Homological multiplicities in representation theory of p-adic groups

Abstract

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-Poincaré 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.

Ext-Analogues of Branching Laws

“…Definition: A complex representation V of G = G(F) is said to be of geometric origin if (1) there is a G-space X V with finitely many G-orbits, (2) a G-equivariant sheaf F V on X V , such that on each G-orbit Y ⊂ X V of the form G/H Y , F V | Y is the equivariant sheaf associated to a finite dimensional representation W Y of H Y , and V ∼ = S(X V , F V ) (cf.…”

“…Towards a proof of finite dimensionality of Ext i in this case, to be made by an inductive argument on n later in the paper, we note that unlike Hom SO n (F) [π 1 , π 2 ], where we will have no idea how to prove finite dimensionality if both π 1 and π 2 are cuspidal, exactly this case we can handle apriori, for i > 0, as almost by the very definition of cuspidal representations, they are both projective and injective objects in the category of smooth representations. Recently, there is a very general finiteness theorem for Ext i [π 1 , π 2 ] (for spherical varieties) by Aizenbud and Sayag in [1]. However, we have preferred to give our own older approach via Bessel models which intervene when analyzing principal series representations of SO n+1 (F) when restricted to SO n (F).…”

Correction to: Homological multiplicities in representation theory of p-adic groups

“…However, [1, Theorem 6.1], whose proof relied upon the wrong [1, Lemma 6.4], requires a different proof which we provide here. We note that the conclusion of [1, Lemma 6.4] holds true for the case of symmetric pairs and, more generally, in the factorizable case (see Chapter 9 of [4]). The rest of [1] requires no modification.…”

“…

Regrettably, [1, Lemma 6.4] is incorrect, even though the statements of the theorems in [1] are correct with no modification required. However, [1, Theorem 6.1], whose proof relied upon the wrong [1, Lemma 6.4], requires a different proof which we provide here.

…”

“…To guarantee the Hausdorff property we will use: Lemma 1.5 Let Z be as in [1,Lemma 6.3]. Let < Z , H < G and put H := H(F).…”