Geometry of second adjointness for \(p\)-adic groups. (With an appendix by Yakov Varshavsky, Roman Bezrukavnikov and David Kazhdan). (Q2792391)

From the introduction: ``Parabolic induction and Jacquet restriction functors play a fundamental role in the representation theory of reductive \(p\)-adic groups. It follows directly from definitions that the parabolic induction functor is right adjoint to the Jacquet functor.''NEWLINENEWLINE ``It has been discovered by Casselman for admissible representations and generalized by Bernstein to arbitrary smooth representations that there is another nonobvious adjointness between the two functors. Namely, the parabolic induction functor turns out to be also \textit{left} adjoint to Jacquet functor with respect to the opposite parabolic (we will refer to this as the second or the Bernstein adjointness). This fact appears in unpublished notes of Bernstein; we reprove it below. Rather than following the original strategy, our approach emphasizes the relation to geometry of the group and related spaces.''NEWLINENEWLINE``The second main result of the paper addresses the question of presenting this map'' (that is, the map responsible for the second adjointness) ``by an explicit correspondence. The answer is that the correspondence giving the map can be expressed in terms of the inverse to the intertwining operator (Radon transform). As an application of this result, we obtain a generalization of a result of Opdam which describes the Plancherel functional restricted to a certain commutative subalgebra in the Hecke algebra.''NEWLINENEWLINE In the appendix, the authors (jointly with Ya. Varshavsky) describe a version of the normal bundle construction which allows them to extend some statements about De Concini-Procesi's wonderful compactification to nonadjoint reductive groups. Further, the authors state that they expect their present methods can be used to obtain a generalization of some of their results replacing a Borel subgroup by a general parabolic subgroup.