Multiplicity in restriction and tempered $L$-packets for $p$-adic groups

Studying the behavior of complex representations of a $p$-adic group restricted to its closed subgroup with the same derived group, we can derive useful information to establish the local Langlands correspondence for the subgroup. In this context, after reviewing the correspondences for $p$-adic inner forms of $SL(n), Sp(4),$ and others, we shall discuss the multiplicity of tempered representations of $p$-adic groups in the restriction, and relate it to its counterpart of corresponding irreducible representations of finite groups in the Langlands dual side, under some assumptions. This relation provides an extensive formula of the multiplicity and certain information on the internal structure of tempered $L$-packets.