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.Here the product is over pairs {Q, Q * } of monic irreducible non-self-reciprocal polynomials in F q 2 [λ] with non-zero constant terms. Theorem 1.2.1 then follows immediately from Theorem 1.2.3 and the explicit formula for the analytic side given in [RTZ13, Proposition 8.2]. Remark 1.2.4. Theorem 1.2.3 is also used to prove the minuscule case of Liu's arithmetic fundamental lemma for Fourier-Jacobi cycles, see [Liu18, Appendix E].Remark 1.2.5. In Theorem 5.2.4 we also 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 also removes the assumption that p n+1 2 in loc. cit. 1.3. Computing the arithmetic intersection. The starting point of the proof of Theorem 1.2.3 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 Vḡ of an explicitly given smooth projective variety V over k, under a finite-order automorphismḡ. It also turns out that Vḡ is an Artinian scheme. Hence Int(g) is given by the k-length of Vḡ.In order to compute the k-length of Vḡ, there are two apparent approaches. One approach, taken in [LZ17], is to explicitly study all the local equations. The other approach, which we take in the current paper, is to compute it using the Lefschetz trace formula. Thus we obtainProof. This follows immediately from Proposition 2.5.1.2.6. Review of regular elements. We recall the definition of regular elements and some standard facts. Let G be a reductive group over k.Definition 2.6.1. An element g ∈ G is called regular, if the centralizer G g of g in G has dimension equal to the rank of G. The set of regular elements is denoted by G reg .If G is semi-simple, the above definition is the same as [Ste65]. In general, one easily checks that g ∈ G is regular in the above sense if and only if the image of g in G ad is regular. Thus we can easily transport the results from [Ste65], which only discusses semi-simple groups, to reductive groups.Theorem 2.6.2. An element g ∈ G is regular if and only if there are only finitely many Borel subgroups of G that contain g.Proof. This follows from [Ste65, Theorem 1.1] applied to G ad .Proposition 2.6.3. Assume G ′ is a reductive group over k that contains G as a closed subgroup. ThenProof. Fix a Borel subgroup B ′ ⊂ G ′ that contains B. By Theorem 2.6.2, it suffices to show that the natural map between flag varie...

Rapoport--Zink spaces for spinor groups with special maximal parahoric level structure

“…Their results are expected to be useful for a similar consideration on Shimura varieties. In this paper, we also prove a variant of the main result of [HLZ19]. See Section 1.3 for more details.…”

“…Moreover, they computed the intersection multiplicities of the GGP cycles in a special case. This calculation was improved by He, Li and Zhu ( [HLZ19]), which successes in generalizing the result of [LZhu18]. Their results are expected to be useful for a similar consideration on Shimura varieties.…”

“…Note that the researches [LZhu18] and [HLZ19] are inspired by the same study as the case of unitary similitude groups GU(1, n − 1), in which the given imaginary quadratic field is inert at p > 2. In [Zha12], Zhang conjectured a presentation of intersection multiplicities of the GGP cycles on the basic Rapoport-Zink spaces with hyperspecial level by means of the first derivatives of orbital integrals.…”

Relative Unitary Rz-Spaces and the Arithmetic Fundamental Lemma

“…Note that in these cases, g is also artinian. Their proof was subsequently simplified by Li and Zhu [12,13] and He, Li and Zhu [8].…”

“…The AFL is proven in the case of dimension n 3; see [23]. In the subsequent work [18], Rapoport, Terstiege and Zhang verify the AFL for arbitrary n and the so-called minuscule group elements g. Their proof was later simplified by Li and Zhu [12,13] and He, Li and Zhu [8].…”