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.
Highlights
The arithmetic fundamental lemma conjecture (AFL) arises from Zhang’s relative trace formula approach toward the arithmetic Gan–Gross–Prasad (AGGP) conjecture for the group U(1, n − 2) × U(1, n − 1), n 2. It relates a derivative of orbital integrals on symmetric spaces to an arithmetic intersection number of cycles on unitary Rapoport–Zink spaces, ω(γ ) · ∂ Orb(γ, 1Sn(OF )) = − Int(g) · log q
We show that each fine Deligne–Lusztig variety
X J,wi is related via parabolic induction to the classical Deligne–Lusztig variety
Summary
In Theorem 5.2.4, we establish an analogous arithmetic intersection formula for GSpin Rapoport–Zink spaces arising from the AGGP conjectures for orthogonal groups. This provides a new proof of the main result of [LZ18], and removes the assumption that p n+1 2 in loc. The starting point of the proof of Theorem 1.2.4 is the observation made in [LZ17, Proposition 4.1.2] that, in the minuscule case, the formal scheme (1.1.2) can be identified with the fixed point scheme V gof an explicitly given smooth projective variety V over k, under a finite-order automorphism g. X J,wi is related via parabolic induction to the classical Deligne–Lusztig variety
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