On the arithmetic fundamental lemma in the minuscule case (Q2860716)

In this paper, the authors prove the so-called arithmetic fundamental lemma which connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of \(p\)-divisible groups of Picard type in the minuscule case.NEWLINENEWLINELet \(p>2\) be a prime. Let \(F\) be a finite extension of \({\mathbb Q}_p\) and \({\mathcal O}_F\) its ring of integers with residue field \(k\) of \(q\) elements. Let \(E\) be an unramified quadratic extension of \(F\) and let \(\eta\) be the quadratic character of \(F^{\times}\) corresponding to \(E/F\). Let \(\sigma\) be the non-trivial element in \(\mathrm{Gal}(E/F)\). Fix the natural inclusion of \(\mathrm{GL}_{n-1}(E)\) in \(\mathrm{GL}_n(E)\) by considering \(F^{n-1}\) as the subspace of \(F^n\) with the last entry trivial, there is an action of \(\mathrm{GL}_{n-1}(E)\) on \(\mathrm{GL}_n(E)\) by conjugation. Let \(v = \{0,0,\dots,0,1\} \in F^n\). An element \(g \in \mathrm{GL}_n(E)\) is said to be regular semi-simple if both the vectors \((g^iv)_{i=0,\dots,n-1}\) and the vectors \((^{t}v g^i)_{i=0,\dots, n-1}\) are linearly independent. Let \(S_n(F) = \{s \in \mathrm{GL}_n(E) \; | \; s\sigma(s) = 1\}\).NEWLINENEWLINELet \(J\) be a Hermitian form on \(E^{n-1}\) over \(F\) of size \(n-1\), then \(J \oplus 1\) is Hermitian form on \(E^n\) of size \(n\) that corresponds to extending \(J\) to \(E^n\) by adding an orthogonal vector \(u\) of length \(1\). Then the natural inclusion \(U(J)(F) \to U(J \oplus 1)(F)\) defines an action of \(U(J)(F)\) on \(U(J \oplus 1)(F)\) by conjugation. The unitary group \(U(J \oplus 1)(F)\) can be considered as a subset of \(\mathrm{GL}_n(F)\) by sending \(u\) to \(v\), and \(E^{n-1}\) to the subspace of vectors with trivial last entry. Two regular semi-simple elements \(\gamma \in S_n(F)\) and \(g \in U(J \oplus 1)(F)\) are said to be match if they are conjugate under \(\mathrm{GL}_{n-1}(E)\). This matching condition defines a bijection between orbit spaces NEWLINE\[NEWLINE[U(J_0 \oplus 1)(F)_{\mathrm{rs}}] \coprod [U(J_1 \oplus 1)(F)_{\mathrm{rs}}] \simeq [S_n(F)_{\mathrm{rs}}]NEWLINE\]NEWLINE which is a result of a previous paper of the third author in the paper [Invent. Math. 188, No. 1, 197--252 (2012; Zbl 1247.14031)].NEWLINENEWLINEFor \(\gamma \in S_n(F)_{\mathrm{rs}}\), \(f \in C^{\infty}_c(S_n(F))\), and \(s \in {\mathbb C}\), define NEWLINE\[NEWLINEO(\gamma,f,s) := \int_{\mathrm{GL}_{n-1}(F)} f(h^{-1}\gamma h)\eta(\mathrm{det} \, h) | \mathrm{det} \,h|^s dhNEWLINE\]NEWLINE and NEWLINE\[NEWLINEO'(\gamma,1_{S_n({\mathcal O}_F)}) = \frac{d}{ds} O(\gamma, 1_{S_n({\mathcal O}_F)}, s) \big |_{s=0}.NEWLINE\]NEWLINENEWLINENEWLINEThe arithmetic fundamental lemma is the following conjecture: \vskip 0.1in \noindent { Conjecture.} For \(\gamma \in S_n(F)_{\mathrm{rs}}\) that matches \(g \in U(J_1 \oplus 1)(F)_{\mathrm{rs}}\), we have NEWLINE\[NEWLINEO'(\gamma,1_{S_n({\mathcal O}_F)}) = -\omega(\gamma) \langle \Delta ({\mathcal N}_{n-1}), (\mathrm{id} \times g)\Delta ({\mathcal N}_{n-1}) \rangle.NEWLINE\]NEWLINE \vskip 0.1in The main theorem of the paper is: \vskip 0.1in \noindent { Theorem.} For \(n \geq 2\), let \({\mathcal N}_n = {\mathcal N}\) and \({\mathcal N}_{n-1} = {\mathcal M}\). Let \(g \in U(J_1 \oplus 1)(F)_{\mathrm{rs}}\). We have the following result. \vskip 0.1in (1) The underlying reduced scheme of the intersection \(\Delta ({\mathcal M}) \cap (\mathrm{id}_{\mathcal M} \times g) \Delta ({\mathcal M})\) has a stratification by Deligne-Lusztig varieties. \vskip 0.1in Assume that \(\mathrm{inv}(g)\) is minuscule, that is, \(\mathrm{inv}(g) = (1^{(m)}, 0^{(n-m)})\), for some \(m \geq 1\). Then the following hold. \vskip 0.1in (2) The intersection of \(\Delta({\mathcal M})\) and \((\mathrm{id}_{\mathcal M} \times g) \Delta ({\mathcal M})\) is proper. Furthermore, the arithmetic intersection product \(\langle \Delta ({\mathcal M}, (\mathrm{id} \times g)\Delta ({\mathcal M} \rangle\) is equal to NEWLINE\[NEWLINE \log q \sum_{x \in (\Delta ({\mathcal M}) \cap (\mathrm{id} \times g) \Delta (\mathcal M))(\bar{k})} \ell({\mathcal O}_{\Delta ({\mathcal M}) \cap (\mathrm{id} \times g) \Delta (\mathcal M), x}).NEWLINE\]NEWLINE \vskip 0.1in (3) The intersection of \(\Delta({\mathcal M})\) and \((\mathrm{id}_{\mathcal M} \times g)\Delta ({\mathcal M})\) is concentrated in the special fiber. \vskip 0.1in (4) The arithmetic fundamental lemma identity holds, provided that \(n \leq 2p-2\). Furthermore, in this case the lengths of the local rings appearing in the second statement are all identical.