Homological multiplicities in representation theory of $p$-adic groups

We study homological multiplicities of spherical varieties of reductive group $G$ over a $p$-adic field $F$. Based on Bernstein's decomposition of the category of smooth representations of a $p$-adic group, we introduce a sheaf that measures these multiplicities. We show that these multiplicities are finite whenever the usual mutliplicities are finite, in particular this holds for symmetric varieties, conjectured for all spherical varieties and known for a large class of spherical varieties. Furthermore, we show that the Euler-Poincar\'e characteristic is constant in families induced from admissible representations of a Levi $M.$ In the case when $M=G$ we compute these multiplicities more explicitly.