Geometry of second adjointness for p-adic groups

We present a geometric proof of second adjointness for a reductive p-adic group. Our approach is based on geometry of the wonderful compactification and related varieties. Considering asymptotic behavior of a function on the group in a neighborhood of a boundary stratum of the compactification, we get a "cospecialization" map between spaces of functions on various varieties carrying a G × G action. These maps can be viewed as maps of bimodules for the Hecke algebra, and the corresponding natural transformations of endo-functors of the module category lead to the second adjointness. We also get a formula for the "cospecialization" map expressing it as a composition of the orispheric transform and inverse intertwining operator a parallel result for D-modules was obtained by Bezrukavnikov, Finkelberg and Ostrik. As a byproduct we obtain a formula for the Plancherel functional restricted to a certain commutative subalgebra in the Hecke algebra generalizing a result by Opdam.