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Geometric construction of Heisenberg-Weil representations for finite unitary groups and Howe correspondences

Naoki Imai, Takahiro Tsushima

Abstract

We give a geometric construction of the Heisenberg-Weil representation of a finite unitary group by the middle étale cohomology of an algebraic variety over a finite field, whose rational points give a unitary Heisenberg group. Using also a Frobenius action, we give a geometric realization of the Howe correspondence for $(\mathit{Sp}_{2n},O_2^-)$ over any finite field including characteristic two. As an application, we show that unipotency is preserved under the Howe correspondence.

Geometric construction of Heisenberg-Weil representations for finite unitary groups and Howe correspondences

Abstract

We give a geometric construction of the Heisenberg-Weil representation of a finite unitary group by the middle étale cohomology of an algebraic variety over a finite field, whose rational points give a unitary Heisenberg group. Using also a Frobenius action, we give a geometric realization of the Howe correspondence for over any finite field including characteristic two. As an application, we show that unipotency is preserved under the Howe correspondence.

Paper Structure

This paper contains 21 sections, 32 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Heisenberg--Weil representation for unitary group
  3. Unitary Heisenberg group
  4. Cohomology of a curve
  5. Geometric construction
  6. Another coordinate
  7. Cohomology of a surface
  8. Construction using another coordinate
  9. Relation with Fermat variety
  10. Unipotent representations of unitary group
  11. Geometric relation
  12. Lusztig induction
  13. Unitary group
  14. Unipotency
  15. Branching formula
  16. ...and 6 more sections

Key Result

Theorem 1

We have an isomorphism as representations of $\mathop{\mathrm{H}}\nolimits(h_n) \rtimes \mathop{\mathrm{U}}\nolimits_n(\mathbb{F}_q)$.

Theorems & Definitions (72)

  • Theorem
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 62 more