Stability of Branching Laws for Highest Weight Modules

Abstract

In this paper, we study the irreducible decomposition of a (C[X], G)-module M for a quasi-affine spherical variety X of a connected reductive algebraic group G over C. We show that for sufficiently large parameters, the decomposition of M with respect to G is reduced to the decomposition of the 'fiber' M/m(x 0 )M with respect to some reductive subgroup L of G. In particular, we obtain a method to compute the maximum value of multiplicities in M . Our main result is a generalization of earlier work by F. Satō in [17]. We apply this result to branching laws of holomorphic discrete series representations with respect to symmetric pairs of holomorphic type. We give a necessary and sufficient condition for multiplicity-freeness of the branching laws.

Uniformly bounded multiplicities, polynomial identities and coisotropic actions

“…He also shows that the multiplicity Hom G ′ (V, V ′ ) is given by Hom L (V ′ , V L ) if the highest weight of V is enough large in some sense. The author generalize these results to quasi-affine spherical varieties in [32].…”

“…We need the following result by Brion-Luna-Vust [13, 0.4]. From the fact, we can see that (B ∩ H) 0 ⊂ L 0 is a Borel subgroup of L 0 and B ∩ H meets every connected component of L (see [32,Proposition 4.3]). In fact, L/(B ∩ H) is isomorphic to the full flag variety of a Levi subgroup of P .…”

Associated symmetric pair and multiplicities of admissible restriction of discrete series

“…To solve this problem involves among other branches of mathematics, algebraic geometry, differential geometry and hard analysis as we can learn from examples and theorems presented in [13], references therein and further work of T. Kobayashi, N. Wallach as well as other researchers. In the book [9], or in [10], [12], [19] as well as in the work of other authors [6], we learn that sometimes the problem of writing the branching law for π is translated into the problem of computing branching law for another pair of groups L ⊂ H 0 and certain irreducible representation of H 0 . From this point of view, in [17], [21], [6] is analyzed the restriction of a family of Zuckerman modules for a real reductive Lie group G and H the connected component of the fix point group of an involution of G. Henceforth, G denotes a connected simple matrix Lie group.…”