Arithmetic intersection on GSpin Rapoport-Zink spaces
Chao Li Yihang Zhu


Impurity scattering and size quantization effects in a single graphene nanoflake
    Mikhail Fonin^1[Email address: mikhail.fonin@uni-konstanz.de]
    December 30, 2023
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  We prove an explicit formula for the arithmetic intersection number of diagonal cycles on GSpin Rapoport–Zink spaces in the minuscule case. This is a local problem arising from the arithmetic Gan–Gross–Prasad conjecture for orthogonal Shimura varieties. Our formula can be viewed as an orthogonal counterpart of the arithmetic-geometric side of the arithmetic fundamental lemma proved by Rapoport–Terstiege–Zhang in the minuscule case.










§ INTRODUCTION




 §.§ Motivation


The arithmetic Gan–Gross–Prasad conjectures (arithmetic GGP) generalize the celebrated Gross–Zagier formula to higher dimensional Shimura varieties (<cit.>, <cit.>). It is a conjectural identity relating the heights of certain algebraic cycles on Shimura varieties to the central derivative of certain Rankin–Selberg L-functions. Let us briefly recall the rough statement of the conjecture. The diagonal embeddings of unitary groups 
    H=U (1,n-1)↪ G=U(1, n-1)×U (1, n)
 or of orthogonal groups 
    H=(2, n-1)↪ G=(2,n-1)×(2,n),
 induces an embedding of Shimura varieties _H↪_G. We denote its image by Δ and call it the diagonal cycle or the GGP cycle on _G. Let π be a tempered cuspidal automorphic representation on G appearing in the middle cohomology of _G. Let  Δ_π be the (cohomological trivialization) of the π-component of Δ. The arithmetic GGP conjecture asserts that the (conditional) Beilinson–Bloch–Gillet–Soulé height of Δ_π   should be given by the central derivative of a certain Rankin-Selberg L-function L(s, π) up to simpler factors 
    ⟨Δ_π,Δ_π⟩∼ L'(1/2, π).


The Gross–Zagier formula <cit.> and the work of Gross, Kudla, Schoen (<cit.>, <cit.>) can be viewed as the special cases n=1 and n=2 in the orthogonal case correspondingly. The recent work of Yuan–Zhang–Zhang (<cit.>, <cit.>)  has proved this conjecture for n=1,2 in the orthogonal case in vast generality.

In the unitary case, W. Zhang has proposed an approach for general n using the relative trace formula of Jacquet–Rallis. The relevant arithmetic fundamental lemma relates an arithmetic intersection number of GGP cycles on unitary Rapoport–Zink spaces with a derivative of orbital integrals on general linear groups. The arithmetic fundamental lemma has been verified for n=1,2 (<cit.>) and for general n in the minuscule case by Rapoport–Terstiege–Zhang <cit.>.

In the orthogonal case, very little is known currently beyond n=1,2 and no relative trace formula approach has been proposed yet. However it is notable that R. Krishna <cit.> has recently established a relative trace formula for the case (2)×(3) and one can hope that his method will generalize to formulate a relative trace formula approach for general (n-1)×(n).

Our goal in this article is to establish an orthogonal counterpart of the arithmetic-geometric side of the arithmetic fundamental lemma in <cit.>, namely to formulate and compute the arithmetic intersection of GGP cycles on  Rapoport–Zink spaces in the minuscule case.



 §.§ The main results
 Let p be an odd prime. Let k=𝔽_p, W=W(k), K=W[1/p] and σ∈(W) be the lift of the absolute p-Frobenius on k. Let n≥4. Let V^♭ be a self-dual quadratic space over _p of rank n-1 and let V=V^♭⊕ℤ_p x_n (orthogonal direct sum) be a self-dual quadratic space over _p of rank n, where x_n has norm 1. Associated to the embedding of quadratic spaces V^♭↪ V we have an embedding of algebraic groups G^♭=(V^♭)↪ G=(V) over ℤ_p. After suitably choosing compatible local unramified Shimura–Hodge data (G^♭, b^♭, μ^♭, C(V^♭))↪ (G, b, μ, C(V)), we obtain a closed immersion of the associated  Rapoport–Zink spaces 
    δ: ^♭↪.
  See <ref> for precise definitions and see <ref> for the moduli interpretation of δ.   The space  is an example of Rapoport–Zink spaces of Hodge type, recently constructed by Kim <cit.> and Howard–Pappas <cit.>. It is a formal scheme over W, parameterizing deformations of a p-divisible group 𝕏_0/k with certain crystalline Tate tensors (coming from the defining tensors of G inside some _N). Roughly speaking, if X^♭ is the p-divisible group underlying a point x∈^♭, then the p-divisible group underlying δ(x) ∈ is given by X= X^♭⊕ X^♭.


 The datum (G,b,μ, C(V)) is chosen such that the space  provides a p-adic uniformization of (𝒮_W)_/𝒮_ss, the formal completion of 𝒮_W along 𝒮_ss, where  𝒮_W is the base change to W of Kisin's integral model (<cit.>) of a  Shimura variety (which is of Hodge type) at a good prime p, and 𝒮_ss is the supersingular locus (= the basic locus in this case) 
  of the special fiber of _W (see <cit.>).




The group J_b(ℚ_p)={g∈ G(K): gb=bσ(g)} is the ℚ_p-points of an inner form of G and acts on  via its action on the fixed p-divisible group 𝕏_0. Let g∈ J_b(ℚ_p). As explained in <ref>, the intersection of the GGP cycle Δ on ^♭×_W and its g-translate leads to study of the formal scheme

    δ(^♭)∩^g,

 where ^g denotes the g-fixed points of .

 We call g∈ J_b(ℚ_p) regular semisimple if 
    L(g):=ℤ_px_n+ℤ_p gx_n+ ⋯+ℤ_p g^n-1x_n
 is a free ℤ_p-module of rank n. Let L(g)^∨ denote the dual lattice of L(g). We further call g minuscule if L(g) ⊂ L(g)^∨ (i.e. the quadratic form restricted to L(g) is valued in _p), and L(g)^∨/L(g) is a 𝔽_p-vector space. See Definition <ref> for equivalent definitions. When g∈ J_b(ℚ_p) is regular semisimple and minuscule, we will show that the formal scheme (<ref>)  is in fact a 0-dimensional scheme of characteristic p. Our main theorem is an explicit formula for its arithmetic intersection number  (i.e., the total W-length of its local rings).

To state the formula, assume g is regular semisimple and minuscule, and suppose ^g is nonempty. Then g stabilizes both L(g)^∨ and L(g) and thus acts on the 𝔽_p-vector space L(g)^∨/L(g). Let P(T)∈𝔽_p[T] be the characteristic polynomial of g acting on L(g)^∨/L(g). For any irreducible polynomial R(T)∈𝔽_p[T], we denote its multiplicity in P(T) by m(R(T)). Moreover, for any polynomial R(T), we define its reciprocal by 
    R^*(T):=T^ R(T)· R(1/T).
 We say R(T) is self-reciprocal if R(T)=R^*(T). Now we are ready to state our main theorem:


  Let g∈ J_b(ℚ_p) be regular semisimple and minuscule. Assume ^g is non-empty. Then
  
  
  * (Corollary <ref>) δ(^♭)∩^g is a scheme of characteristic p.
  
  * (Theorem <ref>) δ(^♭)∩^g is non-empty if and only if P(T) has a unique self-reciprocal monic irreducible factor Q(T)|P(T) such that m(Q(T)) is odd. In this case, p^ℤ\(δ(^♭)∩^g)(k) is finite and has cardinality 
    Q(T)·∏_R(T)(1+m(R(T))),
 where R(T) runs over all non-self-reciprocal monic irreducible factors of P(T). Here, the group p^ acts on  via the central embedding p^↪ J_b(_p), and the action stabilizes δ(^♭)∩^g.
  
  * (Corollary <ref>) Let c = m(Q(T))+1/2. Then 1≤ c ≤ n/2. Assume p>c. Then δ(^♭)∩^g is a disjoint union over its k-points of copies of k[X]/X^c. In particular, the intersection multiplicity at each k-point of δ(^♭)∩^g is the same and equals c. 
  






Along the way we also prove a result that should be of independent interest. In <cit.>, Howard–Pappas define closed formal subschemes _Λ of  for each vertex lattice Λ (recalled in <ref>). 
Howard–Pappas study the reduced subscheme _Λ^red detailedly and prove that they form a nice stratification of ^red. We prove:
[Theorem <ref>]
	_Λ = _Λ^red for each vertex lattice Λ. 
      



 §.§ Novelty of the method

The results Theorem A and Theorem B are parallel to the results in <cit.> for unitary Rapoport–Zink spaces. The main new difficulty in the GSpin case is due to the fact that, unlike the unitary case, the GSpin Rapoport–Zink spaces are not of PEL type. They are only of Hodge type, and as for now they lack full moduli interpretations that are easy to work with directly (see Remark <ref>).

In <cit.>, the most difficult parts are the reducedness of minuscule special cycles <cit.> and the intersection length formula <cit.>. They are the analogues of Theorem B and Theorem A (3) respectively. In <cit.>, they are proved using Zink's theory of windows and displays of p-divisible groups and involve rather delicate linear algebra computation. In contrast, in our method we rarely directly work with p-divisible groups and we completely avoid computations with windows or displays. Instead we make use of what are essentially consequences of Kisin's construction of integral models of Hodge type Shimura varieties to abstractly reduce the problem to algebraic geometry over k. More specifically, we reduce the intersection length computation to the study of a certain scheme of the form S_Λ^g̅ (Proposition <ref>), where S_Λ is a smooth projective k-variety closely related to orthogonal Grassmannians, and g̅ is a certain finite order automorphism of S.  Thus our method overcomes the difficulty of non-PEL type and also makes the actual computation much more elementary. 

It is worth mentioning that our method also applies to the unitary case considered in <cit.>. Even in this PEL type case, our method gives a new and arguably simpler proof of the arithmetic fundamental lemma in the minuscule case. This will be pursued in a forthcoming work.

It is also worth mentioning that the very recent work of Bueltel–Pappas <cit.> gives a new moduli interpretation for  Rapoport–Zink spaces of Hodge type when restricted to p-nilpotent noetherian algebras. Their moduli description is purely group-theoretic (in terms of (G,μ)-displays) and does not involve p-divisible groups. Although we do not use (G,μ)-displays in this article, it would be interesting to see if it is possible to extend the results of this article using their group-theoretic description (e.g., to non-minuscule cases).



 §.§ Strategy of the proofs


Our key observation is that in order to prove these theorems, we only need to understand 𝒪-points of  for very special choices of W-algebras 𝒪.

To prove Theorem B, it turns out that we only need to understand (W/p^2) and (k[ϵ]/ϵ^2). Note that the W-algebras W/p^2 and k[ϵ]/ϵ^2, when viewed as thickenings of k (under reduction modulo p or ϵ respectively), are objects of the crystalline site of k. For such an object , we prove in Theorem <ref> an explicit description of () and more generally an explicit description of 𝒵(), for any special cycle 𝒵 in . Theorem <ref> is the main tool to prove Theorem B, and is also the only place we use p-divisible groups. This result is a Rapoport–Zink space analogue of a result of Madapusi Pera <cit.> for GSpin Shimura varieties. Its proof also relies on loc. cit. and is ultimately  a consequence of Kisin's construction of the integral canonical models of Hodge type Shimura varieties <cit.>. 

To prove the intersection length formula Theorem A (3), let Λ be the vertex lattice L(g)^∨. Theorem B allows us to reduce Theorem A (3) to the problem of studying the fixed-point subscheme of  the smooth k-variety S_Λ≅ p^ℤ\_Λ^red, under the induced action g̅∈(Λ/Λ^∨) of g. Since the fixed point of a smooth k-variety under a group of order coprime to p is still smooth (<cit.>), this point of view immediately explains that when g̅ is semisimple (in which case m(Q(T))=1), the intersection multiplicity must be 1. More generally, we utilize Howard–Pappas's description of S_Λ in <cit.> and reduce the intersection length computation to elementary algebraic geometry of orthogonal Grassmannians over k (Proposition <ref> and Theorem <ref>).

The remaining parts of Theorem A are relatively easier. From Theorem B it is not difficult to deduce Theorem A (1). The set of k-points of  is well understood group theoretically in terms of the affine Deligne–Lusztig set. The point counting formula Theorem A (2) essentially only relies on this description, and we follow the strategy in <cit.> to give a short streamlined proof (Proposition <ref>).











 §.§ Organization of the paper
 In <ref>, we review the structure of GSpin Rapoport–Zink spaces and special cycles. In <ref>, we formulate the arithmetic intersection problem of GGP cycles and prove the point-counting formula for the k-points of the intersection in the minuscule case (Theorem A (2)). In <ref>, we prove reducedness of minuscule special cycles (Theorem B). In <ref>, we deduce from Theorem B that the arithmetic intersection is concentrated in the special fiber (Theorem A (1)) and finally compute the intersection length when p is sufficiently large (Theorem A (3)).



 §.§ Acknowledgments
  We are very grateful to B. Howard, M. Kisin, M. Rapoport and W. Zhang for helpful conversations or comments. Our debt to the two papers <cit.> and <cit.> should be clear to the readers. 



§ GSPIN RAPOPORT–ZINK SPACES


In this section we review the structure of GSpin Rapoport–Zink spaces due to Howard–Pappas <cit.>. We refer to <cit.> for the proofs of these facts.



 §.§ Quadratic spaces and GSpin groups












Let p be an odd prime. Let (V,q) be a non-degenerate self-dual quadratic space over ℤ_p of rank n≥ 3. By definition the Clifford algebra C(V) is the quotient of the tensor algebra V^⊗ by the two sided ideal generated by elements of the form v ⊗ v-q(v). It is free of rank 2^n over ℤ_p. The linear map v↦ -v preserves the quadratic form q on V and induces an involution on C(V). This involution decomposes C(V)=C^+(V) ⊕ C^-(V) into even and odd parts. The image of the injection V↪ C^-(V) generates C(V) as a _p-algebra.

We also have a canonical involution *: C(V)→ C(V), which a ℤ_p-linear endomorphism characterized by (v_1v_2⋯ v_k)^*=v_k⋯ v_2v_1 for v_i∈ V. The spinor similitude group G=(V) is the reductive group over ℤ_p such that for a ℤ_p-algebra R, 
    G(R)={g∈ C^+(V)^×: g V_R g^-1=V_R,    g^*g∈ R^×}.
 The character η_G: G→𝔾_m given by g↦ g^*g is the called spinor similitude.

The conjugation action  g.v=gvg^-1 of G on C(V)  stabilizes V and preserves the quadratic form q. This action thus defines a homomorphism 
    G→(V).
 The kernel of the above morphism is the central 𝔾_m inside G given by the natural inclusion R^×⊂ G(R) for any _p-algebra R. The restriction of η_G on the central 𝔾_m is given by g↦ g^2.
Note that the central _m in G is equal to the identity component of the center of G, and it is equal to the center of G precisely when n is odd.



 §.§ Basic elements in GSpin groups


Let k=𝔽_p, W=W(k) and K=W[1/p]. Let σ∈(W) be the lift of the absolute p-Frobenius on k.  Let D=_ℤ_p(C(V), ℤ_p) be the contragredient G-representation of C(V).

Any b∈ G(K) determines two isocrystals 
    (V_K, Φ=b∘σ),   (D_K, F=b∘σ).
 Denote by 𝕋 the pro-torus over _p of character group . Recall that b∈ G(K) is basic if its slope morphism ν_b: 𝕋_K→ G_K factors through (the identity component) of Z(G_K), i.e., factors through the central 𝔾_m. By <cit.>, b is basic if and only if (V_K,Φ) is isoclinic of slope 0, if and only if (D_K, F) is isoclinic of slope -ν_b∈(𝕋_K,𝔾_m)≅ℚ. The map b↦ν_b gives a bijection between the set of basic σ-conjugacy classes and the set 1/2ℤ. 
Moreover, the ℚ_p-quadratic space 
    V_K^Φ={x∈ V_K: Φ x=x}
 has the same dimension and determinant as V_ℚ_p, and has Hasse invariant (-1)^2ν_b (<cit.>)).



 §.§ Local unramified Shimura–Hodge data


 Since V is self-dual, we know that V_ℚ_p has Hasse invariant +1. 
 In particular V contains at least one hyperbolic plane 
 and  we can pick a ℤ_p-basis x_1,…,x_n of V such that the Gram matrix of the quadratic form q is of the form 
    [ 0 1    ; 1 0      ;     *      ;       *    ;         ⋱  ;           * ]
  We will fix x_1,…, x_n once and for all. Define a cocharacter 
    μ: 𝔾_m→ G,    t↦ t^-1x_1x_2+x_2x_1.
  Pick an explicit element b=x_3(p^-1x_1+x_2)∈ G(ℚ_p), then one can show that b is basic with ν_b=1/2. Thus V_K^Φ has the opposite Hasse invariant -1 (cf. <ref>).

