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Cuspidal endo-support and strong beta extensions

David Helm, Robert Kurinczuk, Daniel Skodlerack, Shaun Stevens

Abstract

Let $G$ be an inner form of a general linear group or classical group over a non-archimedean local field of residual characteristic $p$, assumed odd in the classical case. We prove that every smooth representation of $G$ over an algebraically closed field $R$ of characteristic $\ell\neq p$ contains a maximal semisimple character, i.e., one for which the point in the building of the corresponding centralizer is a vertex. Further, for every endo-parameter adapted to $G$, we define its support, which leads also to the notion of cuspidal endo-support of an irreducible representation, and we relate this to its cuspidal support. We also introduce beta extensions for strong facets in the building of a centralizer, and show these are sufficient for the construction of types. These results are used in a subsequent paper to decompose the category of smooth $R$-representations of $G$.

Cuspidal endo-support and strong beta extensions

Abstract

Let be an inner form of a general linear group or classical group over a non-archimedean local field of residual characteristic , assumed odd in the classical case. We prove that every smooth representation of over an algebraically closed field of characteristic contains a maximal semisimple character, i.e., one for which the point in the building of the corresponding centralizer is a vertex. Further, for every endo-parameter adapted to , we define its support, which leads also to the notion of cuspidal endo-support of an irreducible representation, and we relate this to its cuspidal support. We also introduce beta extensions for strong facets in the building of a centralizer, and show these are sufficient for the construction of types. These results are used in a subsequent paper to decompose the category of smooth -representations of .

Paper Structure

This paper contains 37 sections, 42 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. $m$-realizations and exhaustion in endo-factors
  3. Cuspidal support for endo-parameters
  4. beta extensions for the strong simplicial structure
  5. Structure of the paper
  6. Acknowledgements
  7. Preliminaries
  8. Notation
  9. Smooth ${\rm R}$-representations
  10. Forms of classical groups
  11. Parahoric subgroups for forms of classical groups
  12. Endo-parameters
  13. Semisimple characters and intertwining for ${\widetilde{{\rm G}}}$, following KSS and SkodInnerFormI
  14. Semisimple characters and intertwining for ${\rm G}$, following KSS and SkodInnerFormII
  15. Endo-parameters for $h$
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $\mathfrak{t}$ be an endo-parameter for ${\rm G}$. Then, every representation in $\mathrm{Rep}_{\rm R}(\mathfrak{t})$ is generated by the sum of the isotypic components of all $m$-realizations of $\mathfrak{t}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (107)

  • Theorem 1.1: see Theorem \ref{['thm43']}
  • Theorem 1.2: see Theorem \ref{['thmEssentiallyConjugateClasses']}
  • Theorem 1.3
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • ...and 97 more