LAUSR

For a spherical variety X of a reductive group G over a non-archimedean local field F, and for a smooth representation $$\pi $$π of G we study homological multiplicities $$\dim {\text {Ext}}_{G}^{*}(\mathcal {S}(X),\pi )$$dimExtG∗(S(X),π). Based on Bernstein’s decomposition of the category of smooth representations of G, we introduce a sheaf that measures these multiplicities. We show that these multiplicities are finite whenever the usual mutliplicities $$\begin{aligned} \dim {\text {Hom}}_{G}(\mathcal {S}(X),\pi ) \end{aligned}$$dimHomG(S(X),π)are finite. The latter are known to be finite for symmetric varieties and for many other spherical varieties and conjectured to be finite for all spherical varieties. Furthermore, we show that the Euler–Poincaré characteristic is constant in families parabolically induced from finite length representations of a Levi subgroup $$M