A KÜNNETH THEOREM FOR p-ADIC GROUPS

Let F be a non-Archimedean local field. LetG stand for the F -points of a connected reductive algebraic group defined over F . We will refer to G as a p-adic group, with the understanding that the base field F is fixed once and for all. We let R(G) denote the category of smooth complex representations of G. It is well known [1] that this is an abelian category and has enough projectives and hence, given any two smooth representations π and ρ, we can compute the Ext-groups ExtR(G)(π, ρ). In any homological set-up, it is a fundamental problem to describe the (co)homology of a product of objects in terms of those of the individual constituents. Given two p-adic groups G1 and G2, the Künneth theorem we prove relates extensions for the group G1×G2 to those of G1 and G2. Without further ado, we state the main theorem of this article.