Kirillov's orbit method: the case of discrete series representations (Q2287927)

This paper is concerned with an important problem of noncommutative harmonic analysis: given a unitary irreducible representation \(\pi \) of a Lie group \(G^{\prime }\), how does \(\pi \) decompose when restricted to a closed subgroup \(G\subset G^{\prime }\)? This question is analyzed for the Harish-Chandra discrete series representations of a connected real reductive Lie group \(G^{\prime }\) relative to a connected real reductive subgroup \(G\). The main tool of this study is a unitary irreducible representation \(\pi ^{G^{\prime }}_{\mathcal{O}^{\prime }}\) of the group \(G^{\prime }\) associated to any regular Duflo-admissible elliptic coadjoint orbit \(\mathcal{O}^{\prime }\subset (\mathfrak{g}^{\prime })^{\ast }\).