Abstract
Let G G be a profinite group. A strongly admissible smooth representation Ďą \varrho of G G over C {\mathbb {C}} decomposes as a direct sum Ďą â â¨ Ď â Irr ( G ) m Ď ( Ďą ) Ď \varrho \cong \bigoplus _{\pi \in \text {Irr}(G)} m_\pi (\varrho ) \, \pi of irreducible representations with finite multiplicities m Ď ( Ďą ) m_\pi (\varrho ) such that, for every positive integer n n , the number r n ( Ďą ) r_n(\varrho ) of irreducible constituents of dimension n n is finite. Examples arise naturally in the representation theory of reductive groups over nonarchimedean local fields. In this article we initiate an investigation of the Dirichlet generating function \[ Îś Ďą ( s ) = â n = 1 â r n ( Ďą ) n â s = â Ď â Irr ( G ) m Ď ( Ďą ) ( dim âĄ Ď ) s \zeta _\varrho (s) = \sum \nolimits _{n=1}^\infty r_n(\varrho ) n^{-s} = \sum \nolimits _{\pi \in \text {Irr}(G)} \frac {m_\pi (\varrho )}{(\dim \pi )^s} \] associated to such a representation Ďą \varrho . Our primary focus is on representations Ďą = Ind H G ( Ď ) \varrho = \text {Ind}_H^G(\sigma ) of compact p p -adic Lie groups G G that arise from finite-dimensional representations Ď \sigma of closed subgroups H H via the induction functor. In addition to a series of foundational resultsâincluding a description in terms of p p -adic integralsâwe establish rationality results and functional equations for zeta functions of globally defined families of induced representations of potent pro- p p groups. A key ingredient of our proof is Hironakaâs 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 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 p -adic Kirillov orbit method. Approach (i) is closely related to the notion of Gelfand pairs and works equally well in positive defining characteristic.
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