lambda}=0.The branching law of the restriction {\pi}_{i{\lambda},{\delta}}|_H is purelydiscrete 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 forH=O(p,q) (p>q\geq1). Our emphasis is laid on geometric analysis, which arisesfrom the restriction of `small representations' to various subgroups.

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