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Shintani lifts for Weil representations of unitary groups over finite fields

Naoki Imai, Takahiro Tsushima

TL;DR

This work constructs extended Weil representations for unitary groups over finite fields in a geometric framework and proves they are Shintani lifts of the classical Weil representations. By realizing the extension on the middle $\ell$-adic cohomology of the variety $X_{m,n}$ and leveraging a Gyoja norm map, the authors establish precise trace identities that connect $\mathrm{U}_n(q)$-data with $\mathrm{U}_n(q^m)$–twisted data via $\Gamma$. The paper develops compatibility under orthogonal and parabolic structures, provides a reduction strategy to a primitive reduced case, and completes the proof in the reduced case using tensor induction and base-case analysis. The approach highlights a deep link between Shintani lifting and geometric methods, offering new insight into lifting unitary Weil representations through algebraic geometry over finite fields.

Abstract

We construct extended Weil representations of unitary groups over finite fields geometrically, and show that they are Shintani lifts for Weil representations.

Shintani lifts for Weil representations of unitary groups over finite fields

TL;DR

This work constructs extended Weil representations for unitary groups over finite fields in a geometric framework and proves they are Shintani lifts of the classical Weil representations. By realizing the extension on the middle -adic cohomology of the variety and leveraging a Gyoja norm map, the authors establish precise trace identities that connect -data with –twisted data via . The paper develops compatibility under orthogonal and parabolic structures, provides a reduction strategy to a primitive reduced case, and completes the proof in the reduced case using tensor induction and base-case analysis. The approach highlights a deep link between Shintani lifting and geometric methods, offering new insight into lifting unitary Weil representations through algebraic geometry over finite fields.

Abstract

We construct extended Weil representations of unitary groups over finite fields geometrically, and show that they are Shintani lifts for Weil representations.

Paper Structure

This paper contains 11 sections, 19 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Norm map
  3. Heisenberg--Weil representation
  4. Geometric construction
  5. Compatibility
  6. Orthogonal decomposition
  7. Parabolic subgroup
  8. Support of trace of extended Weil representation
  9. Main theorem
  10. Reduction steps
  11. Proof in the reduced case

Key Result

Theorem 1

There is a unique extension $\widetilde{\rho}_{\mathop{\mathrm{U}}\nolimits_n(q^m),\psi_m}$ of $\rho_{\mathop{\mathrm{U}}\nolimits_n(q^m),\psi_m}$ to $\mathop{\mathrm{U}}\nolimits_n(q^m) \rtimes \Gamma$ such that for any $g \in \mathop{\mathrm{U}}\nolimits_n(q^m)$.

Theorems & Definitions (37)

  • Theorem
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Lemma 4.1: ITGeomHW
  • Lemma 4.2
  • proof
  • Definition 4.3
  • Remark 4.4
  • Proposition 4.5
  • ...and 27 more