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The endoscopic fundamental lemma for unitary Friedberg-Jacquet periods

Spencer Leslie

TL;DR

The paper proves the endoscopic fundamental lemma for the Lie algebra of the symmetric variety $U(2n)/U(n)\times U(n)$, a pivotal step in stabilizing the relative trace formula for $U(n)\times U(n)$-periods on $U(2n)$. The authors implement a multi-layer reduction strategy, connecting the Lie-algebra transfer to Jacquet--Rallis transfer, the Weil representation, and a global comparison of twisted Jacquet--Rallis trace formulas, while carefully handling center factors and transfer factors. A key outcome is an explicit endoscopic transfer statement for the Hecke algebra via a morphism $\xi_{(a,b)}$ and a base-change mechanism, together with a local fundamental lemma for the Jacquet--Rallis setting and a global spectral argument to deduce the desired geometric equality. The results stabilize the elliptic part of the relative trace formula and illuminate connections between endoscopy for symmetric varieties, relative trace formulas, and base-change phenomena, with potential applications to unitary period problems and broader relative functoriality.

Abstract

We prove the endoscopic fundamental lemma for the Lie algebra of the symmetric space $U(2n)/U(n)\times U(n)$, where $U(n)$ denotes a unitary group of rank $n$. This is the first major step in the stabilization of the relative trace formula associated to the $U(n)\times U(n)$-periods of automorphic forms on $U(2n)$.

The endoscopic fundamental lemma for unitary Friedberg-Jacquet periods

TL;DR

The paper proves the endoscopic fundamental lemma for the Lie algebra of the symmetric variety , a pivotal step in stabilizing the relative trace formula for -periods on . The authors implement a multi-layer reduction strategy, connecting the Lie-algebra transfer to Jacquet--Rallis transfer, the Weil representation, and a global comparison of twisted Jacquet--Rallis trace formulas, while carefully handling center factors and transfer factors. A key outcome is an explicit endoscopic transfer statement for the Hecke algebra via a morphism and a base-change mechanism, together with a local fundamental lemma for the Jacquet--Rallis setting and a global spectral argument to deduce the desired geometric equality. The results stabilize the elliptic part of the relative trace formula and illuminate connections between endoscopy for symmetric varieties, relative trace formulas, and base-change phenomena, with potential applications to unitary period problems and broader relative functoriality.

Abstract

We prove the endoscopic fundamental lemma for the Lie algebra of the symmetric space , where denotes a unitary group of rank . This is the first major step in the stabilization of the relative trace formula associated to the -periods of automorphic forms on .

Paper Structure

This paper contains 69 sections, 66 theorems, 491 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Global motivation
  3. Local theory of endoscopy and the main result
  4. Outline of the proof
  5. Acknowledgements
  6. Preliminaries
  7. Invariant theory
  8. Local fields
  9. Groups and Hermitian spaces
  10. Measures and centralizers
  11. Endoscopic theory and reduction
  12. The relative endoscopic fundamental lemma
  13. The Lie algebra of the symmetric variety
  14. Endoscopy for the twisted Lie algebra
  15. Matching
  16. ...and 54 more sections

Key Result

Theorem 1.3

Let $\mathop{\mathrm{End}}\nolimits(\Lambda_n)\subset \mathop{\mathrm{End}}\nolimits(V_n)$ be the compact-open subring of endomorphisms of the lattice $\Lambda_n,$ and let $\mathop{\mathrm{End}}\nolimits(\Lambda_a)\oplus\mathop{\mathrm{End}}\nolimits(\Lambda_b)$ be the analogous subring of $\mathop{

Figures (1)

  • Figure 1: Various spaces and the relations between their orbital integrals. While the notations on the two lower rows are the same, the bottom row deals with stable orbital integrals, while the middle row deals with $\kappa$-orbital integrals.

Theorems & Definitions (134)

  • Conjecture 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Lemma 1.8
  • proof
  • Proposition 2.1
  • ...and 124 more