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On Ichino-Ikeda type formula of Whittaker periods for unitary groups

Kazuki Morimoto

Abstract

Lapid and Mao conjectured Ichino-Ikeda type formula of Whittaker periods for any quasi-split reductive groups and metaplectic groups. In this paper, we prove this formula for any irreducible cuspidal globally generic automorphic representation of quasi-split unitary groups.

On Ichino-Ikeda type formula of Whittaker periods for unitary groups

Abstract

Lapid and Mao conjectured Ichino-Ikeda type formula of Whittaker periods for any quasi-split reductive groups and metaplectic groups. In this paper, we prove this formula for any irreducible cuspidal globally generic automorphic representation of quasi-split unitary groups.
Paper Structure (36 sections, 38 theorems, 300 equations)

This paper contains 36 sections, 38 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Even unitary groups
  4. Odd unitary groups
  5. Weil representation and theta lifts
  6. Pull-back computation for Whittaker periods
  7. Theta lift from $\mathrm{U}(2n-1)$ to $\mathrm{U}(2n)$
  8. Theta lifts from $\mathrm{U}(2m)$ to $\mathrm{U}(2n+1)$ with $m \leq n$
  9. Ichino-Ikeda type formula of Whittaker periods and theta correspondence
  10. Rallis inner product formula
  11. Theta lift from $\mathrm{U}(2n-1)$ to $\mathrm{U}(2n)$
  12. Theta lift from $\mathrm{U}(2n)$ to $\mathrm{U}(2n+1)$
  13. Explicit construction of local theta lift
  14. local Whittaker periods
  15. Theta lift from $\mathrm{U}(2n)$ to $\mathrm{U}(2n+1)$
  16. ...and 21 more sections

Key Result

Theorem 1.1

At a real place $v$, the local identity local identity conj holds for $\pi_v$. At a complex place $v$, we have $c_{\pi_v}c_{\pi_{\bar{v}}} = \omega_{\Pi_v}(\eta)\omega_{\Pi_{\bar{v}}}(\eta^\prime)$ for any $\eta, \eta^\prime \in E_v^\times$ such that $\overline{\eta} = -\eta$ and $\overline{\eta^\pr

Theorems & Definitions (65)

  • Conjecture 1: Conjecture 1.2 and Conjecture 5.1 in LMe
  • Remark 1.1
  • Conjecture 2
  • Remark 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.1
  • Theorem 1.3
  • Remark 1.4
  • ...and 55 more