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On the existence of twisted Shalika periods: the Archimedean case

Zhibin Geng

TL;DR

This work establishes a precise equivalence between the existence of twisted Shalika functionals on irreducible Casselman-Wallach representations of GL_{2n}(K) over archimedean fields and the eta-symplectic type of their L-parameters. Employing a robust Schwartz-homology framework, the authors develop a Hochschild-Serre spectral sequence for nilpotent normals and a Kunneth formula to control homology across products, then analyze S-orbits on a partial flag variety, distinguishing matching from unmatching orbits to isolate the contributing pieces. Key vanishing and finiteness results are proved for orbitwise Schwartz homology, enabling a reduction to low-rank cases (GL_4(R)) and establishing the main A-theorem: nonzero twisted Shalika periods occur exactly when the L-parameter is eta-symplectic (for generic representations). The B-theorem extends these results to parabolic inductions and connects twisted Shalika periods with twisted linear periods via archimedean theta correspondence, while the final analysis of GL^+ restrictions provides explicit epsilon-parameter calculations and clarifies the behavior on the identity component GL^{+}_{2n}(R). The combination of orbit methods, Schwartz homology, and theta correspondence yields a complete archimedean picture linking periods to L-parameters with sharp conditions and constructive proofs.

Abstract

Let $\K$ be an archimedean local field. We investigate the existence of the twisted Shalika functionals on irreducible admissible smooth representations of $\GL_{2n}(\K)$ in terms of their L-parameters. As part of our proof, we establish a Hochschild-Serre spectral sequence for nilpotent normal subgroups and a Kunneth formula in the framework of Schwartz homology. We also prove the analogous result for twisted linear periods using theta correspondence. The existence of twisted Shalika functionals on representations of $\GL_{2n}^{+}(\R)$ is also studied, which is of independent interest.

On the existence of twisted Shalika periods: the Archimedean case

TL;DR

This work establishes a precise equivalence between the existence of twisted Shalika functionals on irreducible Casselman-Wallach representations of GL_{2n}(K) over archimedean fields and the eta-symplectic type of their L-parameters. Employing a robust Schwartz-homology framework, the authors develop a Hochschild-Serre spectral sequence for nilpotent normals and a Kunneth formula to control homology across products, then analyze S-orbits on a partial flag variety, distinguishing matching from unmatching orbits to isolate the contributing pieces. Key vanishing and finiteness results are proved for orbitwise Schwartz homology, enabling a reduction to low-rank cases (GL_4(R)) and establishing the main A-theorem: nonzero twisted Shalika periods occur exactly when the L-parameter is eta-symplectic (for generic representations). The B-theorem extends these results to parabolic inductions and connects twisted Shalika periods with twisted linear periods via archimedean theta correspondence, while the final analysis of GL^+ restrictions provides explicit epsilon-parameter calculations and clarifies the behavior on the identity component GL^{+}_{2n}(R). The combination of orbit methods, Schwartz homology, and theta correspondence yields a complete archimedean picture linking periods to L-parameters with sharp conditions and constructive proofs.

Abstract

Let be an archimedean local field. We investigate the existence of the twisted Shalika functionals on irreducible admissible smooth representations of in terms of their L-parameters. As part of our proof, we establish a Hochschild-Serre spectral sequence for nilpotent normal subgroups and a Kunneth formula in the framework of Schwartz homology. We also prove the analogous result for twisted linear periods using theta correspondence. The existence of twisted Shalika functionals on representations of is also studied, which is of independent interest.
Paper Structure (23 sections, 60 theorems, 208 equations)

This paper contains 23 sections, 60 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the proof
  4. Structure of the paper
  5. Conventions
  6. Preliminaries and notation
  7. Local Langlands correspondence
  8. Nuclear Fréchet spaces
  9. Schwartz homology
  10. Borel's lemma
  11. Schwartz homology
  12. Some vanishing criterion
  13. Orbit decomposition
  14. Homology of standard modules
  15. Homological finiteness
  16. ...and 8 more sections

Key Result

Theorem 1.2

Let $\pi$ be an irreducible Casselman-Wallach representation of ${\mathrm{GL}}_{2n}({\mathbb {K}})$.

Theorems & Definitions (112)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark
  • Theorem 1.4: Theorem \ref{['Thm: archimedean theta shalika linear']}
  • Corollary 1.5
  • Theorem 1.6: Theorem \ref{['Thm: restrict Shalika support']}
  • Lemma 2.1
  • proof
  • Lemma 2.2: Vogan1978gelfand, fang2018godement
  • ...and 102 more