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The FPP Conjecture for Real Reductive Groups

Dougal Davis, Lucas Mason-Brown

Abstract

In this paper, we prove the FPP conjecture, giving a strong upper bound on the unitary dual of a real reductive group. Our proof is an application of the global generation properties of $\mathcal{D}$-modules on the flag variety and their Hodge filtrations.

The FPP Conjecture for Real Reductive Groups

Abstract

In this paper, we prove the FPP conjecture, giving a strong upper bound on the unitary dual of a real reductive group. Our proof is an application of the global generation properties of -modules on the flag variety and their Hodge filtrations.

Paper Structure

This paper contains 8 sections, 8 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. FPP Conjecture
  3. Acknowledgments
  4. Hodge theory and unitarity
  5. Beilinson-Bernstein localization
  6. Cohomological induction and localization
  7. Mixed Hodge modules and localization
  8. Proof of the FPP Conjecture

Key Result

Theorem 1.1

Let $M$ be an irreducible unitary $(\mathfrak{g},K)$-module of real infinitesimal character $\chi_{\lambda}$. Conjugating by $W$, we can assume that $\lambda$ is dominant for $G$. Suppose that there is a simple co-root $\alpha^{\vee}$ such that Then $M$ is cohomologically induced in the weakly good range from a proper Levi subgroup.

Figures (1)

  • Figure 1: The spherical unitary dual of split $G_2({\mathbb{R}})$ ( green), with the bounds given by the Dirac inequality ( blue) and the FPP Conjecture ( red).

Theorems & Definitions (11)

  • Theorem 1.1: FPP Conjecture, see Theorem \ref{['thm:body FPP']} below
  • Theorem 2.1: Beilinson-Bernstein
  • Proposition 2.2
  • proof
  • Theorem 2.3: DV
  • Lemma 2.4
  • Theorem 2.5: DV
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • ...and 1 more