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Generic Mackey Formula for Parahoric Lusztig Functors

Zhihang Yu

TL;DR

The paper addresses the Mackey problem for parahoric Lusztig induction in the deep-level $p$-adic setting, extending Lusztig’s classical result to parahoric contexts by exploiting the geometry of parahoric Lusztig varieties and their cohomology. It develops a robust framework based on generic characters, Howe factorizations, and a deep Bruhat decomposition to establish a generic Mackey formula for $R^{G}_{L,P,r}$, including precise vanishing results for Steinberg-type strata. As an application, it provides an irreducible decomposition for elliptic parahoric Deligne–Lusztig representations, expressing $R^{G}_{T,B,r}(\theta)$ as a finite sum of pairwise non-isomorphic irreducibles via a Howe-factorized ladder of Levi subgroups. The results advance the geometric realization of depth-$r$ supercuspidal representations and deepen connections to the local Langlands program, offering a blueprint for analyzing deep-level representations through parahoric cohomology.

Abstract

Parahoric Lusztig induction gives a broad class of virtual smooth representations of parahoric subgroups in a $p$-adic group, serving as a natural generalization of classical Lusztig induction to the $p$-adic setting. This construction has important applications in the representation theory of p-adic groups. In this paper, we prove the Mackey formula for parahoric Lusztig induction in generic case, which generalizes a classic result of Lusztig in 1976. As an application, we describe the irreducible decomposition of paragoric Deligne-Lusztig representations for the case of elliptic torus.

Generic Mackey Formula for Parahoric Lusztig Functors

TL;DR

The paper addresses the Mackey problem for parahoric Lusztig induction in the deep-level -adic setting, extending Lusztig’s classical result to parahoric contexts by exploiting the geometry of parahoric Lusztig varieties and their cohomology. It develops a robust framework based on generic characters, Howe factorizations, and a deep Bruhat decomposition to establish a generic Mackey formula for , including precise vanishing results for Steinberg-type strata. As an application, it provides an irreducible decomposition for elliptic parahoric Deligne–Lusztig representations, expressing as a finite sum of pairwise non-isomorphic irreducibles via a Howe-factorized ladder of Levi subgroups. The results advance the geometric realization of depth- supercuspidal representations and deepen connections to the local Langlands program, offering a blueprint for analyzing deep-level representations through parahoric cohomology.

Abstract

Parahoric Lusztig induction gives a broad class of virtual smooth representations of parahoric subgroups in a -adic group, serving as a natural generalization of classical Lusztig induction to the -adic setting. This construction has important applications in the representation theory of p-adic groups. In this paper, we prove the Mackey formula for parahoric Lusztig induction in generic case, which generalizes a classic result of Lusztig in 1976. As an application, we describe the irreducible decomposition of paragoric Deligne-Lusztig representations for the case of elliptic torus.

Paper Structure

This paper contains 17 sections, 29 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Background and Motivations
  3. Main Result
  4. Application: Irreducible decomposition of Parahoric Lusztig representations
  5. Structure of the paper
  6. Parahoric Lusztig Induction
  7. Definitions
  8. Properties
  9. Generic Mackey Formula for Parahoric Lusztig Induction
  10. Generic Mackey Formula for Parahoric Lusztig Induction
  11. Generic Characters and Howe factorization
  12. Higher Bruhat decomposition
  13. Steinberg type variety
  14. Some Vanishing results
  15. Proof of \ref{['vansLL']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

(Theorem thm) If $p$ is not a bad prime for $G$ and $p$ does not divide $|\pi_1(G_{\rm der})|$. Then where $\psi, \psi'$ are two $(L, G)$-generic It is a nontriviality condition on the restriction of $\psi$ to $\mathrm{ker}(T^F_r\to T^F_{r-1})$. See Definition generic representations of $L^F_r$, both of them are constituents of $R^{L}_{T, B\cap L,r}(\theta)$ and $\theta$ is an $(L,G)$-generic cha

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Corollary 2.4
  • ...and 45 more