Wei Zhang's Arithmetic Fundamental Lemma for unitary group elements

Let S(E0)S(E_0) be the relevant symmetric space, let U(J1)U(J_1) be the unitary group attached to the odd hermitian form, and let Ω(γ)Ω(γ), Oγ(1S(OE0))\partial O_γ(1_{S(\mathcal{O}_{E_0})}), and Int(g)\mathrm{Int}(g) denote the transfer factor, the derivative of the orbital integral, and the intersection number. Let γS(E0)rsγ\in S(E_0)_{\mathrm{rs}} be regular semi-simple and matching an element gU(J1)g\in U(J_1). Assume either E0=QpE_0=\mathbb{Q}_p or that gg is artinian. Arithmetic Fundamental Lemma. Then

Ω(γ)Oγ(1S(OE0))=Int(g)log(q).Ω(γ)\partial O_γ(1_{S(\mathcal{O}_{E_0})})=-\mathrm{Int}(g)\log(q).

This is the group version of the Arithmetic Fundamental Lemma attributed to Wei Zhang. The surrounding discussion explains that the orbital integral itself vanishes on the nonsplit unitary side; no general resolution is stated in the source.

Sources & referencesView supporting material

Primary source

Andreas Mihatsch, “Relative unitary RZ-spaces and the Arithmetic Fundamental Lemma”, arXiv:1611.06520 (2019).

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