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Archimedean Bernstein-Zelevinsky Theory and Homological Branching Laws

Kaidi Wu, Hongfeng Zhang

TL;DR

The paper advances Archimedean Bernstein-Zelevinsky theory by building a comprehensive filtration framework for real classical groups, including GL$_n$, with a focus on mirabolic restrictions and spectral decompositions. It develops the Casselman-Wallach theory in the Archimedean setting to prove finite-length, Hausdorff, and vanishing properties for derivatives and twisted homology, enabling an Euler-Poincaré characteristic formula for GL and addressing open questions on Ext-vanishing for generic representations. The authors establish a Leibniz law for the highest derivative, derive a unitarity criterion for GL$_n$, and apply the filtration methods to twisted homology in split even orthogonal groups, thereby linking branching laws, Langlands data, and relative harmonic analysis. The framework yields concrete tools for relative Langlands programs (GGP) and the analysis of spectra and nilpotent invariants, with potential implications for Bernstein-Zelevinsky theory beyond the p-adic setting. Overall, the work provides a robust, topologically well-behaved Archimedean analogue of Zelevisnky's filtration, enabling precise control over derivatives, extensions, and unitarity in real groups.

Abstract

We develop the Bernstein-Zelevinsky theory for quasi-split real classical groups and employ this framework to establish an Euler-Poincaré characteristic formula for general linear groups. The key to our approach is establishing the Casselman-Wallach property for the homology of the Jacquet functor, which also provides an affirmative resolution to an open question proposed by A. Aizenbud, D. Gourevitch and S. Sahi. Furthermore, we prove the vanishing of higher extension groups for arbitrary pairs of generic representations, confirming a conjecture of Dipendra Prasad. We also utilize the Bernstein-Zelevinsky theory to establish two additional results: the Leibniz law for the highest derivative and a unitarity criterion for general linear groups. Lastly, we apply the Bernstein-Zelevinsky theory to prove the Hausdorffness and exactness of the twisted homology of split even orthogonal groups.

Archimedean Bernstein-Zelevinsky Theory and Homological Branching Laws

TL;DR

The paper advances Archimedean Bernstein-Zelevinsky theory by building a comprehensive filtration framework for real classical groups, including GL, with a focus on mirabolic restrictions and spectral decompositions. It develops the Casselman-Wallach theory in the Archimedean setting to prove finite-length, Hausdorff, and vanishing properties for derivatives and twisted homology, enabling an Euler-Poincaré characteristic formula for GL and addressing open questions on Ext-vanishing for generic representations. The authors establish a Leibniz law for the highest derivative, derive a unitarity criterion for GL, and apply the filtration methods to twisted homology in split even orthogonal groups, thereby linking branching laws, Langlands data, and relative harmonic analysis. The framework yields concrete tools for relative Langlands programs (GGP) and the analysis of spectra and nilpotent invariants, with potential implications for Bernstein-Zelevinsky theory beyond the p-adic setting. Overall, the work provides a robust, topologically well-behaved Archimedean analogue of Zelevisnky's filtration, enabling precise control over derivatives, extensions, and unitarity in real groups.

Abstract

We develop the Bernstein-Zelevinsky theory for quasi-split real classical groups and employ this framework to establish an Euler-Poincaré characteristic formula for general linear groups. The key to our approach is establishing the Casselman-Wallach property for the homology of the Jacquet functor, which also provides an affirmative resolution to an open question proposed by A. Aizenbud, D. Gourevitch and S. Sahi. Furthermore, we prove the vanishing of higher extension groups for arbitrary pairs of generic representations, confirming a conjecture of Dipendra Prasad. We also utilize the Bernstein-Zelevinsky theory to establish two additional results: the Leibniz law for the highest derivative and a unitarity criterion for general linear groups. Lastly, we apply the Bernstein-Zelevinsky theory to prove the Hausdorffness and exactness of the twisted homology of split even orthogonal groups.

Paper Structure

This paper contains 44 sections, 77 theorems, 415 equations.

Table of Contents

  1. Introduction
  2. Convention and notation
  3. General groups
  4. Real reductive groups
  5. General linear groups
  6. Representation-theoretic conventions
  7. Preliminary
  8. Casselman-Wallach representations
  9. Derivative for quasi-split classical groups
  10. Lie algebra homology
  11. Filtration of a representation
  12. Category $\mathcal{C}\xspace(\mathfrak{g}\xspace,L)$
  13. Schwartz functions on Nash manifolds
  14. Schwartz homology and Euler-Poincaré characteristic
  15. Mirabolic induction and Mackey induction
  16. ...and 29 more sections

Key Result

Theorem 1.1

Let $\pi$ be a Casselman-Wallach representation of $\mathrm{GL}_n(\mathbf{k})$ where $\mathbf{k}=\mathbb {R}\xspace$ or $\mathbb {C}\xspace$. The restriction of $\pi$ to the mirabolic subgroup admits a Bernstein-Zelevinsky filtration.

Theorems & Definitions (162)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1: see Wal92, 11.6.8
  • Corollary 2.2: see AGS15a, Corollary 2.2.5(2)
  • Remark 2.3
  • ...and 152 more