 Fix any δ∈ C(V)^× such that δ^*=-δ. Then ψ_δ(c_1,c_2)=Trd(c_1δ c_2^*) defines a non-degenerate symplectic form on C(V), where Trd: C(V)→ℤ_p is the reduced trace. We have a closed immersion into the symplectic similitude group 
    G↪(C(V), ψ_δ).
 By <cit.>, the tuple (G, b, μ, C(V)) defines a local unramified Shimura–Hodge datum (in the sense of <cit.>). In fact, for the fixed G and μ,  the σ-conjugacy class of b is the unique basic σ-conjugacy class for which  (G, b, μ) is a local unramified Shimura–Hodge datum (cf. <cit.>).

 
   The tuple (G, b, μ, C(V)) is chosen in such a way that the associated Rapoport–Zink space (see below) provides a p-adic uniformization for the supersingular locus of a related  Shimura variety. For more details on the relation with Shimura varieties see <cit.>. 
 
 


 §.§ GSpin Rapoport–Zink spaces

There is a unique (up to isomorphism) p-divisible group 𝕏_0/k such that its (contravariant) Dieudonné module 𝔻(𝕏_0) is given by the W-lattice D_W in the isocrystal D_K. The non-degenerate symplectic form ψ_δ induces a principal polarization λ_0 of 𝕏_0.  Fix a collection of tensors (s_α) on C(V) cutting out G from (C(V)) (including the symplectic form ψ_δ). By <cit.>, we have a GSpin Rapoport–Zink space 
    :=(G, b,μ, C(V), (s_α)).
 It is a formal scheme over W, together with a closed immersion into the symplectic Rapoport–Zink space (𝕏_0,λ_0). Moreover, the formal scheme  itself depends only on the local unramified Shimura–Hodge datum (G,b,μ, C(V)), and not on the choices of the tensors (s_α).

Denote by (X, ρ ,λ) the universal triple over (𝕏_0 , λ_0), where X is the universal p-divisible group, ρ is the universal quasi-isogeny, and λ is the universal polarization. Consider the restriction of this triple to the closed formal subscheme  of (𝕏_0 ,λ_0). We denote this last triple also by (X,ρ ,λ) and call it the universal triple over .


Let _W be the category of W-algebras in which p is nilpotent. 
As a set-valued functor on the category _W, the symplectic Rapoport-Zink space (𝕏_0  , λ_0) has an explicit moduli interpretation in terms of triples (X,ρ,λ). In contrast, the subfunctor defined by  does not have an explicit description. In fact, in <cit.> Howard–Pappas only give a moduli interpretation of  when it is viewed as a set-valued functor on a more restricted category . In this article we do not make use of this last moduli interpretation. All we will need is the global construction of  as a formal subscheme of (𝕏_0, λ_0) due to Howard–Pappas. 




Over , the universal quasi-isogeny ρ respects the polarizations λ and λ_0 up to a scalar c(ρ)∈ℚ_p^× , i.e., ρ^∨∘λ∘ρ=c^-1(ρ)·λ_0 (Zariski locally on _k). Let ^(ℓ)⊆ be the closed and open formal subscheme where _p(c(ρ))=ℓ. 
We have the decomposition into a disjoint union 
    =∐_ℓ∈ℤ^(ℓ).
 In fact each ^(ℓ) is connected and they are mutually (non-canonically) isomorphic. cf. <cit.>.



 §.§ The group J_b


The algebraic group J_b=(V_K^Φ) has ℚ_p-points 
    J_b(ℚ_p)={g∈ G(K): gb=bσ(g)},
 and J_b(_p) acts on  via its action on 𝕏_0 as quasi-endomorphisms. The action of g ∈ J_b(_p) on  restricts to isomorphisms 

    ^(ℓ)^(ℓ + _p (η_b (g))),  ℓ∈

 where η_b : J_b(_p) →_p^× is the spinor similitude. In particular, p^ℤ⊆ J_b(ℚ_p) acts on  and since η_b (p) = p^2, we have an isomorphism 
    p^ℤ\≅^(0)∐^(1).



	In this article we are interested in studying the fixed locus ^g of  under g ∈ J_b (_p). By (<ref>) this is non-empty only when _p (η_b (g)) = 0. Since p^ is central in J_b(_p), one could also study (p^\)^g for g∈ J_b(_p). However by (<ref>), we know that (p^\)^g ≠∅ only if _p (η_b (g)) is even, and in this case 
    (p^\)^g ≅ p^\^g_0,
 where g_0=p^-_p (η_b (g))/2 g. Hence the study of (p^\)^g for general g reduces to the study of ^g for g satisfying _p(η_b(g))=0.





 §.§ Special endomorphisms


Using the injection V↪ C(V)^op, we can view 
    V⊆_ℤ_p(D)
 as special endomorphisms of D: the action of v∈ V on D is explicitly given by 
    (vd)(c)=d(vc),   d∈ D, c∈ C(V).
 
Base changing to K gives V_K⊆_K(D_K). Since the F-equivariant endomorphisms _K,F(D_K) can be identified with the space of quasi-endomorphisms ^0(𝕏_0) of 𝕏_0, we obtain an embedding of _p-vector spaces 
    V_K^Φ↪^0(𝕏_0).
 Elements of V_K^Φ are thus viewed as quasi-endomorphisms of 𝕏_0, and we call them special quasi-endomorphisms.



 §.§ Vertex lattices



A vertex lattice is a ℤ_p-lattice Λ⊆ V_K^Φ such that 
    pΛ⊆Λ^∨⊆Λ.
 We define 
    Ω_0=Λ/Λ^∨.
 Then the quadratic form v↦ p· q(v) makes Ω_0 a non-degenerate quadratic space over 𝔽_p. The type of Λ is defined to be t_Λ:=_𝔽_pΩ_0.

By <cit.>, the type of a vertex lattice is always an even integer such that 2≤ t_Λ≤ t_max , where 
    t_max=
    
      n-2,    if n is even and  (V_ℚ_p)=(-1)^n/2∈ℚ_p^×/(ℚ_p^×)^2, 
    
      n-1,    if n is odd, 
    
      n,    if n is even and (V_ℚ_p)(-1)^n/2∈ℚ_p^×/(ℚ_p^×)^2.


It follows that the quadratic space Ω_0 is always non-split, because otherwise a Lagrangian subspace ℒ⊆Ω_0 would provide a vertex lattice Λ^∨+ℒ⊆ V_K^Φ of type 0 (cf. <cit.>) 



 §.§ The variety S_Λ




  Define 
    Ω=Ω_0 ⊗_𝔽_p k≅Λ_W/Λ_W^∨.
 Let d=t_Λ/2. Let (Ω) be the moduli space of Lagrangian subspaces ℒ⊆Ω. We define S_Λ⊆(Ω) to be the reduced closed subscheme of (Ω) with k-points given as follows:

    S_Λ(k)   ={Lagrangian subspaces ℒ⊆Ω: (ℒ+Φ(ℒ))=d+1}
       ≅{(ℒ_d-1,ℒ_d): ℒ_d ⊆Ω Lagrangian, ℒ_d-1⊆ℒ_d∩Φℒ_d,ℒ_d-1=d-1},

where the last bijection is given by ℒ↦ (ℒ∩Φℒ, ℒ).



More precisely, for any k-algebra R, the R-points S_Λ (R) is the set of pairs (_d-1, _d) such that:


  * _d is a totally isotropic R-submodule of Ω⊗_k R that is an R-module local direct summand of Ω⊗_k R and of local rank d,

  * _d-1 is an R-module local direct summand of Ω⊗_k R and of local rank d-1,

  * _d-1⊂_d ∩Φ_d, where Φ acts on Ω⊗_k R via the p-Frobenius on R. In particular, _d-1 is totally isotropic, and is a local direct summand of _d and of Φ_d. (For the last statement see Remark <ref> below.)


By <cit.>, S_Λ is a k-variety with two isomorphic connected components S_Λ^±, each being projective and smooth of dimension t_Λ/2-1. 
For more details, see <cit.> and <cit.>.


	In the sequel we will frequently use the following simple fact without explicitly mentioning it. Let R be a commutative ring and M a free R-module of finite rank. Suppose M_1, M_2 are submodules of M that are local direct summands. Suppose M_1 ⊂ M_2. Then M_1 is a local direct summand of M_2, and both M_1 and M_2 are locally free.   




 §.§ Structure of the reduced scheme ^red



 For a vertex lattice Λ, we define _Λ⊆ to be locus where ρ∘Λ^∨∘ρ^-1⊆(X), i.e. the quasi-endomorphisms ρ∘ v∘ρ^-1 lift to actual endomorphisms for any v∈Λ^∨. In other words, if we define a locus (𝕏_0,λ_0)_Λ using the same condition inside (𝕏_0, λ_0) (a closed formal subscheme by <cit.>), then _Λ is the intersection of  with (𝕏_0,λ_0)_Λ inside (𝕏_0,λ_0). In particular, _Λ is a closed formal subscheme of . 



  Consider the reduced subscheme ^(ℓ) , red of ^(ℓ). 
By the result <cit.>, the irreducible components of ^(ℓ),red are precisely _Λ^(ℓ),red, where Λ runs through the vertex lattices of the maximal type t_Λ=t_max. Moreover, there is an isomorphism of k-schemes (<cit.>)

    p^ℤ\_Λ^red S_Λ,

 which also induces an isomorphism between _Λ^(ℓ),red and S_Λ^±, for each ℓ∈. In particular, ^red is equidimensional of dimension t_max/2-1.



 §.§ The Bruhat–Tits stratification
 For any vertex lattices Λ_1 and Λ_2, the intersection _Λ_1^red∩_Λ_2^red is nonempty if and only if Λ_1∩Λ_2 is also a vertex lattice, in which case it is equal to _Λ_1∩Λ_2^red (<cit.>). In this way we obtain a Bruhat–Tits stratification on ^red. Associated to a vertex lattice Λ, we define an open subscheme of _Λ^red given by 
    _Λ=^red_Λ-⋃_Λ'⊊Λ_Λ'^red.
  Then 
    ^red=∐_Λ_Λ
 is a disjoint union of locally closed subschemes, indexed by all vertex lattices.



 §.§ Special lattices
 One can further parametrize the k-points in each _Λ using special lattices.


  We say a W-lattice L⊆ V_K is a special lattice if L is self-dual and (L+Φ(L))/L≅ W/pW. 


We have a bijection (<cit.>) 
    p^ℤ\(k){special lattices  L⊆ V_K}.

 To construct this bijection, one uses the fact (<cit.>) that p^ℤ\(k) can be identified with the affine Deligne–Lusztig set
 
    X_G,b,μ^σ(k)={g∈ G(K): g^-1b σ(g)∈ G(W)μ^σ(p)G(W)}/G(W).

The special lattice associated to g∈ G(K) is then given by gμ(p^-1).V_W⊆ V_K. Conversely, given a special lattice L⊆ V_K, then there exists some g∈ G(K) such that gμ(p^-1). V_W=L and g. V_W=Φ(L). The point in (k) then corresponds to the image of g in X_G,b,μ^σ(k). The Dieudonné module of the p-divisible group at this point is given by M=g D_W⊆ D_K and the image of Verschiebung is (F^-1 p) M=g· pμ(p^-1) D_W.

	Suppose x_0∈(k) corresponds to the special lattice L under (<ref>). Let M=𝔻(X_0) ⊂ D_K be the Dieudonné module of the p-divisible group X_0 corresponding to x_0. 
        Then we have (cf. <cit.>) 
    L = v∈ V_K | v (F^-1p)M ⊂ (F^-1p) M ,  Φ L = v∈ V_K| v M ⊂ M.
 Here we view V_K ⊂_K(D_K) as in <ref>.




 §.§ Special lattices and vertex lattices


For any vertex lattice Λ, the bijection (<ref>) induces a bijection 

    p^ℤ\_Λ(k){special lattices  L⊆ V_K: Λ_W^∨⊆ L ⊆Λ_W} = {special lattices  L⊆ V_K: Λ_W^∨⊆ L }
 Sending a special lattice L to ℒ: =L/Λ_W^∨⊆Ω gives a bijection between the right hand side of (<ref>) and S_Λ(k), which is the effect of the isomorphism (<ref>) on k-points.



For each special lattice L⊆ V_K, there is a unique minimal vertex lattice Λ(L)⊆ V_K^Φ such that 
    Λ(L)_W^∨⊆ L⊆Λ(L)_W.
 In fact, let L^(r)=L+Φ(L)+⋯+Φ^r(L). Then there exists a unique integer 1≤ d≤ t_max/2 such that L^(i)⊊ L^(i+1) for i<d, and L^(d)=L^(d+1). Then L^(i+1)/L^(i) all have W-length 1 for i<d, and 
    Λ(L):=(L^(d))^Φ⊆ V_K^Φ
 is a vertex lattice of type 2d and Λ(L)^∨=L^Φ.  

 Notice that Λ(L)_W is the smallest Φ-invariant lattice containing L and Λ(L)^∨_W is the largest Φ-invariant lattice contained in L. It follows that the element of (k) corresponding to a special lattice L lies in _Λ if and only if Λ(L)⊆Λ, and it lies in _Λ if and only if Λ(L)=Λ. Thus we have the bijection

    p^ℤ\_Λ(k){L  special lattices: Λ(L)=Λ}.




 §.§ Deligne–Lusztig varieties


For any vertex lattice Λ, by <cit.>,  p^ℤ\_Λ is a smooth quasi-projective variety of dimension t_Λ/2-1, isomorphic to a disjoint union of two Deligne–Lusztig varieties X_B(w^±) associated to two Coxeter elements w^± in the Weyl group of (Ω_0). Here Ω_0 := Λ/Λ^∨ is the quadratic space over _p defined in Definition <ref>. In particular, the k-variety p^ℤ\_Λ only depends on the quadratic space Ω_0.

Let us recall the definition of X_B(w^±). Let d=t_Λ/2. Let · , · be the bilinear pairing on Ω_0. Since Ω_0 is a non-degenerate non-split quadratic space over 𝔽_p (<ref>), one can choose a basis  e_1,…, e_d, f_d,…, f_1 of Ω such that ⟨ e_i,f_i⟩=1 and all other pairings between the basis vectors are 0, and Φ fixes e_i,f_i for i=1,…,d-1 and interchanges e_d with f_d. This choice of basis gives a maximal Φ-stable torus T⊆(Ω) (diagonal under this basis), and a Φ-stable Borel subgroup B⊇ T as the common stabilizer of the two complete isotropic flags 
    ℱ^±: ⟨ e_1⟩⊆⟨ e_1,e_2⟩⊆⋯⊆⟨ e_1, …,e_d-1, e_d^±⟩,
 where e_d^+:=e_d and e_d^-:=f_d. Let s_i (i=1,…,d-2) be the reflection e_i↔ e_i+1, f_i↔ f_i+1 and let t^± be the reflection e_d-1↔ e_d^±, f_d-1↔ e_d^∓. Then the Weyl group W(T)=N(T)/T is generated by s_1, ⋯, s_d-2, t^+, t^-. We also know that W(T) sits in a split exact sequence 
    0→ (ℤ/2 ℤ)^d-1→ W(T)→ S_d→0.
 Since Φ fixes s_i and swaps t^+ and t^-, we know the d-1 elements s_1,…,s_d-2, t^+ (resp. s_1, …, s_d-2, t^-) form a set of representatives of Φ-orbits of the simple reflections. Therefore 
    w^±:= t^∓ s_d-2⋯ s_2s_1∈ W(T)
 are Coxeter elements of minimal length. The Deligne–Lusztig variety associated to B and the Coxeter element w^± is defined to be 
    X_B(w^±):={g∈(Ω)/B: (g, Φ(g))=w^±},
 where (g,h)∈  B\(Ω)/B≅ W(T) is the relative position between the two Borels gBg^-1 and hBh^-1. The variety X_B(w^±) has dimension d-1. Under the map g↦ gℱ^±, the disjoint union X_B(w^+)∐ X_B(w^-) can be identified with the variety of complete isotropic flags 
    ℱ: ℱ_1⊆ℱ_2⊆⋯⊆ℱ_d
 such that ℱ_i=ℱ_i-1+Φ(ℱ_i-1) and _k( ℱ_d+Φ(ℱ_d))=d+1. The two components are interchanged by an orthogonal transformation of determinant -1. Notice that such ℱ is determined by the isotropic line ℱ_1 by 
    ℱ_i=ℱ_1+Φ(F_1)⋯+Φ^i-1(ℱ_1),
 and is also determined by the Lagrangian ℱ_d by 
    ℱ_i=ℱ_d∩Φ(ℱ_d)∩…∩Φ^d-i(ℱ_d).


