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Asymptotic behavior for twisted traces of self-dual and conjugate self-dual representations of $\mathrm{GL}_n$

Yugo Takanashi, Satoshi Wakatsuki

TL;DR

This work advances the twisted limit multiplicity program for GL_n by analyzing the level-aspect asymptotics of twisted traces for self-dual and conjugate self-dual representations. It employs the Arthur twisted invariant trace formula to isolate main terms, and develops a comprehensive Fourier-analytic framework for twisted orbital integrals, linking them to endoscopic lifts from classical groups. The authors establish explicit asymptotic formulas and automorphic density results in the twisted setting, with globalization applications for constructing self-dual and conjugate self-dual automorphic representations. By exploiting local and global Langlands correspondences and endoscopic transfer, the paper connects geometric orbital data with spectral Plancherel measures, providing concrete measures for twisted endoscopic groups and advancing density theorems for conjugate self-dual GL_n representations. The results have potential impact on automorphic representation counting, globalization strategies, and endoscopy-driven understandings of the automorphic spectrum in the twisted context.

Abstract

In this paper, we study the asymptotic behavior of the sum of twisted traces of self-dual or conjugate self-dual discrete automorphic representations of $\mathrm{GL}_n$ for the level aspect of principal congruence subgroups under some conditions. Our asymptotic formula is derived from the Arthur twisted trace formula, and it is regarded as a twisted version of limit multiplicity formula on Lie groups. We determine the main terms for the asymptotic behavior under different conditions, and also obtain explicit forms of their Fourier transforms, which correspond to endoscopic lifts from classical groups. Its main application is the self-dual (resp. conjugate self-dual) globalization of local self-dual (resp. conjugate self-dual) representations of $\mathrm{GL}_n$. We further derive an automorphic density theorem for conjugate self-dual representations of $\mathrm{GL}_n$.

Asymptotic behavior for twisted traces of self-dual and conjugate self-dual representations of $\mathrm{GL}_n$

TL;DR

This work advances the twisted limit multiplicity program for GL_n by analyzing the level-aspect asymptotics of twisted traces for self-dual and conjugate self-dual representations. It employs the Arthur twisted invariant trace formula to isolate main terms, and develops a comprehensive Fourier-analytic framework for twisted orbital integrals, linking them to endoscopic lifts from classical groups. The authors establish explicit asymptotic formulas and automorphic density results in the twisted setting, with globalization applications for constructing self-dual and conjugate self-dual automorphic representations. By exploiting local and global Langlands correspondences and endoscopic transfer, the paper connects geometric orbital data with spectral Plancherel measures, providing concrete measures for twisted endoscopic groups and advancing density theorems for conjugate self-dual GL_n representations. The results have potential impact on automorphic representation counting, globalization strategies, and endoscopy-driven understandings of the automorphic spectrum in the twisted context.

Abstract

In this paper, we study the asymptotic behavior of the sum of twisted traces of self-dual or conjugate self-dual discrete automorphic representations of for the level aspect of principal congruence subgroups under some conditions. Our asymptotic formula is derived from the Arthur twisted trace formula, and it is regarded as a twisted version of limit multiplicity formula on Lie groups. We determine the main terms for the asymptotic behavior under different conditions, and also obtain explicit forms of their Fourier transforms, which correspond to endoscopic lifts from classical groups. Its main application is the self-dual (resp. conjugate self-dual) globalization of local self-dual (resp. conjugate self-dual) representations of . We further derive an automorphic density theorem for conjugate self-dual representations of .
Paper Structure (64 sections, 74 theorems, 397 equations)

This paper contains 64 sections, 74 theorems, 397 equations.

Table of Contents

  1. Introduction and main theorems
  2. Overview of limit multiplicity problem and our results
  3. Limit formula
  4. $\theta$-Elliptic representations, pseudo-coefficients, and matrix coefficients
  5. Self-dual and conjugate self-dual globalizations
  6. Automorphic density theorem for conjugate self-dual representations of $\mathrm{GL}_n$
  7. Asymptotic formula for twisted traces
  8. Setup
  9. Notations
  10. Twisted orbital integrals
  11. and is even
  12. Asymptotic formula
  13. Proof of Theorem \ref{['thm:asym']}
  14. Preliminaries for the proof
  15. Notations
  16. ...and 49 more sections

Key Result

Theorem 1.2

Assume the above conditions. In particular, fix a place $v_0\in S$, assume either (A1) or (A2), and consider one of Cases (i), (ii), (iii), (iv). Under Case (iii) (resp. (iv)), we also suppose $\mathcal{Q}(\chi_S,j)\neq\emptyset$ (resp. $c(\chi_S,j)\neq 0$) for any large $j$. Then there exists a con where $h_S\in\mathcal{H}(G(F_S))$ is defined by $h_S(g)= f_S(\theta(g^{-1}))$.

Theorems & Definitions (132)

  • Remark 1.1
  • Theorem 1.2
  • proof
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • proof
  • Theorem 1.6
  • proof
  • ...and 122 more