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Uniqueness of Bessel models for $\mathrm{GSpin}$ groups

Pan Yan

TL;DR

This work proves a multiplicity-one result for general Bessel models on $\mathrm{GSpin}$ groups over local fields of characteristic zero, by reducing to the spherical case. The authors construct a local zeta integral $\mathcal{Z}_{\mu}$ associated with a Bessel functional and analyze induced representations $\pi_s'$ of $\mathrm{GSpin}(V')$ to realize $\mathcal{Z}_{\mu}$ in a spherical model, showing nonvanishing and absolute convergence in a suitable region. The main contribution is proving $\dim_{\mathbb{C}} \operatorname{Hom}_{H}(\pi \otimes \pi_0, \xi) \le 1$, unifying Whittaker and spherical uniqueness within a single Bessel-family framework and aligning with local Gan–Gross–Prasad expectations for $\mathrm{GSpin}$ groups. This result paves the way for Euler product factorization of global Rankin–Selberg integrals on $\mathrm{GSpin}$ and provides a foundational step toward the local Gan–Gross–Prasad conjecture in this setting, with implications for both nonarchimedean and archimedean fields.

Abstract

We prove the uniqueness of general Bessel models for $\mathrm{GSpin}$ groups over a local field of characteristic zero. The proof is to reduce it to the spherical case, which has been proved by Emory and Takeda in the non-archimedean case and by Emory, Kim, and Maiti in the archimedean case.

Uniqueness of Bessel models for $\mathrm{GSpin}$ groups

TL;DR

This work proves a multiplicity-one result for general Bessel models on groups over local fields of characteristic zero, by reducing to the spherical case. The authors construct a local zeta integral associated with a Bessel functional and analyze induced representations of to realize in a spherical model, showing nonvanishing and absolute convergence in a suitable region. The main contribution is proving , unifying Whittaker and spherical uniqueness within a single Bessel-family framework and aligning with local Gan–Gross–Prasad expectations for groups. This result paves the way for Euler product factorization of global Rankin–Selberg integrals on and provides a foundational step toward the local Gan–Gross–Prasad conjecture in this setting, with implications for both nonarchimedean and archimedean fields.

Abstract

We prove the uniqueness of general Bessel models for groups over a local field of characteristic zero. The proof is to reduce it to the spherical case, which has been proved by Emory and Takeda in the non-archimedean case and by Emory, Kim, and Maiti in the archimedean case.

Paper Structure

This paper contains 11 sections, 9 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. The $\mathop{\mathrm{GSpin}}\nolimits$ group
  5. The Bessel subgroup $H$
  6. The group $\mathop{\mathrm{GSpin}}\nolimits(V^\prime)$
  7. Induced representations of $\mathop{\mathrm{GSpin}}\nolimits(V^\prime)$
  8. Proof of Theorem \ref{['thm-main']}
  9. A local integral
  10. Proof of Proposition \ref{['prop-integral-converge-1']}
  11. Proof of Theorem \ref{['thm-main']}

Key Result

Theorem 1.1

Let $V, V_0, H, G, \xi$ be as above. For any irreducible admissible representation $\pi$ of $\mathop{\mathrm{GSpin}}\nolimits(V)$ and $\pi_0$ of $\mathop{\mathrm{GSpin}}\nolimits(V_0)$, which are of Casselman-Wallach type if $F$ is archimedean, the following inequality holds:

Theorems & Definitions (17)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 7 more