The bijection (<ref>) induces a bijection 

    p^ℤ\_Λ(k) X_B(w^+)(k)∐ X_B(w^-)(k)

by sending a special lattice L with Λ(L)=Λ to the flag determined by the Lagrangian ℱ_d=L/Λ^∨_W. This bijection is the restriction of the isomorphism (<ref>) on k-points and we obtain the desired isomorphism 

    p^ℤ\_Λ≅ X_B(w^+)∐ X_B(w^-).




 §.§ Special cycles






  For an m-tuple 𝐯=(v_1,…,v_m) of vectors in V_K^Φ, define its fundamental matrix T(𝐯)=(⟨ v_i, v_j⟩)_i,j=1,…, m. We define the special cycle 𝒵(𝐯)⊆ to be the locus where ρ∘ v_i∘ρ^-1∈(X), i.e., all the quasi-endomorphisms ρ∘ v_i∘ρ^-1 lift to actual endomorphisms on X (i=1,…,m). Similar to Definition <ref>, 𝒵(𝐯) is a closed formal subscheme of , which is the intersection  with the analogously defined cycle inside (𝕏_0,λ_0). Since 𝒵(𝐯) only depends on the _p-submodule span__p (𝐯) of V_K^Φ, we also write 𝒵( span__p (𝐯) ).  


	Let x_0 ∈(k) correspond to L under (<ref>). Let 𝐯 be an arbitrary _p-submodule of V_K^Φ. By Remark <ref> we know that x_0 ∈𝒵(𝐯) if and only if 𝐯⊂Φ L, if and only if 𝐯⊂Φ L ∩ L (as 𝐯 is Φ-invariant). 

















    When m=n and T(𝐯) is non-singular, we obtain a lattice 
    L(𝐯)=ℤ_p v_1+⋯ℤ_p v_n⊆ V_K^Φ.
 By the Cartan decomposition, T(𝐯) ∈_n (_p)  (p^r_1, p^r_2,⋯, p^r_n) _n(_p) for a unique non-increasing sequence of integers r_1≥⋯≥ r_n. Note that if we view the matrix T(𝐯)^-1 as a linear operator V_K^Φ→ V_K^Φ using the basis 𝐯, it sends 𝐯 to the dual basis of 𝐯, and in particular it sends any _p-basis of L(𝐯) to a _p-basis of L(𝐯) ^∨. Therefore the tuple (r_1,⋯, r_n) is characterized by the condition that there is a basis e_1,…,e_n of L(𝐯) such that p^-r_1e_1,…, p^-r_ne_n form a basis of L(𝐯)^∨. From this characterization we also see that the tuple (r_1,⋯, r_n) is an invariant only depending on the lattice L(𝐯). We say 𝐯 is minuscule if T(𝐯) is non-singular and r_1=1,   r_n≥0. 



	Suppose m=n and T(𝐯) is non-singular. Then 𝐯 is minuscule if and only if L( 𝐯) ^∨ is a vertex lattice. In this case by definition 𝒵(𝐯) = _L(𝐯) ^∨. 


















§ THE INTERSECTION PROBLEM AND THE POINT-COUNTING FORMULA




 §.§ The GSpin Rapoport–Zink subspace


From now on we assume n≥4. Suppose the last basis vector x_n∈ V has norm 1. Then the quadratic subspace of dimension n-1 
    V^♭=ℤ_px_1+⋯ℤ_px_n-1
 is also self-dual. Let G^♭=(V^♭). Analogously we define the element 
    b^♭=x_3(p^-1x_1+x_2)∈ G^♭(ℚ_p)
 and the cocharacter 
    μ^♭: 𝔾_m→ G^♭,   t↦ t^-1x_1x_2+x_2x_1.
 As in <ref>, we have an associated GSpin Rapoport–Zink space 
    ^♭=(G^♭, b^♭, μ^♭, C(V^♭)).
 The embedding V^♭↪ V induces an embedding of Clifford algebras C(V^♭)↪ C(V) and a closed embedding of group schemes G^♭↪ G over ℤ_p, which maps b^♭ to b and μ^♭ to μ. Thus by the functoriality of Rapoport–Zink spaces (<cit.>), we have a closed immersion  
    δ: ^♭↪
 of formal schemes over W.




 §.§ Relation with the special divisor 𝒵(x_n)




For compatible choices of symplectic forms ψ^♭ on C(V^♭) and ψ on C(V), the closed embedding of group schemes (C(V^♭),ψ^♭)↪(C(V),ψ) induces a closed immersion of symplectic Rapoport–Zink spaces (<ref>) 
    ϕ: (𝕏_0^♭,λ_0^♭)↪(𝕏_0, λ_0).
 Since we have a decomposition of (C(V^♭),ψ^♭)-representations 
    C(V)≅ C(V^♭) ⊕ C(V^♭) x_n,
 we know the moduli interpretation of ϕ is given by sending a triple  (X^♭,ρ^♭,λ^♭) to the p-divisible group X=X^♭⊕ X^♭ with the quasi-isogeny ρ=ρ^♭⊕ρ^♭ and polarization λ=λ^♭⊕λ^♭.

By the functoriality of Rapoport–Zink spaces (<cit.>), we have a commutative diagram of closed immersions

    ^♭@^(->[r]^δ@^(->[d]    @^(->[d]
    (𝕏_0^♭,λ_0^♭) @^(->[r]^ϕ   (𝕏_0, λ_0).

Here the two vertical arrows are induced by the closed immersions (V^♭)↪(C(V^♭),ψ^♭) and  (V)↪(C(V),ψ) (<ref>).


   Diagram (<ref>) is Cartesian, i.e., we have
 
    δ(^♭)=ϕ((𝕏_0^♭,λ_0^♭))∩

 inside (𝕏_0,λ_0).



By flat descent, to show that the closed formal subschemes on the two sides of (<ref>) agree, it suffices to show that they have the same k-points and the same formal completion at every k-point (cf. <cit.>). The claim then follows from the observation that both the k-points and the formal completions have purely group theoretic description. 


In fact, the k-points of ^♭=_G^♭, (𝕏_0^♭,λ_0^♭)=_H and =_G have the group theoretic description as the affine Deligne–Lusztig sets (<ref>) associated to the groups G^♭=(V^♭), H=(C(V^♭), ψ^♭) and G=(V) respectively. Since G^♭=H∩ G inside (C(V)), we know that both sides of (<ref>) have the same k-points. Fix a k-point x∈^♭(k), then by <cit.>, _G^♭,x can be identified with U_G^♭^μ_x,∧, where μ_x: 𝔾_m,W→ G^♭_W gives a filtration that lifts the Hodge filtration for x, U_G^♭^μ_x⊆ G^♭ is the unipotent radical of the opposite parabolic group defined by μ_x (<cit.>) and  U_G^♭^μ_x,∧ is its formal completion along its identity section over W. Similarly, we can identify _H,x and _G,x as U_H^μ_x,∧ and U_G^μ_x,∧. Again because G^♭=H∩ G, we know that the formal completions at x of both sides of (<ref>) agree inside U_(C(V))^μ_x,∧.



  δ(^♭)=𝒵(x_n).



 Let X^♭ be the universal p-divisible group over ^♭ and ρ^♭ be the universal quasi-isogeny. Then it follows from the commutative diagram (<ref>) that the image of (X^♭,ρ)  under δ is given by the p-divisible group (X^♭⊕  X^♭,ρ^♭⊕ρ^♭). Since x_n has norm 1, right multiplication by x_n swaps the two factors C(V^♭) and C(V^♭)x_n.  It follows that the quasi-endomorphism 
    (ρ^♭⊕ρ^♭ )∘ x_n∘ (ρ^♭⊕ρ^♭ )^-1: (X^♭⊕ X^♭)→(X^♭⊕ X^♭)
 (uniquely determined by the rigidity of quasi-isogenies) simply swaps the two factors, which is an actual endomorphism (i.e., swapping) of X^♭⊕ X^♭. By Definition <ref> of 𝒵(x_n), we have δ(^♭)⊆𝒵(x_n).

Conversely, over 𝒵(x_n) the universal p-divisible group X admits an action of C(x_n)^op⊗ C(V), where C(x_n) is the Clifford algebra of the rank one quadratic space ℤ_p x_n. Notice 
    C(x_n)^op⊗ C(V)≅ (C(x_n)^op⊗ C(x_n)) ⊕ (C(x_n)^op⊗ C(V^♭)).
 It follows that over 𝒵(x_n) the universal p-divisible group X admits an action of C(x_n)^op⊗ C(x_n), which is isomorphic to the matrix algebra M_2(ℤ_p). The two natural idempotents of M_2(ℤ_p) then decomposes X as a direct sum of the form X^♭⊕ X^♭. Hence 𝒵(x_n)⊆ϕ((𝕏_0^♭,λ_0^♭))∩. The latter is equal to δ(^♭) by (<ref>) and hence 𝒵(x_n)⊆δ(^♭).





In the following we will only use the inclusion δ(^♭)⊆𝒵(x_n).




 §.§ Arithmetic intersection of GGP cycles



The closed immersion δ induces a closed immersion of formal schemes 
    (𝕀, δ): ^♭→^♭×_W .
 Denote by Δ the image of (𝕀,δ), which we call the GGP cycle.  


The embedding V^♭↪ V also induces an embedding of quadratic spaces V^♭,Φ_K↪ V_K^Φ and hence we can view 
    J_b^♭=(V^♭,Φ_K)↪ J_b
 as an algebraic subgroup over ℚ_p.

For any g∈ J_b(ℚ_p), we obtain a formal subscheme 
    gΔ:=(𝕀× g)Δ⊆^♭×_W ,
 via the action of g on . Our goal is to compute the arithmetic intersection number 
    ⟨Δ,gΔ⟩,
 when g is regular semisimple and minuscule. 


  We say g∈ J_b(ℚ_p) is regular semisimple if the  𝐯(g):=(x_n, gx_n,…, g^n-1x_n) forms a ℚ_p-basis of V_K^Φ. Equivalently, the fundamental matrix T(g):=T(𝐯(g)) is non-singular (Definition <ref>). Equivalently, the stabilizer of g, for the conjugation action of the subgroup J_b^♭, lies in the center (≅𝔾_m) of J_b^♭.
  
  

 We say g is minuscule if 𝐯(g) is minuscule (Definition <ref>).














 §.§ Fixed points


Let g∈ J_b(_p) and let ^g⊆ be the fixed locus of g. Then by definition we have 
    Δ∩ gΔ≅δ(^♭)∩^g.



Let g∈ J_b(ℚ_p) be regular semisimple. We define the lattice 
    L(g):=ℤ_p x_n+⋯ℤ_p g^n-1x_n⊆ V_K^Φ.
  




 Inside  both the formal subschemes ^g and δ(^♭) are stable under p^. Moreover, under the bijection (<ref>), we have

  
  
  * p^ℤ\δ(^♭(k))≅{L=L^♭⊕ W x_n: L^♭⊆ V^♭_K  special lattices}.
  
  * p^ℤ\δ(^♭(k))≅{L special lattices: x_n ∈ L }.
  
  * p^ℤ\^g(k)≅{L  special lattices: gL=L}.
  
  * p^ℤ\(δ(^♭(k))∩^g(k))≅{L  special lattices: gL=L, L⊇ L(g)_W}.
  



	Since p^ is central in J_b(_p), we know ^g is stable under p^. The morphism δ: ^♭→ is equivariant with respect to the natural inclusion J_b^♭ (_p) → J_b (_p), and the morphism J_b^♭→ J_b restricts to the identity between the centers _m of J_b^♭ and of J_b. It follows that δ is equivariant for the p^ action, and so δ(^♭) is stable under p^. We now prove the statements (1) to (4).   
  
  
  * For a point L^♭∈ p^ℤ\^♭(k), we can write L^♭=h^♭μ^♭(p^-1).V^♭_W⊆ V^♭_K, for some h^♭∈ G^♭(K). Then its image under δ is given by L=hμ(p^-1).V_W⊆ V_K, where h is the image of h^♭ in G(K). By V=V^♭⊕ℤ_p x_n and the compatibility between h,μ and h^♭, μ^♭, we know that L=L^♭⊕ W x_n. 
  
  * Suppose L is a special lattice with x_n∈ L. Since x_n has norm 1, we know that L=L' ⊕ Wx_n is the direct sum of Wx_n and its orthogonal complement L' in L. One can check L'⊆ V_K^♭ is also a special lattice. This finishes the proof in view of item (1). 
  
  * This is clear since ^g(k) is the fixed locus of g.
  
  * For a point L∈ p^ℤ\(δ(^♭(k))∩^g(k)), by items (1) (3), we have L=L^♭⊕ W x_n and gL=L. It follows from x_n∈ L that gx_n,…, g^n-1x_n∈ L, and so L⊇ L(g)_W. Conversely, if a point L∈(k) satisfies gL=L and L ⊃ L(g)_W, then L ∈ p^ℤ\(δ(^♭(k))∩^g(k)) by items (2) and (3)




We say a vertex lattice Λ is  a g-vertex lattice if gΛ=Λ and Λ⊆ L(g)^∨. Denote the set of all g-vertex lattices by (g). In general, if a vertex lattice Λ satisfies gΛ=Λ, then g induces an action on Ω_0=Λ/Λ^∨, which further induces an action g̅ on _Λ^red and _Λ. We denote the fixed locus of g̅ on _Λ by _Λ^g̅. 



  
    p^ℤ\(δ(^♭)∩^g)(k)=∐_Λ∈(g)p^ℤ\_Λ^g̅ (k).




By Lemma <ref>,  it suffices to show the k-points of the right hand side are in bijection with special lattices L such that gL=L and L⊇ L(g)_W. Notice that any special lattice L is self-dual, so the condition L⊇ L(g)_W is equivalent to the condition L⊆ L(g)_W^∨. Since Λ(L)_W is the minimal Φ-invariant lattice containing L (<ref>), and L(g)_W^∨ is Φ-invariant, we know that the condition L⊆ L(g)_W^∨ is equivalent to the condition Λ(L)⊆ L(g)^∨. The result now follows from taking g̅-invariants and g-invariants of the two sides of the bijection (<ref>).




 §.§ Fixed points in a Bruhat–Tits stratum


Let Λ be a vertex lattice and Ω_0=Λ/Λ^∨ (<ref>). By the isomorphism (<ref>), p^ℤ\_Λ is disjoint union of two isomorphic Deligne–Lusztig varieties X_B(w^±) associated to the Coxeter elements w^± for (Ω_0). Write X:=X_B(w^±). To compute p^ℤ\_Λ^g̅, it suffices to compute the g̅-fixed points X^g̅.


We say a semisimple element g̅∈(Ω_0) is regular 
if Z^∘(g̅), the identity component of the centralizer of g̅ in (Ω_0), is a (necessarily maximal) torus[Note the difference with Definition <ref>. The conflict of the usage of the word "regular" should hopefully not cause confusion.]. 
  





  Let Λ be a vertex lattice and let g̅∈(Ω_0)(𝔽_p).
  
  
  *  X^g̅ is non-empty if and only if g̅ is semisimple and contained in a maximal torus of Coxeter type.
  
  *  X^g̅ is non-empty and finite if and only if g̅ is regular semisimple and contained in a maximal torus of Coxeter type. In this case, the cardinality of X^g̅ is given by t_Λ/2.
  



Recall that a maximal torus T' is of Coxeter type if T'=h T h^-1 for some h∈(Ω_0) such that h^-1Φ(h) lifts to a  Coxeter element w in the Weyl group W(T)=N(T)/T. In other words, T' is conjugate to T over k but its Frobenius structure is given by w·Φ. For the Coxeter element w=w^± constructed in <ref>, we know that an element (λ_1,…,λ_d,λ_d^-1,…,λ_1^-1) of T(k) is fixed by w·Φ if and only if 
    (λ_1,λ_2,…,λ_d-1, λ_d)=(λ_d^∓ p,λ_1^p …, λ_d-2^p,λ_d-1^± p).
  It follows that a semisimple element g̅∈(Ω_0)(𝔽_p) is contained in a maximal torus of Coxeter type if and only if the eigenvalues of g̅ on Ω_0 ⊗ k belong to a single Galois orbit.








 
  * Suppose X^g̅ is non-empty. Then it is a general fact about Deligne–Lusztig varieties that g̅ must be semisimple (<cit.>). Let T(w)⊆(Ω_0) be a torus of Coxeter type (associated to w=w^+ or w^-) and B(w)⊇ T(w) be a Borel. Assume g̅ is semisimple. Then we know from <cit.> that X^g̅ is a disjoint union of Deligne-Lusztig varieties X_T'⊆ B' for the group G'=Z^∘(g̅) and the pairs 
    (T', B')=(h T(w) h^-1, h B(w) h^-1∩ G'),
 where h runs over classes G'(𝔽_p)\(Ω_0)(𝔽_p) such that g̅∈ h T(w)h^-1. Therefore X^g̅ is non-empty if and only if there exists h∈(Ω_0)(𝔽_p) such that g̅∈ h T(w) h^-1, if and only if g̅ is contained in a maximal torus of Coxeter type (as so is T(w)).
 
