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Rationality properties of complex representations of reductive p-adic groups

David Kazhdan, Maarten Solleveld, Yakov Varshavsky

Abstract

For a reductive group G over a non-archimedean local field, we compare smooth representations over C with smooth representations over Qbar (an algebraic closure of Q). For example, we show that tensoring with C over Qbar preserves irreducibility of representations. We also show that an elliptic G-representation (in the sense of Arthur) can be realized over Qbar if and only if its central character takes values in Qbar. That applies in particular to all essentially square-integrable G-representations. We also study the action of the automorphism group of C/Q on complex G-representations. We prove that the classes of essentially square-integrable representations and of elliptic representations are stable under Aut(C/Q).

Rationality properties of complex representations of reductive p-adic groups

Abstract

For a reductive group G over a non-archimedean local field, we compare smooth representations over C with smooth representations over Qbar (an algebraic closure of Q). For example, we show that tensoring with C over Qbar preserves irreducibility of representations. We also show that an elliptic G-representation (in the sense of Arthur) can be realized over Qbar if and only if its central character takes values in Qbar. That applies in particular to all essentially square-integrable G-representations. We also study the action of the automorphism group of C/Q on complex G-representations. We prove that the classes of essentially square-integrable representations and of elliptic representations are stable under Aut(C/Q).

Paper Structure

This paper contains 16 sections, 41 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Results
  4. Brief outline of the paper
  5. Cuspidal representations
  6. Irreducible $G$-representations over ${\overline{\mathbb Q}}$
  7. Schur's lemma and admissibility
  8. Comparison with $G$-representations over $\mathbb C$
  9. Essentially square-integrable representations
  10. Some general results
  11. Clozel's result about the Galois action
  12. Elliptic representations
  13. Intertwining operators and $\mu$-functions
  14. R-groups and elliptic representations
  15. Realization of elliptic representations over ${\overline{\mathbb Q}}$
  16. ...and 1 more sections

Key Result

Theorem 1.1

(see Proposition prop:2.9 and Theorem thm:3.15) For any irreducible ${\overline{\mathbb Q}}$-representation $V$ of the group $G$ the complex representation $V \otimes_{\overline{\mathbb Q}} {\mathbb C}$ is irreducible. The functor $\otimes_{\overline{\mathbb Q}} \mathbb C$ provides a bijection from

Theorems & Definitions (79)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • ...and 69 more