Fine Deligne-Lusztig varieties and Arithmetic Fundamental Lemmas

Abstract

We prove a character formula for some closed fine Deligne-Lusztig varieties. We apply it to compute fixed points for fine Deligne-Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport-Zink spaces arising from the arithmetic Gan-Gross-Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic.

Final, published version

References in corpus (12)

• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
• Rapoport-Zink spaces for spinor groups
• On the Arithmetic Fundamental Lemma in the minuscule case
• On the arithmetic transfer conjecture for exotic smooth formal moduli spaces
• On the supersingular locus of the GU(2,2) Shimura variety
• Arithmetic diagonal cycles on unitary Shimura varieties
• The $G$-stable pieces of the wonderful compactification
• Arithmetic intersection on GSpin Rapoport-Zink spaces
• $G$-stable pieces and partial flag varieties
• Weil representation and Arithmetic Fundamental Lemma
• Remarks on the arithmetic fundamental lemma
• The basic locus of the unitary Shimura variety with parahoric level structure, and special cycles

Cited by in corpus (6)

• Kudla--Rapoport cycles and derivatives of local densities
• Weil representation and Arithmetic Fundamental Lemma
• On supersingular loci of Shimura varieties for quaternionic unitary groups of degree $2$
• Deligne-Lusztig varieties and basic EKOR strata
• Rapoport--Zink spaces for spinor groups with special maximal parahoric level structure
• Fourier-Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)