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Classification of $\mathrm{GL}_{n}(\mathbb{C})$-Representations Distinguished by $\mathrm{GL}_n(\mathbb{R})$

Basudev Pattanayak, Kaidi Wu, Hongfeng Zhang

TL;DR

The paper delivers a complete classification of GL_n(R)-distinguished irreducible representations of GL_n(C) in the generic and unitary regimes by translating distinction into conditions on Langlands parameters under a Weyl-involution. It develops a method to explicitly construct and meromorphically continue distinguished period integrals, proving non-vanishing on the distinguished minimal K-type and relating these periods to degenerate Whittaker models. The results connect distinction to Asai L-functions and enable branching-law insights via seesaw identities in theta correspondence, including precise statements for local factors like epsilon-factors. The work provides a robust framework combining Archimedean Langlands, Bernstein–Zelevinsky derivatives, Schwartz homology, and parabolic induction to establish both necessary and sufficient conditions for distinction, with explicit building blocks and test vectors. Overall, it advances relative Langlands theory at archimedean places and offers tools for further study of periods, L-functions, and harmonic analysis on GL_n-manifolds and symmetric spaces.

Abstract

This paper provides a complete classification of $\mathrm{GL}_n(\mathbb{R})$-distinguished irreducible representations of $\mathrm{GL}_n(\mathbb{C})$ when the representations are either generic or unitary. Additionally, for each such $\mathrm{GL}_n(\mathbb{R})$-distinguished representation, we explicitly construct the associated period and prove its non-vanishing on the distinguished minimal $K$-type. Furthermore, we offer some applications to the branching problem using theta correspondence.

Classification of $\mathrm{GL}_{n}(\mathbb{C})$-Representations Distinguished by $\mathrm{GL}_n(\mathbb{R})$

TL;DR

The paper delivers a complete classification of GL_n(R)-distinguished irreducible representations of GL_n(C) in the generic and unitary regimes by translating distinction into conditions on Langlands parameters under a Weyl-involution. It develops a method to explicitly construct and meromorphically continue distinguished period integrals, proving non-vanishing on the distinguished minimal K-type and relating these periods to degenerate Whittaker models. The results connect distinction to Asai L-functions and enable branching-law insights via seesaw identities in theta correspondence, including precise statements for local factors like epsilon-factors. The work provides a robust framework combining Archimedean Langlands, Bernstein–Zelevinsky derivatives, Schwartz homology, and parabolic induction to establish both necessary and sufficient conditions for distinction, with explicit building blocks and test vectors. Overall, it advances relative Langlands theory at archimedean places and offers tools for further study of periods, L-functions, and harmonic analysis on GL_n-manifolds and symmetric spaces.

Abstract

This paper provides a complete classification of -distinguished irreducible representations of when the representations are either generic or unitary. Additionally, for each such -distinguished representation, we explicitly construct the associated period and prove its non-vanishing on the distinguished minimal -type. Furthermore, we offer some applications to the branching problem using theta correspondence.
Paper Structure (35 sections, 42 theorems, 167 equations)

This paper contains 35 sections, 42 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminary
  4. Notation
  5. Archimedean Bernstein-Zelevinsky derivative
  6. Archimedean local Langlands correspondence
  7. Non-vanishing on the distinguished minimal $K$-type
  8. Generic calculation of distribution kernel
  9. Meromorphic continuation
  10. Branching laws of $\mathrm{U}(n)$
  11. Description of double cosets $P\backslash G/H$ and Schwartz homology
  12. MVW-involution and higher homology vanishing
  13. Main result
  14. Necessary condition
  15. Reduction to the integral infinitesimal character case
  16. ...and 20 more sections

Key Result

Theorem 1.1

Assume that $\pi$ is an irreducible representation of $\mathrm{GL}_n(\mathbb{C})$, which corresponds to the Langlands parameters $\lambda=\oplus_{j=1}^n\lambda_j$, where $\lambda_j=\varkappa_{m_{j},s_{j}}$ for some $m_j \in \mathbb{Z} \text{ and }s_j \in \mathbb{C}$.

Theorems & Definitions (87)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1
  • Theorem 2.1: AGS15aAGS15b
  • Theorem 2.2: Sahi and Baruch
  • Lemma 2.3
  • proof
  • Theorem 2.4: Langlands classification for $\mathrm{GL}_n(\mathbb{C})$ ZN
  • ...and 77 more