  * By part (1) we know that X^g̅ is further finite if and only if all X_T'⊆ B' are zero dimensional, if and only if all B'=hBh^-1∩ G' are tori. This happens exactly when G'=Z^∘(g̅) itself is a torus, i.e., when g̅ is regular. In this case, G' is a maximal torus of Coxeter type in (Ω_0) and the cardinality of X^g̅ is equal to the cardinality of N(T(w))(𝔽_p)/T(w)(𝔽_p). The latter group is isomorphic to (N(T(w))/T(w))^Φ by Lang's theorem and hence is isomorphic to the Φ-twisted centralizer of w in the Weyl group W(T)=N(T)/T: 
    Z_Φ(w):={x∈ W(T): xw=wΦ(x)}.
 The cardinality of Z_Φ(w) is known as the Coxeter number of the group (Ω_0), which is equal to d=t_Λ/2 since (Ω_0) is a non-split even orthogonal group (<cit.>).  




 §.§ Point-counting in the minuscule case
 Let g∈ J_b(ℚ_p) be regular semisimple and minuscule. Then Ω_0(g):=L(g)^∨/L(g) is a 𝔽_p-vector space (see Definition <ref>), and hence L(g)^∨ is a vertex lattice.


If ^g is non-empty, then g fixes some vertex lattice and so we know that the characteristic polynomial of g has _p-coefficients. It follows that L(g) is a g-stable lattice, from which it also follows easily that L(g)^∨ is g-stable. Hence by definition L(g)^∨ is a g-vertex lattice. The induced action of g on Ω_0(g), denoted by g̅∈(Ω_0(g))(𝔽_p), makes Ω_0(g) a g̅-cyclic 𝔽_p-vector space. It follows that the minimal polynomial of g̅ is equal to its characteristic polynomial.


From now on we assume ^g is non-empty. Let g̅∈(Ω_0(g))(𝔽_p) be as in Remark <ref>.

 


For any polynomial R(T), we define its reciprocal to be 
    R^*(T):=T^ R(T)· R(1/T).
 We say R(T) is self-reciprocal if R(T)=R^*(T).




 Let P(T)∈𝔽_p[T] be the characteristic polynomial of g̅∈(Ω_0(g)). Then P(T) is self-reciprocal. For any monic irreducible factor Q(T) of P(T), we denote by m(Q(T)) to be the multiplicity of Q(T) appearing in P(T).    

  


Assume ^g is non-empty. Then 
p^ℤ\(δ(^♭)∩^g)(k) is non-empty if and only if P(T) has a unique self-reciprocal monic irreducible factor Q(T) such that m(Q(T)) is odd. In this case, p^ℤ\(δ(^♭)∩^g)(k) is finite and has cardinality 
    Q(T)·∏_R(T)(1+m(R(T))),
 where R(T) runs over all non-self-reciprocal monic irreducible factors of P(T).



  By Proposition <ref>, we know that p^ℤ\(δ(^♭)∩^g)(k) is non-empty if and only if p^ℤ\_Λ^g is non-empty for some Λ∈(g). For any Λ∈(g), by definition we have a chain of inclusions of lattices 
    L(g)⊆Λ^∨⊆Λ⊆ L(g)^∨,
 which induces a filtration of 𝔽_p-vector spaces, 
    0⊆Λ^∨/L(g)⊆Λ/L(g)⊆Ω_0(g).
 It follows that the map Λ↦Λ^∨/L(g) gives a bijection

    (g)≅{totally isotropic g̅-invariant subspaces U⊆Ω_0(g)}.
 By the bijection (<ref>), (g) is non-empty if and only if there is a totally isotropic g̅-invariant subspace U of Ω_0(g). Such a subspace U induces a filtration
  
    0⊆ U⊆ U^⊥⊆Ω_0(g).

 Since U and U^⊥ are g̅-invariant, we obtain a decomposition of the characteristic polynomial
  
    P(T)=P_1(T)Q(T)P_2(T)

  where P_1(T), Q(T), P_2(T) are respectively the characteristic polynomials of g̅ acting on the associated graded U, U^⊥/U and Ω_0(g)/U^⊥. Notice the non-degenerate quadratic form on Ω_0(g) identifies Ω_0(g)/U^⊥ with the linear dual of U, from which we know that P_2(T)=P_1^*(T). Similarly, we know that Q(T)=Q^*(T), i.e., Q(T) is self-reciprocal.

  Let Λ=L(g)+U^⊥ be the g-vertex lattice corresponding to U under the bijection (<ref>) and let Ω_0=Λ/Λ^∨ and g̅_0∈(Ω_0)(𝔽_p) be the induced action of g̅ on Ω_0. By Remark <ref>, the minimal polynomial of g̅ is equal to its characteristic polynomial P(T). Thus the minimal polynomial of g̅_0 is equal to its characteristic polynomial Q(T)  under the decomposition (<ref>). If g̅_0 is semisimple, then its eigenvalues are distinct. If g̅ is further contained in a torus of Coxeter type, then we know that its eigenvalues belong to a single Galois orbit (Remark <ref>), so Q(T) is irreducible. Conversely, if Q(T) is irreducible, then clearly g̅_0 is semisimple and contained in a torus of Coxeter type. Hence we know that g̅_0 is semisimple and contained in a torus of Coxeter type if and only if Q(T) is irreducible.

  Therefore by  Proposition <ref> (<ref>), _Λ^g̅_0 is non-empty if and only if Q(T) is irreducible.  In this case, g̅_0 is indeed regular semisimple and the cardinality of p^ℤ\_Λ^g̅_0 is equal 2·#X^g̅_0 (due to two connected components), which is equal to _𝔽_pΩ_0= Q(T) by Proposition <ref> (<ref>).

  Since P_2(T)=P_1^*(T), we know the multiplicity of R(T) in P_1(T)P_2(T) is even for any self-reciprocal factor R(T). Hence Q(T) is the unique self-reciprocal monic irreducible factor of P(T) such that m(Q(T)) is odd. Finally, the factorizations  (<ref>) with P_2(T)=P_1^*(T) corresponds bijectively to the filtrations (<ref>). The proof is now finished by noticing that the number of such factorizations is exactly given by  
    ∏_R(T)(1+m(R(T))),
  where R(T) runs over all monic irreducible factors of P(T) such that R(T) R^*(T).









§ THE REDUCEDNESS OF MINUSCULE SPECIAL CYCLES



 §.§ Review of Faltings's explicit deformation theory

We review Faltings's explicit deformation theory of p-divisible groups <cit.> 7, following <cit.> and <cit.>. 

We allow k to be an arbitrary perfect field of characteristic p>2.
We use the notations X_0, (s_α), G, μ to denote more general objects:

	
  * X_0 is an arbitrary p-divisible group over k. 
	
  * Define M_0 : = 𝔻 (X_0) (W). It is equipped with the Frobenius, denoted by ϕ_M_0. We will interchangeably think of ϕ_M_0 as a W-linear map ϕ_W^* M_0 → M_0 or as a ϕ_W-semi-linear map M_0 → M_0, where ϕ_W is the Frobenius on W. This convention about semi-linear maps will also be adopted in the following for other Frobenii. 
	
  * G ⊂(M_0) is a reductive subgroup (over W) cut out by a family of tensors (s_α). 
	
  * We assume (s_α) are ϕ_M_0-invariant, when viewed as tensors over M_0 [1/p]. (cf. <cit.> Footnote 6 on P 17.)
	
  * We assume the Hodge filtration on 𝔻(X_0) (k) = M_0 ⊗ _W k is G⊗ _W k-split, in the sense that it admits a splitting whose corresponding cocharacter μ_0 : _m →(M_0 ⊗ _W k) factors through G⊗ _W k. We fix the choice of such a μ_0. 
	
  * Fix the choice of a lift μ : _m → G of μ_0.
	
  * Let ^1 M_0 ⊂ M_0 be the filtration defined by μ. This gives rise to a p-divisible group X_0,W over W lifting X_0.
	  

Let U^o (resp. U^o_G) be the opposite unipotent in (M_0) (resp. G) defined by μ. Let R (resp. R_G) be the complete local ring at the identity section of U^o (resp. U^0_G). Let ι: R → R_G be the natural quotient map.  

Choose an isomorphism of W-algebras: 
    _R: R  W [[ t_1,⋯, t_n]].
 This defines a Frobenius ϕ_R on R, by the usual Frobenius on W and t_i ↦ t_i ^p. Note that ϕ_R depends on the choice of _R. 

Define M = M _0⊗ _W R. Define the Frobenius ϕ_M by the composition: 

    M = M_0 ⊗ _W R  M  M,
 where u ∈ U^o(R) ⊂ (M) = (M_0 ⊗ _W R) is the tautological section[Write this out explicitly].

Define ^1 M = ^1 M_0 ⊗ _W R. At this point we have obtained a triple (M, ^1 M, ϕ_M). In general, there is at most one connection ∇ on M such that ϕ_M is horizontal. If exists, it is integrable and topologically nilpotent. (cf. <cit.> 4.3.1., 4.3.2.) 
Now Faltings's theory implies that such a connection ∇ exists for (M, ϕ_M), and the resulting quadruple (M, ^1 M, ϕ_M, ∇) corresponds to a p-divisible group X _R over R[In general there is an equivalence from the category of such quadruples to the category of p-divisible groups over R, cf. <cit.> 4.1.], such that X_R ⊗ _R R/(t_1,⋯, t_n) is canonically isomorphic to X_0,W and such that X_R is a versal deformation of X_0. See <cit.> 4.5 for details.     


Before we proceed, let's recall some terminology about connections mentioned in the above paragraph, and fix some notations:

	
  * a connection ∇ on M means a map ∇: M → M⊗_R Ω̂^1_R satisfying the usual linearity and Leibniz conditions. Here Ω̂^1_R : = _eΩ^1_R/p^e is the "module of p-adically continuous 1-differentials" on R. For our specific R, we have Ω̂_R^1 = Ω_R/W ^1 ≅ W[[t]] dt_1 ⊕⋯⊕ W[[t]] d t_n. To check this, it suffices to check that Ω^1_R/p^e = Ω^1_(R/p^e) / (W/p^e). Since k = W/pW is perfect, W/p^e as an abelian group is generated by elements of the form w^p^e, w∈ W/p^e. It follows that for any w ∈ W/p^e, we have dw = 0 in Ω_R/p^e. This proves the claim that Ω̂_R = Ω_R/W. 
	
  * If ∇ is a connection on M as above, write Θ_i : = ∇_∂/∂ t_i : M → M. Thus we have 
    ∇(m) = ∑_i=1 ^n Θ_i (m) dt_i,  ∀ m ∈ M.
 Integrability of ∇ is equivalent to the condition Θ_i Θ_j = Θ_j Θ_i,  ∀ i,j. Topological nilpotence of ∇ is equivalent to the condition that for each 1≤ i ≤ n and each m ∈ M, there exists N∈ such that Θ_i^N' (m) ∈ pM, for all ∋ N' ≥ N. 
	
  * Suppose ∇ is integrable and topological nilpotent. Define Θ_i as in the previous item. For a multi-index K= (K_1,⋯, K_n) ∈_≥ 0 ^⊕ n, write Θ_K: = Θ_1^K_1⋯Θ_n^K_n. Define the differential-equation-solving map: 
	
    ξ: M_ 0 → M_0 ⊗ _W W[1/p] [[ t_1,⋯, t_n]]
 by the formula 
	
    ξ (m) : = ∑_K∈_≥ 0 ^⊕ n Θ_K (m) (-t) ^K/K!.
 Now we claim that ξ sends m_0 ∈ M_0 to the unique element in the RHS which is horizontal under ∇ (where we use _R to view R as a subring of W[1/p] [[t_1,⋯, t_n]]), and which specializes to m_0 under t_i ↦ 0. In fact, that ξ(m_0) is horizontal follows from direct computation, and the uniqueness statement follows from the observation that ξ sends a W-basis of M_0 to a W[1/p] [[ t_1,⋯, t_n]]-basis of the RHS, which is true because the determinant of these vectors is an element in W[1/p] [[ t_1,⋯, t_n]] whose constant term is not zero in W[1/p]. Moreover, by the formula for ξ we see that ξ has image in M_0 ⊗ _W R', where R' is the subring of W[1/p] [[t]] defined as R' = ∑_K a_K t ^K | a_K ∈ W[1/p], K! a_K ∈ W. 
	
  * Let ∇ be an arbitrary connection on M. Define Θ_i as before. We now explain what it means to say that ϕ_M is horizontal with respect to ∇. Write M̃: = M + p^-1^1 M. Thus ϕ_M induces an R-isomorphism ϕ_M : ϕ_R ^* M̃ M. We will write a typical element in ϕ_R^* M̃ = M̃⊗ _R, ϕ_R R as m⊗ r, with m∈M̃ and r ∈ R. The connection ∇ induces a connection ∇̃ on ϕ_R^* M̃, characterized by 
	
    ϕ_R^* M̃⊗_R Ω̂_R^1 ∋∇̃( m⊗ 1) = ∑_i (Θ_i(m) ⊗ 1) ⊗ d (ϕ_R t_i) = ∑_i (Θ_i(m) ⊗ 1) ⊗ p t_i^p-1d t_i.
 The horizontality of ϕ_M is the requirement that under the isomorphism ϕ_M : ϕ_R^* M̃ M, the connections ∇̃ and ∇ on the two sides correspond. 



We choose an isomorphism of W-algebras 

    _R_G: R_G  W[[τ_1,⋯, τ_r]]
 and get a Frobenius ϕ_R_G on R_G similarly as in the discussion for R. We assume that _R and _R_G are chosen in such a way that the natural map ι: R → R_G is compatible with ϕ_R and ϕ_R_G. 

The quadruple (M, ^1 M , ϕ_M, ∇) specializes along ι: R → R_G to a quadruple (M_R_G, ^1 M_R_G, ϕ_M_R_G, ∇_M_R_G) over R_G in the naive way, where in particular ϕ_M_R_G is defined to be the composition 
    ϕ_R_G ^* M_R_G = ϕ_R_G ^* ι^* M ≅ι^* ϕ_R^* M ι^* M = M_R_G .
 

We remark that in general, without assuming that ιϕ_R_G = ϕ_R ι, we can still define the specialization of (M, ^1 M, ϕ_M,∇) along ι, but in the definition of ϕ_M_R_G we need to replace the canonical isomorphism ϕ_R_G ^* ι^* M ≅ι^* ϕ_R^* M in the above with an isomorphism whose definition uses ∇, cf. <cit.> P 143, or <cit.> 4.3, <cit.> P 16. 

In any case, the quadruple (M_R_G, ^1 M_R_G, ϕ_M_R_G, ∇_M_R_G) over R_G corresponds to the p-divisible group ι^* X_R = : X_R_G over R_G. Morally the pair (R_G, X_R_G) is a versal deformation of X_0 viewed as a p-divisible with addition G-structure. 

Note that there are mistakes in <cit.> 1.5.2 - 1.5.4, which are corrected in <cit.> Erratum. By the corrected statement in <cit.> 1.5.4 (cf. <cit.> E.1), ∇_M_R_G has coefficients in G. In particular, the tensors (ι^* (s_α⊗ 1)) on M_R_G are parallel with respect to ∇_M_R_G. 

Consider an arbitrary W-algebra homomorphism x: R → W. Consider X_x : = x^* X_R, which is a p-divisible group over W together with a canonical isomorphism X_x ⊗ _W k ≅ X_0. By Grothendieck-Messing theory, X_x corresponds to a filtration  ^1_x M_0 ⊂ M_0 = 𝔻(X_0) (W) lifting the filtration in M_0 ⊗ _W k. We describe ^1_x M_0 as follows: 

Recall that the differential-equation-solving map: 

    ξ: M_ 0 → M_0 ⊗ _W W[1/p] [[ t_1,⋯, t_n]]
 has image in M_0 ⊗ _W R'. Since x : R  = W[[t_1,⋯, t_n]] → W necessarily sends each t_i into the ideal pW, which has divided powers, we can compose to get a map 
    (1⊗ x)∘ξ : M_0 → M_0,
 which we call the parallel transport map from t_i= 0 to t_i= x(t_i), denoted by g_x. We know that g_x is a W-module automorphism of M_0. 


	We have ^1_x M_0 = g_x^-1^1 M_0.


