On branching rules of depth-zero representations

Nevins, Monica

doi:10.1016/j.jalgebra.2014.03.016

Cited by 3 publications

(2 citation statements)

References 21 publications

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Section: 4mentioning

confidence: 99%

“…They used direct representation-theoretic considerations and, as a consequence, deduced a formula for the associated zeta function. It is interesting to conduct similar investigations for supercuspidal representations of G; Nevins [45] has made steps in this direction. 1.4.2. In this article we consider, for simplicity, only smooth representations of profinite groups G over the complex field C. More generally, following unpublished work of González-Sánchez, Jaikin-Zapirain, and Klopsch, one could derive analogues of several of our results also for representations over fields F of characteristic 0 that are not necessarily algebraically closed.…”

Section: 4mentioning

confidence: 99%

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

Kionke

Klopsch

2019

Trans. Amer. Math. Soc.

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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.

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Section: 4mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

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

Kionke

Klopsch

2019

Trans. Amer. Math. Soc.

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On the Unicity of Types for Toral Supercuspidal Representations

Latham

Nevins

2019

Progress in Mathematics

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For tame arbitrary-length toral, also called positive regular, supercuspidal representations of a simply connected and semisimple p-adic group G, constructed as per Adler-Yu, we determine which components of their restriction to a maximal compact subgroup are types. We give conditions under which there is a unique such component, and then present a class of examples for which there is not, disproving the strong version of the conjecture of unicity of types on maximal compact open subgroups. We restate the unicity conjecture, and prove it holds for the groups and representations under consideration under a mild condition on depth.

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Typical representations via fixed point sets in Bruhat–Tits buildings

Latham

Nevins

2021

Represent. Theory

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For a tame supercuspidal representation π \pi of a connected reductive p p -adic group G G , we establish two distinct and complementary sufficient conditions, formulated in terms of the geometry of the Bruhat–Tits building of G G , for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of G G which is not inertially equivalent to π \pi . The consequence is a set of broadly applicable tools for addressing the branching rules of π \pi and the unicity of [ G , π ] G [G,\pi ]_G -types.

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On branching rules of depth-zero representations

Cited by 3 publications

References 21 publications

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

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

On the Unicity of Types for Toral Supercuspidal Representations

Typical representations via fixed point sets in Bruhat–Tits buildings

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