On multiplicity in restriction for $p$-adic groups

Kwangho Choiy

Published 2013-06-26, updated 2016-06-15Version 4

We study the multiplicity occurring when irreducible smooth representations of $\widetilde{\bold G}(F)$ are restricted to $\bold G(F)$ in a general setting, where $\widetilde{\bold G}$ is a connected reductive algebraic group over a $p$-adic field $F$ of characteristic 0 and $\bold G$ is its closed $F$-subgroup sharing the same derived group. We first illuminate various quantitative aspects of the multiplicity for discrete series representations in the case of $\widetilde{\bold G} =\rm{GL}_m(D)$ and $\bold G = \rm{SL}_m(D),$ where $D$ is a central division algebra of dimension $d^2$ over $F.$ We then investigate parallel phenomena occurring in restrictions of representations between the connected reductive $F$-groups in the general setting and their component groups, so-called $\mathcal{S}$-groups, under the assumptions of the local Langlands conjecture and internal structure of $L$-packets. We also obtain the equality of multiplicities in the both sides, and provide a general formula of the multiplicity in the restriction of irreducible smooth representations of $\widetilde{\bold G}(F)$ to $\bold G(F).$ This formula is given in terms of dimensions of irreducible representations of their $\mathcal{S}$-groups and generalizes Hiraga and Saito's result in 2012 for the case of $\rm{GL}_m(D)$ and $\rm{SL}_m(D).$