	Consider the following morphism in the cristalline site of R/p:
	
    R = W[[t_1,⋯, t_n]] → R,  t_i ↦ t_i + x(t_i).

	This morphism together with the crystal structure of 𝔻(X_R/p) defines an identification 
	
    j_x: M= 𝔻(X_R/p) (R) 𝔻(X_R/p) (R) = M,
 which we know is determined by parallel transport on M.
 The lemma now follows from the fact that the identification of 
    𝔻 (X_0) (W)= 𝔻(X_R) (R)⊗ _R, t_i↦ 0 W = M_R ⊗ _R, t_i↦ 0 W = M_0
 with 
    𝔻 (X_x) (W) = 𝔻(X_R)(R) ⊗_R,x W= M_R ⊗_R ,x W = M_0 ⊗ _W R ⊗ _R,x W = M_0,
 given by the natural identification 𝔻(X_0) (W) 𝔻(X_x) (W), is equal to j_x ⊗_R, t_i↦ 0 W.


	Suppose x : R→ W is a W-algebra homomorphism that factors through ι : R → R_G. Then _x^1 M_0 = g_x^-1^1 M_0, for some g_x ∈ G(W) ⊂(M_0). 


Since (ι^*( s_α⊗ 1)) are parallel under ∇_M_R_G = ι^* ∇, we know that the parallel transport map g_x discussed above fixes s_α. But that means g_x ∈ G(W). 





 §.§ The analogue of a result of Madapusi Pera on special cycles



	Let  be an arbitrary [1/2]-algebra. Assume  is local. Let 𝐋 be a finite free -module equipped with the structure of a self-dual quadratic space over . By an isotropic line in 𝐋 we mean a direct summand of rank one on which the quadratic form is zero.  

We start with a general lemma on Clifford algebras.

	Let  and 𝐋 be as in Definition <ref>. Let C(𝐋) be the associated Clifford algebra. Let ξ∈𝐋 be an -generator of an isotropic line. Let (ξ) be the kernel of the endomorphism of C(𝐋) given by left multiplication by ξ. Then for any v∈𝐋, left multiplication by v preserves (ξ) if and only if v is orthogonal to ξ.  


	Assume v is orthogonal to ξ. Then vξ  = -ξ v, so v preserves  (ξ). 
	
	Conversely, assume v preserves (ξ). Write q for the quadratic form and , the corresponding bilinear pairing. Since ξ is a direct summand of 𝐋, there exists an -module homomorphism 𝐋→ sending ξ to 1. Since 𝐋 is self-dual, we know that there exists ζ∈𝐋 representing such a homomorphism. Namely we have 
    ζ, ξ = 1.
 
	It immediately follows that we have an -module direct sum 𝐋 = ξ^⊥⊕ζ. Replacing ζ by ζ - q(ζ)/2ξ, we may arrange that ζ is isotropic.  We have 
	
    q(ζ +ξ) = 2 ζ, ξ = 2,
 and in C(𝐋) we have 
	
    q(ζ +ξ) = ζξ + ξζ.
 Hence in C(𝐋) we have 
    ξζ + ζξ =2
Write 
    v = v_1 + λζ,
 with v_1 ∈ξ^⊥ and λ∈. By the first part of the proof we know that v_1 preserves (ξ). Therefore λζ preserves (ξ). Note that ξ∈ (ξ) as ξ is isotropic. It follows that, in C(𝐋), 
	
    0 λζ (ξ) ξ (λζ) ξ(<ref>)λ (2-ζξ) ξξ 2λξ.
 This is possible only when λ =0, and hence we have v= v_1 ∈ξ^⊥.


The next result is a Rapoport–Zink space analogue of <cit.> which is in the context of special cycles on  Shimura varieties. We only state a weaker analogue as it is sufficient for our need. The proof builds on loc. cit. too. We first introduce some definitions.


Denote by y_00 the distinguished k-point of  corresponding to 𝕏_0 and the identity quasi-isogeny. Let y_0 ∈ (k) be an arbitrary element. Let L be the special lattice corresponding to y_0 under (<ref>).  When y_0 = y_00, we have Φ L = V_W (cf. the discussion below (<ref>)). In this case define ^1  (Φ L) _k to be the one-dimensional subspace of V_k defined by the cocharacter μ of G_W and the representation G_k →(V_k). For general y_0, let g ∈ X_G, b ,μ^σ (k) be associated to y_0. Then Φ L = g V_W and g induces a map V_k → (Φ L)_k (cf. loc. cit.).  Define ^1 (Φ L) _k to be the image of ^1 V_k under the last map. 


By our explicit choice of μ in <ref>, the submodule _p x_2 in V is of weight 1 with respect to μ, and ⊕_1≤ i ≤ n, i≠ 2_px _i is of weight 0 with respect to μ, so ^1 V_k = k x_2.


	^1 (Φ L)_k is the orthogonal complement in (Φ L) _k of (L ∩Φ L) _k. However we will not need this description in the sequel. 



Let 𝒞 be the category defined as follows:

	
  * Objects in 𝒞 are triples (, → k, δ), where  is a local artinian W-algebra, → k is a W-algebra map, and δ is a nilpotent divided power structure on (→ k). 
	
  * Morphisms in 𝒞 are W-algebra maps that are compatible with the structure maps to k and the divided power structures.  

In the following we will abuse notation to write  for an object in 𝒞.


Let y_0 ∈(k) be an arbitrary element. Let 𝐯 be as in Definition <ref> such that the special cycle 𝒵: = 𝒵 (𝐯) contains y_0. In particular 𝐯⊂ L∩Φ L by Remark <ref>. Let _y_0 and 𝒵_y_0 be the formal completions of  and 𝒵 at y_0 respectively. 

For any ∈𝒞 there is a bijection  
    f_: _y_0 ()
 such that the following properties hold. Here we equip (Φ L)_ with the -bilinear form obtained by extension of scalars of the W-bilinear form on Φ L.

	
  * f_ is functorial in ∈𝒞 in the following sense. Let ' ∈𝒞 be another object of 𝒞 and let ϕ : →' be a morphism in 𝒞. Then we have a commuting diagram. 
	
    _y_0 () [d]^f_[r]    _y_0 (') [d]^f_'
     (f_ ) [r]     (f_')
 Here the top horizontal map is the natural map induced by ϕ, and the bottom horizontal map is given by base change along ϕ. 
	
  * f_ restricts to a bijection 
	 
    f_, 𝐯:  𝒵_y_0 ()

    




  
	
The existence and construction of the bijection f_ and the property (1) are consequences of <cit.> and the global construction of  in <cit.> using the integral model of the  Shimura variety. We explain this more precisely below.

 Consider  
    = _U_p U^p,
 the canonical integral model over _(p) of the Shimura variety associated to the  Shimura datum associated to a quadratic space V_ over , at a suitable level U^p away from p and a hyperspecial level U_p at p. See <cit.> or <cit.> for more details on this concept. By <cit.>, we may assume that the following package of data:

	
  * the Shimura datum associated to V_,
	
  * the Kuga-Satake Hodge embedding (cf. <cit.>) of the Shimura datum into a  Shimura datum,
	
  * the chosen hyperspecial level at p,
	
  * an element x_00∈_U^p_U^p U_p (k),
 
induces, in the fashion of <cit.>, the local unramified Shimura-Hodge datum that we used to define . Let  be the formal scheme over _p obtained from p-adic completion of , and let _W be the base change to W of . Then as in <cit.>, we have a morphism of formal schemes over W:
 
    Θ: →_W.
	
 We know that Θ maps y_00 to the k-point of _W induced by x_00. 
Moreover, let 
    x_0: = Θ(y_0) ∈ _W (k) = (k)
 and let U be the formal completion of  at x_0 (or, what amounts to the same thing, the formal completion of _W at x_0). By the construction of  in <cit.>, we know that Θ induces an isomorphism _y_0U. 

In <cit.>, two crystals 𝐇_, 𝐋_ are constructed on _k. (In fact <cit.> works over _p, but we always base change from _p to k.) Here 𝐇_ is by definition the first relative crystalline cohomology of the Kuga-Satake abelian scheme over _k in the sense of loc. cit.[See footnote <ref>.]The specialization of 𝐇_ over k via x_00 is identified with the Dieudonné module C_W, which is the covariant Diedonné module of the p-divisible group 𝕏_0 considered in this article (and <cit.>) and the contravariant Diedonné module of the Kuga-Satake abelian variety at x_00 considered in <cit.>.[Due to different conventions, the Kuga-Satake abelian scheme (and p-divisible group) considered by Madapusi Pera in <cit.> is different from that considered by Howard-Pappas in <cit.>. In fact they are dual to each other.]
 Moreover, the embedding V ↪__p (C) has a cristalline realization, which is a sub-crystal 𝐋_ of (𝐇_). For details see <cit.>. Among others, 𝐋_ has the following structures: 

	
  * Its specialization 𝐋_, x_0 to any x_0 ∈ (k), viewed as a W-module, has the structure of a W-quadratic space. 
	
  * 𝐋_, x_0⊗_W k contains a canonical isotropic line ^1 (𝐋_, x_0⊗_W k). 
	

By the definition of Θ and the definition of the parametrization of (k) by the affine Deligne-Lusztig set (cf. <cit.>), we know that when y_0 ∈ (k) corresponds to the special lattice L under (<ref>), the following statements are true: 

	
  *  There is an isomorphism of Dieudonné modules (gC)_W 𝐇_, x_0.
	
  *  There is a W-linear isometry (Φ L)_W 𝐋_,x_0 under which ^1 (Φ L)_k is identified with 
    ^1 (𝐋_, x_0⊗_W k).
 
	
  * We have a commutative diagram:
	
    Φ L  @^(->[r]  [d]    _W ((gC)_W) [d] 
    𝐋_, x_0@^(->[r]     _W(𝐇_, x_0),
 where 
        
 	
  * the right vertical map is induced by the map in <ref>.          
 	
  * the left vertical map is the map in <ref>. 
 	
  * the bottom horizontal map arises from the fact that 𝐋_ is a sub-crystal of (𝐇_).
 

In the rest of the proof we make the identifications in <ref> and <ref> above and omit them from the notation. Abbreviate 𝐇: = 𝐇_ ,x_0 and 𝐋 := 𝐋_, x_0.

Now in <cit.> Madapusi Pera constructs a bijection 

    U()   .

Moreover by the construction given in loc. cit. the above bijection is functorial in ∈𝒞. We define f_ as the above bijection precomposed with the isomorphism Θ: _y_0 U. 
 
 It remains to prove property (2). Note that 𝐇 = gC_W is the covariant Dieudonné module of the p-divisible group X_y_0 over k determined by y_0∈(k). Given y ∈ _y_0 () lifting y_0, by Grothendieck-Messing theory (for covariant Dieudonné modules) we know that y ∈𝒵_y_0 if and only if the image of 𝐯 in _ (𝐇_) stabilizes ^1 𝐇_⊂𝐇_, where ^1 𝐇_ is the Hodge filtration corresponding to the deformation from k to  of the X_y_0 determined by y. Now, as is stated in the proof of <cit.>[Madapusi Pera defines ^1 𝐇_ using the contravariant Grothendieck-Messing theory of the p-divisible group of the Kuga-Satake abelian scheme in his sense, which is the same as the covariant Grothendieck-Messing theory of the p-divisible group over U transported via Θ from the universal p-divisible group over  in the sense of Howard-Pappas.], we know that ^1 𝐇_ is the kernel in 𝐇_ of any -generator ξ of the isotropic line f_(y). Here ξ∈𝐋_ is viewed as an element of _(𝐇_). By Lemma <ref>, 𝐯 preserves ^1 𝐇_ = ξ if and only if 𝐯 is orthogonal to ξ (inside 𝐋_). Thus y ∈𝒵_y_0 if and only if f_ (y) is orthogonal to the image of 𝐯 in 𝐋_ = (Φ L) _.
  




	Consider the bijection f_, 𝐯 for =k. Since the source of this bijection is non-empty, it follows that ^1 (Φ L)_k is orthogonal to the image of 𝐯 in (Φ L) _k. This observation also follows from the Remark <ref> as 𝐯⊂ L ∩Φ L. 




 §.§ Reducedness of minuscule special cycles


	Let Λ be a _p-lattice in V_K^Φ with p^i Λ⊂Λ^∨⊂Λ for some i ∈_≥ 1. (Equivalently, Λ^∨ has invariant (r_1,⋯, r_n) such that i≥ r_1≥ r_2≥⋯≥ r_n ≥ 0.) Then the special cycle 𝒵(Λ^∨) defined by Λ ^∨ has no (W/p^i+1)-points. In particular, taking i=1 we see that _Λ (W/p^2) = ∅ for any vertex lattice Λ, or equivalently 𝒵 (𝐯) (W/p^2) = ∅ for any minuscule 𝐯. 


	Suppose there exists x∈𝒵(Λ^∨ ) (W/p^i+1). Let x_0 ∈𝒵(Λ^∨ ) (k) be induced by x under the reduction map W/p^i+1→ W/p =  k. Under (<ref>) x_0 determines a special lattice L. By Remark <ref>, Λ ^∨_W ⊂ L ∩Φ L. Note that W/p^i+1→ k is a surjection whose kernel admits nilpotent divided powers. By Theorem <ref>, the existence of the lift x of x_0 implies that there exists an isotropic line ℒ  (over W/p^i+1) in (Φ L)_ W/p^i+1 lifting ^1 (Φ L)_k and such that ℒ is orthogonal to the image of Λ^∨ in (Φ L)_W/p^i+1. Let l ∈Φ L be a lift of a generator of ℒ. Then l,λ∈ p^i+1 W for all λ∈Λ^∨. It follows that p^-(i+1) l ∈Λ_W. Hence p^-1  l ∈ p^i Λ_W ⊂ (Λ^∨)_W ⊂Φ L, i.e. l ∈ p Φ L. This contradicts with the fact that ℒ maps to a non-zero line in (Φ L)_k.    




  §.§.§ 

	Let u ∈ V_K^Φ -0. Suppose x_0 ∈𝒵(u)(k). Let T= k[ϵ]/ϵ^2 be the ring of dual numbers over k. We equip T with the map T→ k, ϵ↦ 0, which has its kernel (ϵ) admitting nilpotent divided powers (in a unique way). Thus Theorem <ref> can be applied to =T. 
	
	Let 𝒯_x_0_k and 𝒯_x_0𝒵(u)_k be the tangent spaces at x_0 to _k = ×_ W  k and to 𝒵(u) _k = 𝒵(u) × _ W k respectively. We will always take the point of view that 𝒯_x_0_k is the preimage of x_0 under the reduction map (T) →(k). Similarly for 𝒯_x_0𝒵(u) _k. We compute 𝒯_x_0_k and 𝒯_x_0𝒵(u)_k explicitly in the following. The result is given in Corollary <ref>. 
	
	Let L be the special lattice associated to x_0 under (<ref>). Since x_0 ∈𝒵(u) (k), we have u ∈ L ∩Φ L by Remark <ref>. Let u̅ be the image of u in (Φ L)_k. Let ^1 (Φ L)_k be as in Definition <ref>. By Remark <ref> we know that u̅ is orthogonal to ^1 (Φ L)_k. 
	
	 Define 𝒟 to be the set of isotropic lines in (Φ L)_T lifting ^1 (Φ L)_k. Define 𝒟_u to be the subset of 𝒟 consisting of lines which are in addition orthogonal to the image of u in (Φ L)_T. Let 
	 
    𝒢= f_T: 𝒯_x_0_k 𝒟.
 be the bijection given in Theorem <ref>. By the same theorem it restrict to a bijection 
	
    𝒯_x_0𝒵(u)_k 𝒟_u 
    .
	
	
 We identify (Φ L)_T with (Φ L)_k ⊗ _k T. Fix a k-generator v_0 of ^1 (Φ L)_k.  
 Define a map 
    ℱ̃: (Φ L)_k  → (Φ L)_T
 
    w↦span_T  v_0 ⊗_k 1 + w⊗_k ϵ .
  
	
	
		ℱ̃ factors through (Φ L)_k / ^1 (Φ L )_k, and its image consists of T-module direct summands of (Φ L)_T of rank one.
	
	
		For any λ∈ k, we have 
		
    v_0 ⊗ 1 + (w+λ v_0)⊗ϵ = (1+ λϵ ) (v_0 ⊗ 1 + w⊗ϵ),
 and 1+λϵ∈ T^×. Hence ℱ̃ factors through (Φ L)_k / ^1 (Φ L) _k. For any w∈ (Φ L)_k, we know that ℱ̃ (w) is a free module of rank one by definition. It remains to show that ℱ̃ (w) is a direct summand of (Φ L)_T. Let A be a k-vector space complement of ^1(Φ L) _k inside (Φ L)_k. We easily check that the following T-submodule of (Φ L)_T is a T-module complement of ℱ̃(w): 
		
    span_T v' ⊗ 1 + w⊗ϵ | v' ∈ A.

