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On the twisted Osborne conjecture

Chang Huang

TL;DR

This work proves a twisted Osborne-type identity for a real reductive group $\mathbf G$ with a finite-order automorphism $\tau$, relating the twisted character $\Theta^{\tau}(\pi)$ of a $\tau$-stable Casselman-Wallach representation to an alternating sum of twisted $\mathfrak n$-homology characters of the $N$-part of a real parabolic and a twisted Weyl denominator $D^{\tau}_{\mathfrak n}$. The authors develop a comprehensive framework: establish tau-stable data (maximal compact subgroups, real parabolics, and infinitesimal characters), obtain local expressions and ${\mu}$-isotypic components to isolate the key contributions, and apply coherent continuation together with a very anti-dominant regime to reduce the problem to induced modules. A detailed analysis of induced modules through Hirai-type distribution formulas, twisted Weyl integration, and asymptotic estimates on regular and non-regular elements yields a robust mechanism to compare twisted characters with their Jacquet (or $\mathfrak n$-homology) counterparts. The results have potential applications to Arthur packets and twisted endoscopy, with a parallel p‑adic picture, and they establish the main compatibility required for constructing and understanding endoscopic transfer in the twisted setting. The methodology blends representation-theoretic tools (Langlands classification, Jacquet modules, Harish-Chandra theory) with explicit harmonic-analysis techniques on twisted spaces, extending HS83 and BC13 to a unified, twist-aware framework.

Abstract

We aim to prove a twisted version of the Osborne conjecture obtained by Hecht and Schmid in their 1983 Acta Mathematica paper. Bergeron and Clozel (2013) have considered a special case, and we generalize their method to our setting.

On the twisted Osborne conjecture

TL;DR

This work proves a twisted Osborne-type identity for a real reductive group with a finite-order automorphism , relating the twisted character of a -stable Casselman-Wallach representation to an alternating sum of twisted -homology characters of the -part of a real parabolic and a twisted Weyl denominator . The authors develop a comprehensive framework: establish tau-stable data (maximal compact subgroups, real parabolics, and infinitesimal characters), obtain local expressions and -isotypic components to isolate the key contributions, and apply coherent continuation together with a very anti-dominant regime to reduce the problem to induced modules. A detailed analysis of induced modules through Hirai-type distribution formulas, twisted Weyl integration, and asymptotic estimates on regular and non-regular elements yields a robust mechanism to compare twisted characters with their Jacquet (or -homology) counterparts. The results have potential applications to Arthur packets and twisted endoscopy, with a parallel p‑adic picture, and they establish the main compatibility required for constructing and understanding endoscopic transfer in the twisted setting. The methodology blends representation-theoretic tools (Langlands classification, Jacquet modules, Harish-Chandra theory) with explicit harmonic-analysis techniques on twisted spaces, extending HS83 and BC13 to a unified, twist-aware framework.

Abstract

We aim to prove a twisted version of the Osborne conjecture obtained by Hecht and Schmid in their 1983 Acta Mathematica paper. Bergeron and Clozel (2013) have considered a special case, and we generalize their method to our setting.

Paper Structure

This paper contains 29 sections, 28 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Application
  3. $p$-adic analogy
  4. Existen results
  5. Preliminaries
  6. Notation
  7. $\tau$-regular elements
  8. Twisted Weyl denominator
  9. $\tau$-stable maximal compact subgroup
  10. $\tau$-stable real parabolic subgroup
  11. $\tau$-stable infinitesimal character
  12. Reduction
  13. Local expression
  14. $\mu$-component
  15. Coherent continuation
  16. ...and 14 more sections

Key Result

Theorem 1.1

Over a sufficiently large subset of $M$, it holds where $D_{\mathfrak n}^\tau(m) = {\mathrm{det}}_{\mathfrak n} (1 - m\tau )$ is invertible. Here, the subscript ${\mathfrak n}$ means that the determinant is taken for endomorphism $\operatorname{Id}_{\mathfrak n} - \operatorname{Ad}(m) \circ \tau$ over ${\mathfrak n}$. This subset, denoted by $M^-$,

Theorems & Definitions (34)

  • Theorem 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Theorem 1.4: Casselman, Rogawski
  • Remark 1.2.1
  • Theorem 1.5: Xu
  • Theorem 1.6
  • Lemma 2.1
  • Remark 2.2.1
  • Lemma 2.2
  • ...and 24 more