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Fundamental Lemma for Rank One Spherical Varieties of Classical Types

Zhaolin Li

TL;DR

This paper proves a fundamental lemma for rank-one spherical varieties of classical types within the relative Langlands program. It constructs and analyzes non-standard test measures, transfer operators built from Weil representations, and explicit basic vectors encoding local L-functions, then matches the geometric side of relative trace formulas to the Kuznetsov side across types $A_{n-1}$, $D_n$, $B_n$, and $C_n$. The results yield explicit local transfer factors and Tamagawa-normalized constants that connect the $L$-value data ${L}( ho,s)$ to basic vectors $f_{L_X}$, enabling the desired functorial transfer on the spectral side. Overall, the work advances the rank-one transfer program by providing uniform fundamental-lemma statements with concrete constants across the classical families, reinforcing the relative Langlands picture for spherical varieties. The global implications include paving the way for a full relative trace formula comparison and deeper understanding of automorphic spectra under $L$-group morphisms.

Abstract

According to the relative Langlands functoriality conjecture, an admissible morphism between the $L$-groups of spherical varieties should induce a functorial transfer of the corresponding local and global automorphic spectra. Via the relative trace formula approach, two basic problems are the fundamental lemma and the local transfer on the geometric side of the relative trace formulas. In this paper, we consider the rank-one spherical variety case, where the admissible morphism between the $L$-groups is the identity morphism, in which case, Y. Sakellaridis has already established the local transfer (\cite{Sak21}). We formulate the statement of the fundamental lemma for the general rank-one spherical variety case and prove the fundamental lemma for the rank-one spherical varieties of classical types.

Fundamental Lemma for Rank One Spherical Varieties of Classical Types

TL;DR

This paper proves a fundamental lemma for rank-one spherical varieties of classical types within the relative Langlands program. It constructs and analyzes non-standard test measures, transfer operators built from Weil representations, and explicit basic vectors encoding local L-functions, then matches the geometric side of relative trace formulas to the Kuznetsov side across types , , , and . The results yield explicit local transfer factors and Tamagawa-normalized constants that connect the -value data to basic vectors , enabling the desired functorial transfer on the spectral side. Overall, the work advances the rank-one transfer program by providing uniform fundamental-lemma statements with concrete constants across the classical families, reinforcing the relative Langlands picture for spherical varieties. The global implications include paving the way for a full relative trace formula comparison and deeper understanding of automorphic spectra under -group morphisms.

Abstract

According to the relative Langlands functoriality conjecture, an admissible morphism between the -groups of spherical varieties should induce a functorial transfer of the corresponding local and global automorphic spectra. Via the relative trace formula approach, two basic problems are the fundamental lemma and the local transfer on the geometric side of the relative trace formulas. In this paper, we consider the rank-one spherical variety case, where the admissible morphism between the -groups is the identity morphism, in which case, Y. Sakellaridis has already established the local transfer (\cite{Sak21}). We formulate the statement of the fundamental lemma for the general rank-one spherical variety case and prove the fundamental lemma for the rank-one spherical varieties of classical types.

Paper Structure

This paper contains 16 sections, 36 theorems, 313 equations.

Table of Contents

  1. Introduction
  2. Relative trace formulas and fundamental lemmas
  3. The organization of the paper
  4. Notations
  5. Acknowledgement
  6. Preliminaries
  7. Test measures for the Kuznetsov formula
  8. Basic functions and non-standard test measures
  9. Transfer operators for spherical varieties of rank $1$
  10. $\epsilon$-Hermitian modules and unitary groups
  11. Weil representations
  12. Transfer operators from Weil representations
  13. Fundamental lemma of type $A_{n-1}$
  14. Fundamental lemma of type $D_n$
  15. Fundamental lemma of type $B_{n}$
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let ${\mathrm {X}}$ be a rank one spherical variety of classical type, ${\mathfrak {C}}_{{\mathrm {X}}}=({\mathrm {X}}\times{\mathrm {X}})\mathbin{/\mkern-6mu/}{\mathrm {G}}$. Fix the isomorphism ${\mathfrak {C}}_{{\mathrm {X}}}\cong{\mathbb {A}}^1$ such that ${\mathrm {X}}\times{\mathrm {X}}\righta Under the transfer operator in Sak21, we have where

Theorems & Definitions (76)

  • Theorem 1.1
  • Conjecture 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3: Theorem 1.3.1 in Sak21
  • Lemma 2.4
  • Definition 2.5
  • ...and 66 more