	
	 The map ℱ̃ induces a bijection of sets: 
		
    ℱ: (  ^1 (Φ L)_k)^⊥ / ^1 (Φ L)_k  𝒟.
 Moreover, ℱ restricts to a bijection 
    u̅,  ^1 (Φ L)_k ^⊥ /^1 (Φ L)_ k 𝒟_u.

	
	Since v_0,v_0 = 0 ∈ k, the condition that ℱ̃(w) is isotropic is equivalent to w, v_0 = 0 ∈ k. Since v_0 is orthogonal to u̅, the condition that ℱ̃(w) is orthogonal to the image of u in (Φ L)_T is equivalent to w , u̅ = 0 ∈ k. 

 
Let 𝒢 be as in (<ref>) and let ℱ be as in Corollary <ref>. The map 
    𝒢^-1∘ℱ: (^1 (Φ L)_k)^⊥/^1 (Φ L)_k →𝒯_x_0_k
  is k-linear. 


	The proof is a routine check, using the functorial property stated in Theorem <ref>.
	
	We first recall the k-vector space structure on 𝒯_x_0_k, from the point of view that 𝒯_x_0_k is the preimage of x_0 under the map (T) →(k). 
	
Scalar multiplication: Given a tangent vector v ∈𝒯_x_0_k corresponding to v_T ∈(T) and given a scalar λ∈ k, the tangent vector λ v corresponds to the following element (λ v)_T of (T): the image of v_T under (T) (T). We see that (λ v)_T is indeed a preimage of x_0.

Addition: Let v_1, v_2∈𝒯_x_0_k be two tangent vectors. Let T_i = k[ϵ_i]/ϵ _i ^2, i=1,2 be two copies of T. We represent v_i as an element (v_i)_T_i in (T_i) that reduces to x_0 ∈(k), for i=1,2. Let T̃ be the fiber product of T_1 and T_2 over k, in the category of k-algebras. Namely, T̃ = k[ϵ_1,ϵ_2]/(ϵ_1,ϵ_2) ^2. Let δ be the k-algebra map 

    δ : T̃→ T,  ϵ_1↦ϵ, ϵ_2 ↦ϵ.
  
  By the fact that T̃ is the fiber product of T_1 and T_2, there is a canonical bijection 
 
    (T_1) ×(T_2) (T̃).

Denote by v_1 +̃ v_2 the image of ((v_1)_T_1, (v_2) _T_2 ) in (T̃ ) under the above bijection. Then the tangent vector v_1+v_2 corresponds to the following element (v_1+v_2)_T of (T): the image of v_1 +̃ v_2 under δ_*: (T̃) →(T). This last element is indeed a preimage of x_0. 
 
 We now check that 𝒢^-1∘ℱ is k-linear. We first check the compatibility with scalar multiplication. For any λ∈ k and w∈(  ^1 (Φ L)_k)^⊥, we have ℱ̃ (w) = span_Tv_0 ⊗ 1 + w⊗ _k ϵ and ℱ̃ (λ w) = span_Tv_0 ⊗ 1 + λ w⊗ _k ϵ. Let m_λ denote the map T→ T, ϵ↦λϵ. Then we have ℱ̃ (w) ⊗ _T,m_λ T = ℱ̃ (λ w) as submodules of (Φ L)_T. By the functoriality in  stated in Theorem <ref>, we know that for all d∈𝒟, the element 𝒢 ^-1 (  d⊗ _T,m_λT )∈(T) is equal to the image of 𝒢^-1 (d) under (T) (T). It follows that (𝒢^-1∘ℱ)(λ w) is equal to λ times the tangent vector (𝒢^-1∘ℱ)( w).
 
 We are left to check the additivity of 𝒢^-1∘ℱ. Let w_1, w_2 ∈ (  ^1 (Φ L)_k)^⊥. Let 𝒟_i, ℱ_i,𝒢_i be the analogues of 𝒟, ℱ, 𝒢 respectively with T replaced by T_i, for i=1,2. Also let f_T̃ be as in Theorem <ref> (with = T̃, where (T̃→ k) is equipped with the unique nilpotent divided power structure.) Let d_i : = ℱ_i (w_i),   i =1,2. Then d_i = span_T_i (v_0⊗ 1 + w_i ⊗ϵ_i).
  We easily see that the assertion (𝒢^-1∘ℱ) (w_1 +w_2) = (𝒢^-1∘ℱ)(w_1) + (𝒢^-1∘ℱ)(w_2)
follows from the following claim:

Claim. Under (<ref>), the element 
(𝒢_1^-1 (d_1),𝒢_2^-1 (d_2) )
 is sent to the element 
 
    f_T̃^-1 (span_T̃v_0⊗ 1 + w_1⊗ϵ_1 + w_2⊗ϵ_2).


 
 We now prove the claim. Let d̃ be such that the element (𝒢_1^-1 (d_1),𝒢_2^-1 (d_2) ) is sent under (<ref>) to f_T̃^-1 (d̃).  Thus d̃ is an isotropic line in (Φ L) _T̃. By the functoriality stated in Theorem <ref> and the functorial definition of (<ref>), we see that d̃ is characterized by the condition that d̃⊗ _T̃ T_i = d_i,   i = 1,2, where the tensor product is with respect to the the structure map T̃→ T_i expressing T̃ as the fiber product of T_1,T_2 (i.e. reduction modulo ϵ_j for j ≠ i). Using this characterization of d̃, we see that d̃ is as predicted in the claim.


	The tangent space 𝒯_x_0_k is isomorphic to 
    (^1 (Φ L)_k)^⊥/^1 (Φ L)_k.
 Under this isomorphism, the subspace 𝒯_x_0𝒵(u)_k of 𝒯_x_0_k is identified with 
	
    u̅,  ^1 (Φ L)_k ^⊥ /^1 (Φ L)_ k.



	This follows from Corollary <ref>, Lemma <ref>, and the bijectivity of 𝒢^-1 asserted in Theorem <ref>. 


	Let Λ⊂ V^Φ _K be a vertex lattice. Let L be a self-dual W-lattice in V_K such that Λ^∨_W ⊂ L ⊂Λ_W. Let A be the image of Λ^∨_W in L_k. Then the following statements hold.
	
		
  * _k Λ_W/ L =  _k L /Λ_W^∨. Here both spaces are vector spaces over k because p Λ_W ⊂Λ_W^∨⊂ L and p L ⊂ p Λ_W ⊂Λ_W^∨.  
		
  * A ⊃ A^⊥. Here A^⊥ is the orthogonal complement of A in L_k.
	


	(1) Consider the W-bilinear pairing 
    Λ_W ×Λ_W → W
 
    (x,y) ↦ p x,y,
 where , is the K-bilinear form on V_K^Φ⊗ __p K = V_K. We get an induced k-quadratic space structure on Λ_W/Λ_W^∨. The image of L in Λ_W/Λ_W^∨ is equal to the orthogonal complement of itself, i.e. it is a Lagrangian subspace. Claim (1) follows.
	
	(2) By definition A^⊥ is the image in L_k of the W-submodule pΛ_W^∨∨ = p Λ_W of L.  We have pΛ_W ⊂Λ_W^∨, so A^⊥ lies in the image of Λ_W^∨ in L_k, which is A.   


 Let Λ⊂ V_K^Φ be a vertex lattice of type t (so t≥ 2 is even). For all x_0 ∈_Λ (k), we have 
	
    _k 𝒯_x_0_Λ,k  = t/2-1.


 
	Let L be the special lattice associated to x_0 under (<ref>), and let ^1 (Φ L)_k be as in Definition <ref>. Then Λ^∨ _W ⊂ L ∩Φ L. Denote by A the image of Λ_W^∨ in (Φ L)_k. Then A is orthogonal to ^1 (Φ L)_k by Remark <ref>. By Corollary <ref>, we have an isomorphism of k-vector spaces 
    𝒯_x_0_Λ,k≅A, ^1 (Φ L)_k ^⊥ / ^1 (Φ L)_k .
  Since A is orthogonal to ^1 (Φ L)_k, we have A ⊃^1 (Φ L)_k by Lemma <ref> applied to the self-dual W-lattice Φ L. Therefore 𝒯_x_0_Λ,k≅ A^⊥ / ^1 (Φ L)_k. Since the bilinear pairing on (Φ L)_k is non-degenerate, we have _k 𝒯_x_0_Λ,k = _k (Φ L)_k - _k A - 1 = _k (Φ L/Λ_W^∨) -1. By claim (1) in Lemma <ref> (applied to Φ L), we have _k (Φ L/Λ_W^∨) = t/2.




	Let Λ⊂ V^Φ _K be a vertex lattice. The formal scheme _Λ×_ W k is regular.

Let t be the type of Λ.	Denote X: = _Λ ^red and Y: = _Λ×_ W k. Then X is a formal subscheme of Y over k. Recall from <ref> that X is a smooth k-scheme of dimension t/2-1. It follows that for all x_0 ∈ Y(k), the complete local ring of Y at x_0 is of dimension ≥ t/2-1. By Proposition <ref>, the tangent space of Y at x_0 has k-dimension equal to t/2 -1. Hence Y is regular at x_0.        



	Let Λ⊂ V^Φ _K be a vertex lattice. Then _Λ = _Λ^red and is of characteristic p.


 _Λ does not admit W/p^2-points (Proposition <ref>) and its special fiber is regular (Corollary <ref>). It follows from <cit.> that _Λ is equal to its special fiber. Being regular itself, _Λ is reduced.




§ THE INTERSECTION LENGTH FORMULA





 §.§ The arithmetic intersection as a fixed point scheme

Recall from <ref> that we are interested in computing the intersection of ^g and δ(^♭), for g∈ J_b (_p).


	Assume g∈ J_b(ℚ_p) is regular semisimple. Then δ(^♭)∩^g is contained in 𝒵(𝐯(g)), where 𝐯 (g) = ( x_n, gx_n,⋯, g^n-1 x_n).



	By Lemma <ref>, we have δ(^♭)⊆𝒵(x_n). Hence δ(^♭)∩^g⊆𝒵(x_n)∩^g⊆𝒵(gx_n) by the definition of special cycles. Repeating this procedure we obtain 
    δ(^♭)∩^g⊆𝒵(x_n)∩𝒵(gx_n)∩⋯∩𝒵(g^n-1x_n)=𝒵(𝐯(g)).
 


Assume g∈ J_b(_p) is regular semi-simple and minuscule. Then 
    δ(^♭)∩^g ⊂ _L(g) ^∨ = _L(g) ^∨ ^red.
 In particular, δ(^♭)∩^g is a scheme of characteristic p.


	The first statement is an immediate consequence of Remark <ref>, Theorem <ref>, and Proposition <ref>. Now both δ (^♭) and ^g are closed formal subschemes of , so δ (^♭)∩^g is a closed formal subscheme of the scheme _L(g) ^∨ = _L(g) ^∨ ^red of characteristic p. Hence δ (^♭)∩^g is its self a scheme of characteristic p. 



  §.§.§ 

In the rest of this section we will fix g ∈ J_b(_p) regular semisimple and minuscule, and assume ^g ≠∅. Take Λ:=L(g)^∨. Then Λ is a vertex lattice stable under g, cf. Remark <ref>. We are interested in computing the intersection length of δ(^♭) and ^g around a k-point of intersection. Recall the isomorphism (<ref>) between p^\_Λ ^red (which we now know is just p^\_Λ) and S_Λ. Recall from <ref> that S_Λ is a projective smooth variety over k of dimension t_Λ /2 -1. We write d = t_Λ/2. Let Ω_0 = Λ / Λ^∨ and Ω = Ω_0 ⊗ __p  k = Λ_W / Λ_W^∨. Let , be the k-bilinear form on Ω (cf. <ref>). Let 𝔾 = (Ω) , 𝔾_0 = (Ω_0). Let g̅ be the induced action of g on Ω. Then g̅∈𝔾_0(_p) ⊂𝔾(k). Recall that 
    S_Λ (k) = ℒ⊂Ω,  (ℒ + Φℒ) =d+1

    = (ℒ_d-1 ,ℒ_d) | ℒ_d ⊂Ωℒ_d-1 = d-1, ℒ_d-1⊂ℒ_d ∩Φℒ_d .


There is a natural action of g̅ on S_Λ via its action on Ω. On R-points g̅ sends (ℒ_d-1, _d) to (g̅_d-1, g̅_d). The latter is indeed a point of S_Λ because g̅Φ = Φg̅ by the fact that g̅∈𝔾_0 (_p). The following proposition allows us to reduce the study of intersection multiplicities to the study of the non-reduced structure of S_Λ ^g̅. 


 p^ℤ\(δ (^♭)∩^g)≅ S_Λ^g̅.



In view of Theorem <ref>, Corollary <ref> and the observation that the isomorphism (<ref>) induces an isomorphism p^\ (_Λ^red )^g  S_Λ ^g̅, it suffices to show 

    (p^\_Λ^red ) ∩  ( p^\δ(^♭)) = (p^\_Λ^red ).
 Since both p^\_Λ^red and p^\δ(^♭) are closed formal subschemes of p^\ and since p^\_Λ^red is a reduced scheme, it suffices to check that 
    p^\_Λ ^red (k) ⊂ p^\δ (^♭) (k).
 Now the left hand side consists of special lattices L containing Λ^∨, and the right hand side consists of special lattices L containing x_n (cf. (<ref>) and Lemma <ref>). We finish the proof by noting that by definition x_n ∈Λ^∨ = L(g). 





Proposition <ref> reduces the intersection problem to the study of S_Λ^g̅. 



 §.§ Study of S_Λ^g̅


We continue to use the notation in <ref>. 
We adopt the following notation from <cit.>.

	Let (d-1) (resp. (d)) be the moduli space of totally isotropic subspaces of Ω of dimension d-1 (resp. d). For a finite dimensional vector space W over k and an integer l with 0≤ l ≤ W, we write (W,l) for the Grassmannian classifying l-dimensional subspaces of W. Thus for j ∈d-1, d and any k-algebra R, we have 
	
    (j) (R) = .

	Also 
    (W,l) (R) = .

	Let (d-1, d) ⊂(d-1) ×(d) be the subscheme defined by the incidence relation. Namely, it is the locus where the (d-1)-dimensional isotropic subspace is contained in the d-dimensional isotropic subspace. 

Recall from loc. cit. that (d-1,d) has two connected components (d-1, d) ^±, and the projection to the first factor (d-1, d) →(d-1) restricts to isomorphisms 
    (d-1, d)^+ (d-1)
 
    (d-1, d)^- (d-1).

Also recall from loc. cit. that the natural morphism S_Λ↪(d-1, d) is a closed embedding, and S_Λ^± : = S_Λ∩(d-1, d) ^± give the two connected components of S_Λ. In the following, we denote by S_Λ^0


Following the notation of <cit.>, we have parabolic subgroups P_0 , P^+, P^- of 𝔾. Here P_0 is the stabilizer of the standard (d-1)-dimensional isotropic subspace e_1,⋯, e_d-1, and P^+ (resp. P^-) is the stabilizers of the standard d-dimensional isotropic subspace e_1,⋯, e_d (resp. e_1,⋯, e_d-1, f_d). We have P_0  = P^+ ∩ P^-. The Frobenius Φ interchanges P^± and stabilizes P_0. Let (d-1) (resp. (d)) be the moduli space of isotropic subspaces of Ω of dimension d-1 (resp. d). Using e_1,⋯, e_d-1 one identifies (d-1) with 𝔾/P_0. Using e_1,⋯, e_d (resp. e_1,⋯, e_d-1, f_d) one identifies (d) with 𝔾/P^+ (resp. 𝔾/P^-).

Consider 
    i^+ =  (π ^+ ,π^-) : 𝔾/ P_0 →𝔾/P^+ ×_k 𝔾/P^-
 
    g↦ (g,g)
 which is a closed embedding. We let Γ^+_Φ⊂𝔾/P^+ ×𝔾/P^- be the graph of Φ: 𝔾/ P^+ →𝔾/ P^-. Then by loc. cit. we have 
    S_Λ ^+      (i^+) ^-1Γ_Φ ^+ ⊂𝔾/P_0
     S_Λ∋ℒ = h e_1,⋯, e_d     h


 Under our identifications 𝔾/P_0 = (d-1) and 𝔾/P^± =  (d), the morphism i^+  : /P_0 →𝔾/P^+ ×_k 𝔾/P^- is equivalent to the morphism (d-1) → (d) ×_k (d) which sends ℒ_d-1∈ (d-1) to the pair (ℒ_d^+ ,ℒ_d^-), such that the two Lagrangians ℒ_d^+, ℒ_d^- are the (unique two) Lagrangians containing ℒ_d, and such that there exists g∈𝔾 such that g e_1,⋯, e_d = ℒ_d^+, ge_1,⋯, e_d-1 , f_d = ℒ_d^-. (A priori exactly one of the two possible orderings (ℒ_d^+ ,ℒ_d^-) and (ℒ_d^- ,ℒ_d^+) satisfies the last condition.) See <cit.> for more details.



