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Semi-Lie arithmetic fundamental lemma for the full spherical Hecke algebra

Evan Chen

TL;DR

This paper advances the arithmetic fundamental lemma (AFL) program by formulating and proving a semi-Lie analogue on the full spherical Hecke algebra, focusing on the $n=2$ case. It develops a detailed basechange framework via the Satake isomorphism to connect inputs on GL$_n(E)$ and the unitary side, and provides explicit formulas for the weighted orbital integrals on the semi-Lie side, including a complete derivation of the derivative at $s=0$ and several corollaries. The work establishes a first nontrivial instance of the semi-Lie AFL for the full spherical Hecke algebra, proving injectivity of the derivative map and analyzing the kernel and image, while also linking the analytic side to the geometric side through Lubin–Tate and Gross–Keating-type computations. The results illuminate the large-image phenomenon in the semi-Lie setting and lay groundwork for broader generalizations to higher $n$, with a clear geometric interpretation via Rapoport–Zink spaces and their intersection theory. Overall, the paper closes a crucial gap in the AFL program by extending the inhomogeneous AFL to the semi-Lie framework and deriving concrete, testable formulas in the simplest nontrivial case, bridging harmonic analysis, $p$-adic geometry, and arithmetic intersection theory.

Abstract

As an analog to the Jacquet-Rallis fundamental lemma that appears in the relative trace formula approach to the Gan-Gross-Prasad conjectures, the arithmetic fundamental lemma was proposed by Wei Zhang and used in an approach to the arithmetic Gan-Gross-Prasad conjectures. The Jacquet-Rallis fundamental lemma was recently generalized by Spencer Leslie to a statement holding for the full spherical Hecke algebra. In the same spirit, Li, Rapoport, and Zhang have recently formulated a conjectural generalization of the arithmetic fundamental lemma to the full spherical Hecke algebra. This paper formulates another analogous conjecture for the semi-Lie version of the arithmetic fundamental lemma proposed by Yifeng Liu. Then this paper produces explicit formulas for particular cases of the weighted orbital integrals in the two conjectures mentioned above, and proves the first non-trivial case of the conjecture.

Semi-Lie arithmetic fundamental lemma for the full spherical Hecke algebra

TL;DR

This paper advances the arithmetic fundamental lemma (AFL) program by formulating and proving a semi-Lie analogue on the full spherical Hecke algebra, focusing on the case. It develops a detailed basechange framework via the Satake isomorphism to connect inputs on GL and the unitary side, and provides explicit formulas for the weighted orbital integrals on the semi-Lie side, including a complete derivation of the derivative at and several corollaries. The work establishes a first nontrivial instance of the semi-Lie AFL for the full spherical Hecke algebra, proving injectivity of the derivative map and analyzing the kernel and image, while also linking the analytic side to the geometric side through Lubin–Tate and Gross–Keating-type computations. The results illuminate the large-image phenomenon in the semi-Lie setting and lay groundwork for broader generalizations to higher , with a clear geometric interpretation via Rapoport–Zink spaces and their intersection theory. Overall, the paper closes a crucial gap in the AFL program by extending the inhomogeneous AFL to the semi-Lie framework and deriving concrete, testable formulas in the simplest nontrivial case, bridging harmonic analysis, -adic geometry, and arithmetic intersection theory.

Abstract

As an analog to the Jacquet-Rallis fundamental lemma that appears in the relative trace formula approach to the Gan-Gross-Prasad conjectures, the arithmetic fundamental lemma was proposed by Wei Zhang and used in an approach to the arithmetic Gan-Gross-Prasad conjectures. The Jacquet-Rallis fundamental lemma was recently generalized by Spencer Leslie to a statement holding for the full spherical Hecke algebra. In the same spirit, Li, Rapoport, and Zhang have recently formulated a conjectural generalization of the arithmetic fundamental lemma to the full spherical Hecke algebra. This paper formulates another analogous conjecture for the semi-Lie version of the arithmetic fundamental lemma proposed by Yifeng Liu. Then this paper produces explicit formulas for particular cases of the weighted orbital integrals in the two conjectures mentioned above, and proves the first non-trivial case of the conjecture.

Paper Structure

This paper contains 115 sections, 42 theorems, 430 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Briefhistoryandmotivationforthearithmeticfundamentallemma
  3. TheGGPconjectures,andthefundamentallemmaofJacquet-Rallis
  4. ThearithmeticGGPconjectures,andthearithmeticfundamentallemma
  5. Thesemi-LieversionoftheAFLproposedbyLiu
  6. GeneralizationsofFLandAFLtothefullsphericalHeckealgebra
  7. FormulationofAFLconjecturestothefullsphericalHeckealgebra
  8. TheinhomogeneousversionofthearithmeticfundamentallemmaforsphericalHeckealgebrasproposedbyLi-Rapoport-Zhang
  9. AproposedarithmeticfundamentallemmaforsphericalHeckealgebrasinthesemi-Liecase
  10. Aproposedlargeimageconjecture
  11. Results
  12. Formulasfortheorbitalsideofthesemi-LieAFLconjecturefor$n=2$
  13. Kernelandimageresultsforthesemi-Lieorbitalintegralwhen$n=2$
  14. Thegeometricsideofthesemi-LieAFLconjecturefor$n=2$
  15. Roadmap
  16. ...and 100 more sections

Key Result

Theorem 1.5

Let satisfytherequirementsinlem:semi_lie_params. Let$r\ge0$. If$v(e)<0$or$v(b)+v(c)<-2r$,then holdsidenticallyforall$s\in\mathbb C$. Otherwisedefine where Also,if$v(d-a)<v(e)-r$and$v(b)+v(c)>2v(d-a)$,thenadditionallydefine Otherwisedefine$\cc_{(\gamma,\mathbf u,\mathbf v^\top)}(k)=0$. Thenwehave

Figures (3)

  • Figure 1: Thehistorybehindthefundamentallemmaanditsarithmeticcounterpart. Unlabeledarrowsdenotegeneralizations.
  • Figure 2: Figurecorrespondingto\ref{['lem:no_mastercard']}.
  • Figure 3: Sketchofthefunctionsin\ref{['thm:semi_lie_formula']}. Theboxednumbersindicatevaluesof$k$.

Theorems & Definitions (118)

  • Conjecture 1.1: InhomogeneousversionoftheAFLforthefullsphericalHeckealgebra, ref:AFLspherical
  • Conjecture 1.2: Semi-LieversionoftheAFLforthefullsphericalHeckealgebra
  • Remark 1.3
  • Conjecture 1.4: Largeimageconjecturefor$(S_n(F)\times V'_n(F))_{\mathrm{rs}}^-$
  • Theorem 1.5: Explicitorbitalintegralon$S_2(F)\times V'_2(F)$
  • Corollary 1.5: Derivativeat$s=0$for$S_2(F)\times V'_2(F)$
  • Corollary 1.5: Thespecialcase$\partial\Orb(\guv,\mathbf{1}_{K'_{S,\le r}}+\mathbf{1}_{K'_{S,\le(r-1)}})$
  • Example 1.6: Examplesof\ref{['cor:semi_lie_combo']}
  • Theorem 1.7: $\partial\Orb$isinjectiveevenforfixed$\gamma\in S_2(F)$
  • Theorem 1.8: Thekernelof$\partial\Orb$islargeforfixed$(\uu,\vv^\top)\in V'_2(F)$
  • ...and 108 more