We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished feature here is that even a 'small' deformation of a discrete subgroup may destroy proper discontinuity of its action. In order to understand the local structure of the deformation space of discontinuous groups, we introduce the concepts from a group theoretic perspective, and focus on 'stability' and 'local rigidity' of discontinuous groups. As a test case, we give an explicit description of the deformation space of ℤk acting properly discontinuously on ℝk+1 by affine nilpotent transformations. Our method uses an idea of 'continuous analogue' and relies on the criterion of proper actions on nilmanifolds.

Kobayashi's multiplicity-free theorem asserts that irreducible unitary highest-weight representations π are multiplicity-free when restricted to every symmetric pair if π is of scalar type. The aim of this paper is to find the "classical limit" of this multiplicity-free theorem in terms of the geometry of two coadjoint orbits, for which the correspondence is predicted by the Kirillov–Kostant–Duflo orbit method.For this, we study the Corwin–Greenleaf multiplicity function [Formula: see text] for Hermitian symmetric spaces G/K. First, we prove that [Formula: see text] for any G-coadjoint orbit [Formula: see text] and any K-coadjoint orbit [Formula: see text] if [Formula: see text]. Here, 𝔤 = 𝔨 + 𝔭 is the Cartan decomposition of the Lie algebra 𝔤 of G.Second, we find a necessary and sufficient condition for [Formula: see text] by means of strongly orthogonal roots. This criterion may be regarded as the "classical limit" of a special case of the Hua–Kostant–Schmid–Kobayashi branching laws of holomorphic discrete series representations with respect to symmetric pairs.

We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished feature here is that even a 'small' deformation of a discrete subgroup may destroy proper discontinuity of its action. In order to understand the local structure of the deformation space of discontinuous groups, we introduce the concepts from a group theoretic perspective, and focus on 'stability' and 'local rigidity' of discontinuous groups. As a test case, we give an explicit description of the deformation space of Z k acting properly discontinuously on R k+1 by affine nilpotent transformations. Our method uses an idea of 'continuous analogue' and relies on the criterion of proper actions on nilmanifolds.

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