Suppose x_0 ∈ S_Λ ^+ (k) corresponds to (ℒ_d-1, ℒ_d). We have 
    𝒯 _x_0 S_Λ ^+ = 𝒯_e  ( Stab _Φℒ_d𝔾 / Stab _ℒ_d-1𝔾).
 
Suppose in addition that x_0 is fixed by g̅. Then g̅∈Stab _ℒ_d-1𝔾. We have

    𝒯_x_0 ( S_Λ ^+)^g̅ = 𝒯_e (Stab _Φℒ_d𝔾 / Stab _ℒ_d-1𝔾)^g̅,

where the action of g̅ on Stab _Φℒ_d𝔾 / Stab _ℒ_d-1𝔾 is through left multiplication or conjugation, which are the same.

 
 	Through (<ref>) we regard S_Λ ^+ as a sub-variety of 𝔾/ P_0, and write points of it as [h], h∈𝔾. 
 	Assume x_0 = [h_0] ∈ S_Λ^+ (k), where h_0 ∈𝔾(k). We identify the tangent space of 𝔾/P_0 at [h_0] with 𝔾 /  P_0 using left multiplication by h_0. We compute 𝒯_x_0 S_Λ^+ as a subspace of 𝔾/ P_0. Because the tangent map of Φ: 𝔾/ P^+ →𝔾/ P^- is zero, by (<ref>) we have 
    𝒯_x_0 S_Λ ^+ = X∈𝔾/ P_0|  Im (X) = 0 ∈𝔾/  P^- =  P^- /  P_0.

 	
 	Now suppose that x_0 is fixed by g̅. Then we have 
 	
    h_0^-1g̅ h_0 =: p_0 ∈ P_0(k).
 
 	Under (<ref>), the tangent action of g̅ on the 𝒯_x_0 S_Λ ^+ corresponds to the action of (p_0) on P^- /  P_0. Consequently the tangent space of (S_Λ ^+)^g̅ at x_0 is
 	
    𝒯_x_0 (S^+_Λ) ^g̅ = (  P^-/  P_0)^ p_0.
 
 	Now by the identifications we have made, the formulas (<ref>) and (<ref>) are respectively equivalent to (<ref>) and (<ref>).
 
 
We now study the right hand sides of (<ref>) and (<ref>).

 
	Let x_0 = (ℒ_d-1 ,ℒ_d) ∈ S_Λ ^+ (k). The Grassmannian (Φℒ_d , d-1), with the natural action by Stab _Φℒ_d𝔾 and with the distinguished point ℒ_d-1∈ (Φℒ_d , d-1)(k), realizes the quotient Stab _Φℒ_d𝔾 / Stab _ℒ_d-1𝔾. 

Since Φℒ_d is a Lagrangian subspace of Ω, we know that the algebraic group Stab _Φℒ_d𝔾 acts on (Φℒ_d , d-1) through its Levi quotient group (Φℒ_d). Moreover the isotropic subgroup at ℒ_d-1∈ (Φℒ_d , d-1) is equal to Stab _ℒ_d-1𝔾. Now (Φℒ_d) is a quotient of Stab _Φℒ_d𝔾, and  (Φℒ_d , d-1) is a quotient of (Φℒ_d). The lemmas follows, for instance by <cit.>.



	Let x_0 = (ℒ_d-1 ,ℒ_d) ∈ S_Λ ^+ (k). Then we have an identification
	
    𝒯_x_0  S_Λ ^+ ≅ (ℒ_d-1, Φℒ_d / ℒ_d-1).
 Suppose in addition that x_0  ∈ S_Λ ^+ (k)^g̅. Then g̅∈𝔾 stabilizes ℒ_d, Φℒ_d, and ℒ_d-1. We have an identification  
	
    𝒯_x_0 ( S_Λ ^+ )^g̅≅ (ℒ_d-1, Φℒ_d / ℒ_d-1)^g̅.



	This follows from Proposition <ref> and Lemma <ref>.








 

Let x_0 = (ℒ_d-1 ,ℒ_d) ∈ S_Λ ^+ (k)^g̅. Let λ, c be as in Definition <ref>. Then the tangent space 𝒯_x_0 (S_Λ ^+) ^g̅ has dimension at most one over k. It has dimension one if and only if c > 1. 


By Corollary <ref>, we have 
	
    𝒯_x_0 ( S_Λ ^+ )^g̅≅ (ℒ_d-1, Φℒ_d / ℒ_d-1)^g̅≅ (ℒ_d-1 ^*) ^λ h^t,

	where 
    h: = (g̅ |_ℒ_d-1 )^-1∈(ℒ_d-1).
 Namely, 𝒯_x_0 ( S_Λ ^+ )^g̅ is the eigenspace of eigenvalue λ^-1 of h^t acting on ℒ_d-1 ^*. Since g̅|_Φℒ_d has the property that to each eigenvalue there is at most one Jordan block, the same holds for g̅|_ℒ_d-1, h, and h^t. Moreover λ^-1 is an eigenvalue of h^t if and only if λ is an eigenvalue of g̅|_ℒ_d-1, if and only if c>1. It follows that the eigenspace of eigenvalue λ^-1 of h^t has dimension at most one, and it has dimension one if and only if c>1.

Next we study the local lengths of (S_Λ ^+)^g̅.

 
 
 	If A is a finite dimensional k-vector space, we write A for the affine space over k defined by A. Thus for a k-algebra R we have A (R) = A⊗_k R. 
 
 
 	Let V_1⊂ V_2 ⊂ V_3 be k-vector spaces. Let A be a subspace of (V_1, V_3) and B be a subspace of (V_2, V_3). Write (A× B)^comp for the subspace of A× B consisting of compatible elements, i.e. elements (ϕ,ψ) ∈ A× B such that ψ|_V_1 = ϕ. Write A×^compB for (A× B)^comp. 
 
 Let ℒ_d,ℳ_d be Lagrangian subspaces of Ω such that Ω = ℒ_d ⊕ℳ_d. we write _ (ℒ_d, ℳ_d) for the space of anti-symmetric k-linear maps ℒ_d →ℳ_d. Here we say ϕ: ℒ_d →ℳ_d is anti-symmetric if the bilinear form _d ×_d → k, (x,y) ↦x, ϕ y is anti-symmetric.
 	 
 	 
 	 

  §.§.§ 

Recall that in general, if A is a finite dimensional vector space over k and B is a subspace, then we can construct a Zariski open of the Grassmannian (A, B) as follows. Choose a subspace C of A such that A = B ⊕ C. Then there is an open embedding ι_B,C: _k (B, C)→ (A,  B) which we now describe. For any k-algebra R and any R-point ϕ of _k(B,C), we view ϕ as an element of _k(B, C) ⊗ R =  _R(B⊗ R, C⊗ R). Then ι_B,C maps ϕ to the R-point of (A,  B) corresponding the following R-submodule of A: 

    x+ ϕ(x)| x ∈ B⊗ R.

For details see for instance <cit.>. In the following we will think of _k(B,C) as a Zariski open of (A, B), omitting ι_B,C from the notation.
 	 
 	 
 
 Let ℒ_d,ℳ_d be complementary Lagrangian subspaces of Ω over k. Then 
    (d) × _(Ω, d) (_d, ℳ_d) = _ ( ℒ_d ,ℳ_d) .
 In particular, the k-point _d in (d) has an open neighborhood of the form _ ( ℒ_d ,ℳ_d).
 
 
 	Let R be a k-algebra and ϕ an R-point of (_d, ℳ_d). Then the submodule (<ref>) (for B = _d) is Lagrangian if and only if for all x∈ B⊗ R, 
    x+ ϕ (x) , x + ϕ(x) =0.
 But we have x , x = ϕ (x), ϕ(x) = 0 since _d⊗ R and ℳ_d ⊗ R are both Lagrangian. Hence (<ref>) is Lagrangian if and only if x, ϕ(x) =0 for all x∈ℳ_d ⊗ R.
 	  


  §.§.§ 

It follows from the assumptions we made on g̅∈𝔾(k) in <ref> that its characteristic polynomial on Ω is equal to its minimal polynomial on Ω (cf. Remark <ref>). In general this property is equivalent to the property that in the Jordan normal form all the Jordan blocks have distinct eigenvalues. From now on we let x_0  = (ℒ_d-1 ,ℒ_d) ∈ S_Λ (k) be an element fixed by g̅. Then Φℒ_d ⊂Ω is also stable under g̅. If we identify Ω/ Φℒ_d with (Φℒ_d)^* (the k-vector space dual) using the bilinear form on Ω, the action of g̅ on Ω/ Φℒ_d is equal to the inverse transpose of g̅|_Φℒ_d. It follows that the minimal polynomial (resp. characteristic polynomial) of g̅ on Ω is equal to the minimal polynomial (resp. characteristic polynomial) of g̅|_Φℒ_d times its reciprocal. Hence g̅|_Φℒ_d has equal minimal and characteristic polynomial, too.


	Let λ be the (nonzero) eigenvalue of g̅ on the one-dimensional Φℒ_d/ℒ_d-1. Let c be the size of the unique Jordan block of eigenvalue λ of g̅|_Φℒ_d.  

 

  §.§.§ 

Let x_0 = (ℒ_d-1 ,ℒ_d ) ∈ S_Λ (k)^g̅ as in <ref>. Define 

    Y: = (Φ_d, d-1 ) × _k  (d).

 Let ℐ⊂ Y be the sub-functor defined by the incidence relation, i.e. for a k-algebra R
  
    ℐ (R) =  (_d-1 ' , _d') ∈ (Φ_d, d-1)  (R) × (d) (R)  | _d-1' ⊂'_d.

   The pair (_d-1 ,_d ) defines a k-point in ℐ, which we again denote by x_0. It is well known that the incidence sub-functor of (Φ_d, d-1) ×(Ω, d) is represented by a closed subscheme, and it follows that ℐ is a closed subscheme of Y.

Since x_ 0 = (_d-1, _d) ∈ S_Λ (k) is fixed by g̅, we have a natural action of g̅ on Y, stabilizing ℐ and fixing x_0 ∈ℐ. Let 
    ℛ̃: = _ℐ ,x_0,  ℛ: = _ℐ^g̅, x_0,  𝒮̃: = _S_Λ , x_0,  𝒮: = _S_Λ^g̅ , x_0
 be the local rings at x_0 of ℐ, ℐ^g̅, S_Λ, S_Λ^g̅ respectively. Let 
    ℛ̃_p : = ℛ̃ / 𝔪_ℛ̃^p,  ℛ_p : = ℛ/𝔪_ℛ^p,  𝒮̃_p : = 𝒮̃/𝔪_𝒮̃^p,  𝒮_p: = 𝒮/𝔪_𝒮^p

be the above four local rings modulo the p-th powers of their respective maximal ideals.

The following lemma expresses the observation that ℐ^g̅ may serve as a model for S_Λ^g̅ locally around x_0. 

		
		
  * There is a k-algebra isomorphism ℛ̃ _p ≅𝒮̃ _p, equivariant for the g̅-action on both sides. 
	
  * There is a k-algebra isomorphism ℛ_p ≅𝒮_p. 
	 

	We first show (1). Let (_d-1' , _d') be the tautological pair over 𝒮̃ _p for the moduli problem S_Λ, and let (_d-1” , _d”) be the tautological pair over ℛ̃ _p for the moduli problem ℐ. Note that 
    Φ_d' = (Φ_d) ⊗𝒮̃ _p
 as submodules of Ω⊗_k 𝒮̃ _p because Φ: 𝒮̃ _p →𝒮̃ _p factors through the reduction map 𝒮̃ _p → k. It follows that (_d-1',_d') defines a point in ℐ (𝒮̃_p) lifting x_0 ∈ℐ(k). Similarly, 
    Φ_d” = (Φ_d) ⊗ℛ̃ _p
 as submodules of Ω⊗_k ℛ̃ _p, and hence (_d-1”,_d”) defines a point in S_Λ (ℛ̃_p) lifting x_0 ∈𝒮_Λ(k). The point in ℐ(𝒮̃_p) and the point in S_Λ (ℛ̃ _p) constructed above give rise to inverse k-algebra isomorphisms between ℛ̃ _p and 𝒮̃ _p, which are obviously g̅-equivariant. 
	
	(2) follows from (1), since ℛ_p (resp. 𝒮_p) is the quotient ring of ℛ̃_p (resp. 𝒮̃ _p) modulo the ideal generated by elements of the form r - g̅· r with r∈ℛ̃_p (resp. r∈𝒮̃_p).   
 

	The above proof also shows that _S_Λ ^+, x_0/𝔪_x_0^l is isomorphic to _ℐ , x_0 /𝔪 _x_0 ^l for 1≤ l ≤ p. In particular, S_Λ ^+ and ℐ have isomorphic tangent spaces at x_0. By Lemma <ref>, we see that the tangent space of ℐ at x_0 is isomorphic to ( (_d-1, span_k (w_d)) ×_ (_d,  span_k (w_1,⋯, w_d)))^comp. The last vector space is isomorphic to (_d-1, span_k (w_d)) via projection to the first factor. From this we recover Corollary <ref>. 





 §.§ Study of ℐ^g̅

Next we study ℐ^g̅ by choosing certain explicit coordinates on ℐ. Choose a k-basis v_1,⋯, v_d, w_1,⋯ , w_d of Ω, such that 
 
 	
  * _d-1 is spanned by v_1,⋯, v_d-1.
 	
  * _d is spanned by v_1,⋯, v_d.
 	
  * Φ_d is spanned by v_1,⋯, v_d-1 , w_d.
 	
  * v_i, v_j = w_i , w_j = 0, v_i,w_j = δ_ij.
 
 We will denote 
    v̂_i : = 
     v_i,  1≤ i≤ d-1 
     
     w_d,   i=d

 Also denote 
 
    ℳ_d : = span_k (w_1,⋯, w_d).


 For 1≤ i ≤ d-1, define an element ϕ_i ∈ (_d-1 ,  span_k (w_d) ) by 
 
    ϕ_i (v_j) = δ_ij  w_d.

 Then ϕ_1,⋯, ϕ_d-1 is a basis of (_d-1 ,  span_k (w_d) ).


By <ref> and Lemma <ref>, there is a Zariski open neighborhood of x_0 in Y, of the form 
    𝒰: =  (_d-1 ,   span_k (w_d) )×_ (_d,ℳ_d) .

 

	
		
  * Let R be a k-algebra. Let y ∈𝒰(R), corresponding to 
    (ϕ,ψ) ∈ (_d-1 ,       span_k(w_d))⊗ R ⊕_ (_d, ℳ_d) ⊗ R.
 We view ϕ∈_R(_d-1⊗ R, span_R (w_d)) and ψ∈ _R (_d ⊗ R, ℳ_d ⊗ R). Then y is in ℐ if and only if ψ|__d-1⊗ R = ϕ. 
		
  * The projection to the first factor 𝒰→ (_d-1 ,span_k (w_d)) restricts to an isomorphism 
		
    𝒰∩ℐ (_d-1 ,span_k (w_d)).

	

 

	(1) We know that y is in ℐ if and only if for all v∈_d-1⊗ R, there exists v'∈_d⊗ R, such that 
    v+ ϕ(v)  = v'+ ψ(v')
 as elements of Ω⊗ R. Decompose v' = v'_1 + v'_2 with v'_1 ∈_d-1⊗ R and v'_2 ∈span_R (v_d). Then the above equation reads 
	
    v-v'_1  = v'_2 + (ψ(v') - ϕ(v)).
 Since v-v'_1 ∈_d-1⊗ R,  v_2' ∈span_R (v_d),  ψ(v')-ϕ(v) ∈ℳ_d ⊗ R, the above equation holds if and only if v= v_1' ,   v_2' = 0,  ϕ(v) = ψ(v). 
	Hence y∈ℐ if and only if for all v ∈_d-1⊗ R we have ψ(v) = ϕ(v). This proves (1). 
	
	(2) By (1), we know that 𝒰∩ℐ is the affine subspace of 𝒰 associated to the linear subspace of 
    (_d-1 , span_k (w_d)) ×_ (_d, ℳ_d)
 consisting of pairs (ϕ,ψ) such that ψ|__d-1 = ϕ. Call this subspace A. We only need to show that projection to the first factor induces an isomorphism A  (_d-1 ,span_k  (w_d)).
	
