Pseudo-dual pairs and branching of Discrete Series

Abstract

For a semisimple Lie group $G$, we study Discrete Series representations with admissible branching to a symmetric subgroup $H$. This is done using a canonical associated symmetric subgroup $H_0$, forming a pseudo-dual pair with $H$, and a corresponding branching law for this group with respect to its maximal compact subgroup. This is in analogy with either Blattner's or Kostant-Heckmann multiplicity formulas, and has some resemblance to Frobenius reciprocity. We give several explicit examples and links to Kobayashi-Pevzner theory of symmetry breaking and holographic operators. Our method is well adapted to computer algorithms, such as for example the Atlas program.

arXiv commentTo appear in "Symmetry in Geometry and Analysis -- Festschrift for Toshiyuki Kobayashi''. M. Pevzner und H. Sekiguchi eds., Progress in Mathematics, Springer 2024