Zeta functions associated to admissible representations of compact 𝑝-adic Lie groups

Abstract

Let G be a profinite group. A strongly admissible smooth representation ̺ of G over C decomposes as a direct sum ̺ ∼ = π∈Irr(G) m π (̺) π of irreducible representations with finite multiplicities m π (̺) such that for every positive integer n the number r n (̺) of irreducible constituents of dimension n is finite. Examples arise naturally in the representation theory of reductive groups over non-archimedean local fields. In this article we initiate an investigation of the Dirichlet generating functionOur primary focus is on representations ̺ = Ind G H (σ) of compact p-adic Lie groups G that arise from finite-dimensional representations σ of closed subgroups H via the induction functor. In addition to a series of foundational results -including a description in terms of p-adic integrals -we establish rationality results and functional equations for zeta functions of globally defined families of induced representations of potent pro-p groups. A key ingredient of our proof is Hironaka's resolution of singularities, which yields formulae of Denef-type for the relevant zeta functions.In some detail, we consider representations of open compact subgroups of reductive p-adic groups that are induced from parabolic subgroups. Explicit computations are carried out by means of complementing techniques: (i) geometric methods that are applicable via distance-transitive actions on spherically homogeneous rooted trees and (ii) the p-adic Kirillov orbit method. Approach (i) is closely related to the notion of Gelfand pairs and works equally well in positive defining characteristic.

Groups acting on rooted trees and their representations on the boundary

“…The definition given in the introduction suffices for our purposes. For more information on zeta functions of induced representations the reader should consult [25].…”

“…which will be called the zeta function of the representation (ρ, V ρ ); for the details we refer to [25]. As observed above, the zeta function of the regular representation equals ζ G (s − 1).…”

“…In this case the boundary representation ρ ∂,C contains, apart from the trivial representation, exactly one irreducible summand of dimension |L n | − |L n−1 | for all n ∈ N. The account in [8] is based on unitary representations and can be translated via the remarks in Section 2.3 to apply to ρ ∂,C . A direct argument is contained in [25,Thm. 6.3].…”

“…Facing this difficulty it might be fruitful to break the problem into smaller pieces by counting only certain subsets of all representations. In the recent preprint [25] B. Klopsch and the author developed such a more flexible approach to representation zeta functions which is based on the following observation. The representation zeta function at s − 1 is…”

“…As observed above, the zeta function of the regular representation equals ζ G (s − 1). Here the results of [25] will not be used. It is sufficient to know that this setting provides a wealth of examples, where explicit formulae for zeta functions of representations can actually be determined; see [25,Sect.…”