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On unipotent representations of ramified $p$-adic groups

Maarten Solleveld

Abstract

Let G be any connected reductive group over a non-archimedean local field. We analyse the unipotent representations of G, in particular in the cases where G is ramified. We establish a local Langlands correspondence for this class of representations, and we show that it satisfies all the desiderata of Borel as well as the conjecture of Hiraga, Ichino and Ikeda about formal degrees. This generalizes work of Lusztig and of Feng, Opdam and the author, to reductive groups that do not necessarily split over an unramified extension of the ground field. We also interpret our results in terms of rigid inner twists of G.

On unipotent representations of ramified $p$-adic groups

Abstract

Let G be any connected reductive group over a non-archimedean local field. We analyse the unipotent representations of G, in particular in the cases where G is ramified. We establish a local Langlands correspondence for this class of representations, and we show that it satisfies all the desiderata of Borel as well as the conjecture of Hiraga, Ichino and Ikeda about formal degrees. This generalizes work of Lusztig and of Feng, Opdam and the author, to reductive groups that do not necessarily split over an unramified extension of the ground field. We also interpret our results in terms of rigid inner twists of G.

Paper Structure

This paper contains 15 sections, 35 theorems, 146 equations.

Table of Contents

  1. List of ramified simple groups
  2. $\mathbf{B\text{\bf -} C_n}$
  3. $\mathbf{C\text{\bf -} BC_n}$
  4. $\mathbf{C\text{\bf -} B_n}$
  5. $\mathbf{ {}^2 B\text{\bf -} C_n}$
  6. $\mathbf{ {}^2 C\text{\bf -} B_{2n}}$
  7. $\mathbf{ {}^2 C\text{\bf -} B_{2n+1}}$
  8. $\mathbf{F_4^I}$
  9. $\mathbf{G_2^I}$
  10. Matching of unipotent representations
  11. Matching of Hecke algebras
  12. Comparison of Langlands parameters
  13. Supercuspidal unipotent representations
  14. A local Langlands correspondence
  15. Rigid inner twists

Key Result

Theorem 1

Let $\mathcal{G}$ be a connected reductive group over a non-archimedean local field $F$ and write $G = \mathcal{G}(F)$. There exists a bijection We can construct such a bijection for every group $G$ of this kind, in a compatible way. The resulting family of bijections has the following properties: Moreover the properties (a), (c), (k) and (l) uniquely determine the surjection up to twisting by

Theorems & Definitions (64)

  • Theorem 1
  • Conjecture 2
  • Theorem 3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • ...and 54 more