	 Note that if ψ∈_ (_d, ℳ_d), then ψ is determined by ψ|__d-1. This is because for each 1≤ i ≤ d, we have 
    ψ v_d, v_i = 
    - v_d, ψ v_i ,    i ≤ d-1
    
    0,   i =d
 which means that ψ(v_d) is determined by ψ|__d-1. Conversely, given ϕ∈ (_d-1 , span_k (w_d)) we can construct ψ∈_ (_d, ℳ_d) such that ψ|__d-1 = ϕ as follows. For 1≤ j ≤ d-1, define ψ  (v_j) to be ϕ(v_j). Define ψ(v_d) to be the unique element of ℳ_d satisfying (<ref>). In this way we have defined a linear map ψ: _d→ℳ_d such that ψ|__d-1 = ϕ. We now check that ψ is anti-symmetric. We need to check that for all 1≤ i ≤ j ≤ d, we have ψ v_j , v_i = - ψ v_i, v _j. If j = d, this is true by (<ref>). Suppose j<d. Then ψ v_j , v_i = ψ v_i, v_j = 0 because ψ v_j , ψ v_i ∈span_k (w_d) and w_d, _d-1 = 0. Thus ψ is indeed antisymmetric. It follows that A  (_d-1, span_k (w_d)).

  
 

  §.§.§ 

 Now assume that x_0 ∈ S_Λ^g̅(k)  . Then g̅ stabilizes _d, Φ_d, _d-1. The natural action of g̅ on Y does not stabilize 𝒰 in general, but we have a natural identification 
 
    g̅·𝒰≅ (_d-1, span_k (g̅ w_d))×_ (_d, g̅ℳ_d)⊂ Y.

 Let R be a k-algebra. If y ∈𝒰(R) is given by (ϕ,ψ), where ϕ∈ (_d-1, span_k ( w_d))⊗ R and ψ∈_ (_d, ℳ_d ) ⊗ R, then g̅ (y ) ∈g̅·𝒰 corresponds under (<ref>) to 
    (g̅∘ϕ∘ (g̅|__d-1)^-1, g̅∘ψ∘ (g̅|__d)^-1 ).
 Denote the last pair by (^g̅ϕ, ^g̅ψ). Then y = g̅ y if and only if the following two conditions hold. 
    ∀ u ∈_d-1, ∃ u' ∈_d-1⊗ R,    u + ϕ (u)  = u' + ^g̅ϕ (u') 
    ∀ v ∈_d, ∃ v' ∈_d⊗ R,   v+ ψ (v) = v'+ ^g̅ψ(v')

From now on we assume x_0 = (_d-1, _d)∈ S_Λ ^g̅ (k).

	Write the matrix over k of g̅ acting on Φ_d under the basis v̂_1,⋯, v̂_d (cf. <ref>) as 
	
    [ H_1 H_2; H_3 H_4 ],
 where H_1 is of size (d-1) × (d-1), H_2 is of size (d-1) × 1, H_3 is of size 1× (d-1), and H_4∈ k. 


Since g̅ stabilizes _d-1, we have H_3=0



	Let R be a k-algebra and let y =(ϕ,ψ) ∈𝒰(R). Represent ϕ as an R-linear combination ϕ = ∑_i=1 ^d-1 r_i ϕ_i of the ϕ_i's (cf. (<ref>)), where r_i ∈ R. Write r⃗ for the row vector (r_1,⋯, r_d-1).  
	
		
  * View ϕ as an element of (Φ_d, d-1)(R). It is fixed by g̅|_Φ_d if and only if 
		
    r⃗ ( H_1 + H_2 r⃗) = H_4 r⃗.

		
    r_i = h_dd r_i ∑_l=1^d-1 h^ld r_l +   h_dd∑_ l =1^d-1 h^li r_l ,   1≤ i ≤ d-1.
 	
		
  * Assume that y ∈ℐ(R) and that ϕ∈ (Φ_d, d-1) is fixed by g̅|_Φ_d. Then ψ, viewed as an element of (d)(R), is fixed by g̅. In other words, y is fixed by g̅ in this case.
	

 
	(1)
	First we identify (Φ_d)⊗ R with R^d-1 using the basis v̂_1,⋯, v̂_d. 
	As a point of (Φ_d, d-1), ϕ corresponds to the following submodule of (Φ_d)⊗ R: the image, i.e. column space, of the R-matrix 
	
    [ I_d-1     0;    r⃗     0 ].

Hence ϕ∈ (Φ_d, d-1) is fixed by g̅|_Φ_d if and only if the following two R-matrices have the same column space:

    A_1 : = [ I_d-1     0;    r⃗     0 ] A_2:= [ H_1 H_2; H_3 H_4 ][ I_d-1     0;    r⃗     0 ].

Note that since [ H_1 H_2; H_3 H_4 ] is invertible, A_1 and A_2 have the same column space if and only if the column space of A_2 is contained in that of A_1.
Since H_3 =0 (cf. Remark <ref>), we have

    A_2=  [ H_1 + H_2 r⃗            0;       H_4 r⃗            0 ].

But we easily see that the column space of [ H_1 + H_2 r⃗            0;       H_4 r⃗            0 ] is contained in that of [ I_d-1     0;    r⃗     0 ] if and only if (<ref>) holds. 

	(1)
Condition (<ref>) is equivalent to the condition that for each 1≤ i ≤ d-1, there exist a_1,⋯, a_d-1∈ R such that

    v_i + ϕ(v_i) = ∑_t=1 ^d-1 a_t ( v_t + ^g̅ϕ ( v_t)).
 
We have ϕ(v_i) = r_i  w_d. 
We compute, for 1≤ t ≤ d-1, 

    ^g̅ϕ ( v_t) = g̅ϕg̅^-1 v_t = ∑_ l = 1^d-1g̅ϕ h^lt v_l = ∑ _l=1^d-1 h^ltg̅ r_l w_d = ∑_l=1^d-1 r_l h^lt∑ _m=1^d h_mdv̂_m .
 

Hence the RHS of (<ref>) is equal to 

    [ ∑_t=1 ^d-1 a_t ∑_ l =1^d-1 r_l h^lt h_dd] w_d + ∑_j=1^d-1[a_j +  ∑_t=1 ^d-1 a_t ∑_ l =1^d-1 r_l h^lt h_jd] v_j .


Thus (<ref>) is equivalent to 

    δ_ij = a_j +  ∑_t=1 ^d-1 a_t ∑_ l =1^d-1 r_l h^lt h_jd ,   1≤ j ≤ d-1 
    
    r_i = ∑_t=1 ^d-1 a_t ∑_ l =1^d-1 r_l h^lt h_dd.

Note that h_dd∈ k^×, so we may substitute h_dd^-1 times the second equation into the first, and get 
    a_j = δ_ij - h_dd^-1 r_i h_jd.

Substituting these values of a_j back to the second equation we get the condition
 
    r_i = ∑_t=1 ^d-1 (h_ddδ_it - r_i h_td) ∑_ l =1^d-1 r_l h^lt,   1≤ i ≤ d-1,
 	
 Conversely, if (<ref>) holds, the values (<ref>) of a_j satisfy (<ref>).
 	
 Now, using ∑_t=1 ^d-1 h_td h^lt = δ _ld - h_dd h^ld =- h_dd h^ld for 1≤ l ≤ d-1 to simplify the RHS of (<ref>), we see that (<ref>) is equivalent to (<ref>). 

(2)
 Let (d-1,d) be the incidence subscheme of (d-1) ×(d). Consider the natural morphism f: ℐ→(d-1,d), (_d-1', _d') ↦ (_d-1', _d'). Note that 𝒰∩ℐ is connected because it is a linear subspaces of the affine spaces 𝒰 (cf. Lemma <ref>). Thus (g̅·𝒰) ∩ℐ = g̅ (𝒰∩ℐ) is also connected. Since 𝒰∩ℐ and (g̅·𝒰) ∩ℐ share a common k-point, namely x_0, we see that that f(𝒰∩ℐ) and f((g̅·𝒰)∩ℐ ) are in one connected component of (d-1, d). We have y ∈𝒰∩ℐ and g̅ y ∈ (g̅·𝒰)∩ℐ. In particular f(y) and f(g̅ y) are R-points of the aforementioned connected component of (d-1, d). Recall from <cit.> that (d-1, d) has two connected components, and each is isomorphic to (d-1) via the projection to the first factor. Our assumptions imply that f(y), f(g̅ y) have the same image in (d-1). It follows that f(y) = f(g̅ y). But by definition f is injective on R-points, so y = g̅ y. 


We first claim that under the assumptions (<ref>) is automatic for v∈_d-1. Indeed, for v ∈_d-1, by (<ref>) we can find v' ∈_d-1⊗ R such that v+ ϕ (v) = v' + ^g̅ϕ (v'). By Lemma <ref>, we have ψ|__d-1⊗ R = ϕ, and therefore 
    v + ψ (v) = v+ ϕ(v) = v' + g̅ϕg̅^-1 v' g̅^-1 v' ∈_d-1⊗ R v' + g̅ψg̅^-1 v' = v' + ^g̅ψ  (v'),
 thus (<ref>). It remains to study the condition <ref> for v = v_d. 
By Lemma <ref>, we have 

    ψ v_j = r_j w_d,   1≤ j ≤ d-1
 

    ψ v_d = -∑_ j=1 ^d-1 r_j w_j.
 Now condition (<ref>) for v = v_d is equivalent to the existence of a_1,⋯, a_d ∈ R such that 

    v_d - ∑_j=1 ^d-1 r_j w_j = ∑_k=1 ^d a_k ( v_k + ^g̅ψ (v_k)).

Denote the left (resp. right) hand side of (<ref>) by ℒ (resp. ℛ). 
For 1≤ j ≤ d-1, we compute 

    ℛ, v_j = ∑_k=1^d a_k g̅ψg̅^-1 v_k , v_j=  ∑_k=1^d a_k ψg̅^-1 v_k , g̅^-1 v_j = ∑_k=1^d a_k ψ∑_l=1^d g^lk v_l  , ∑_m=1^d-1 g^mj v_m


    =∑_k=1^d a_k    g^dkψ v_d , ∑_m=1^d-1 g^mj v_m = -∑_k=1^d a_k    g^dk∑ _l=1^d-1 r_l w_l , ∑_m=1^d-1 g^mj v_m =-∑_k=1^d a_k   g^dk∑ _l=1^d-1 r_l g^lj.

Also 
    ℒ, v_j = -r_j.

Using (<ref>), we have 

    ℒ - ℛ, v_j = [∑_k=1^d a_k   g^dk∑ _l=1^d-1 r_l g^lj]- r_j = [∑_k=1^d a_k   g^dk    h_dd^-1 (r_j + r_j r_d) ]- r_j  .

We compute 

    ℛ, v_d =∑_k=1^d a_k ψ∑_l=1^d g^lk v_l  , ∑_m=1^d g^md v_m = [∑_k=1^d a_k ψ∑_l=1^d g^lk v_l  , ∑_m=1^d-1 g^md v_m] + [∑_k=1^d a_k ψ∑_l=1^d g^lk v_l  ,  g^dd v_d]


    =[-∑_k=1^d a_k   g^dk∑ _l=1^d-1 r_l g^ld] + [∑_k=1^d a_k ∑_l=1^d-1 g^lk  r_l w_d  ,  g^dd v_d] = [-∑_k=1^d a_k   g^dk∑ _l=1^d-1 r_l g^ld] + [∑_k=1^d a_k g^dd∑_l=1^d-1 r_l g^lk] .

Also 

    ℒ, v_d = 0

Using (<ref>), we have 

    ℛ -ℒ, v_d = ℛ, v_d =  [-∑_k=1^d-1 a_k   g^dk∑ _l=1^d-1 r_l g^ld] - [ a_d   g^dd∑ _l=1^d-1 r_l g^ld]+ [∑_k=1^d-1 a_k g^dd∑_l=1^d-1 r_l g^lk]  + [a_d g^dd∑_l=1^d-1 r_l g^ld]
 

    = [-∑_k=1^d-1 a_k   g^dk∑ _l=1^d-1 r_l g^ld]+ [∑_k=1^d-1 a_k g^dd∑_l=1^d-1 r_l g^lk] (<ref>),   h_dd = g^dd[-∑_k=1^d-1 a_k   g^dk∑ _l=1^d-1 r_l g^ld]+ [∑_k=1^d-1 a_k (r_k + r_k r_d)   ]


















We compute, for 1≤ k ≤ d-1, 

    ^g̅ψ (v_k)  = ^g̅ϕ ( v_k) (<ref>)∑_l=1^d-1 r_l g^lk∑ _m=1^d h_mdv̂_m ,
 
and 
    ^g̅ψ (v_d) = g̅ψg̅^-1 v_d = ∑_l=1^d g^ldg̅ψ v_l = ∑_l=1^d-1  r_l g^ld∑ _m=1^d h_mdv̂_m  - g^dd∑_j=1^d-1 r_j (gw_j) .

Hence the RHS of (<ref>) is equal to 

    ∑_k=1 ^d a_k v_k + ∑ _k=1^d a_k  ∑_l=1^d-1  r_l g^ld∑ _m=1^d h_mdv̂_m - g^dd∑_j=1^d-1 r_j (gw_j)
 

    = ∑_j=1^d-1[a_j + ∑ _k=1^d a_k  ∑_l=1^d-1  r_l g^lk h_jd] v_j + [a_d v_d]  + [ ∑ _k=1^d a_k  ∑_l=1^d-1  r_l g^lk  h_dd] w_d - g^dd∑_j=1^d-1 r_j (gw_j).

Therefore RHS of (<ref>) pairs with v_j (1≤ j ≤ d-1) to give 

    - g^dd∑_j=1^d-1 r_j gw_j, v_j =  - g^dd∑_j=1^d-1 r_j w_j, ∑_ l=1^d-1 g^lj v_l=   - g^dd∑_j=1^d-1 r_j g^jj.

On the other hand, the LHS of (<ref>) pairs with v_j (1≤ j ≤ d-1) to give r_j. Therefore (<ref>) implies that 



	Assume x_0 ∈ S_Λ^g̅(k). Then the local ring ℛ = _ℐ^g̅, x_0 of ℐ ^g̅ at x_0, is isomorphic to the local ring at the origin of the  subscheme of ^d-1_k defined by the equations (<ref>), where _k^d-1 has coordinates r_1,⋯ , r_d-1. Moreover, explicitly we have 
	
    ℛ≅ k[X]/ X^c.



	The first claim follows from Lemma <ref> and Proposition <ref>. To compute ℛ explicitly, we may and shall assume that the bases chosen in <ref> are such that the matrix H_1 is already in its (upper-triangular) Jordan normal form. Recall from Definition <ref> that all the Jordan blocks have distinct eigenvalues. Let J_d_1 (λ_1), ⋯, J_d_s-1 (λ_s-1) be the Jordan blocks that have eigenvalues different from λ. Let λ_s = λ and let J_d_s (λ_s) be the Jordan block of eigenvalue λ_s that appears in H_1, where we allow d_s =0. Then d_s =c-1. Moreover, we assume that J_d_1 (λ_1),⋯, J_d_s (λ_s) appear in the indicated order. Note that H_4 =λ. 
	Write H_1 = (h_ij)_1≤ i,j ≤ d-1. The equations (<ref>) become 
	

    r_i-1 h_i-1, i  + ( h_i,i- λ +r⃗ H_2) r_i = 0 ,  2≤ i ≤ d-1 
    
    ( h_1,1- λ +r⃗ H_2) r_1 = 0

		Note that when h_i,i is not in the Jordan block J_d_s (λ_s), we have h_i,i -λ∈ k^×, so the element h_i,i - λ + r⃗ H_2 is a unit in the local ring _^d-1, 0. Hence for i ≤ d_1 + d_2 + ⋯ + d_s-1 = d-c, each r_i is solved to be a multiple of r_i-1 and this multiple eventually becomes zero when this procedure is iterated. In other words, the ideal in _^d-1, 0 defining ℛ is generated by 
    r_1, r_2,⋯, r_d-c,    (r⃗ H_2)r_d-c+1,      (r⃗ H_2) r_i + r_i-1 (d-c+1 < i ≤ d-1).

		When c=1, we have ℛ≅ k as expected. Assume now c ≥ 2. Let h_1,⋯, h_c-1 be the last c-1 entries of the (d-1)× 1-matrix H_2. Make the change of variables 
		
    X_i = r_d-c +i,   1≤ i ≤ c-1 , 
     
    A  = r⃗ H_2.

		Then we have  
		
    ℛ≅(k [X_1,⋯, X_c-1, A] / (  A-∑_ i =1^c-1 h_i X_i,   AX_1, X_1 + AX_2,  X_2 +AX_3,⋯, X_c-2 +AX_c-1))_(X_1,⋯, X_c-1)

		By eliminating the variables X_1,…,X_c-2, we obtain that 
    ℛ≅(k[X_c-1, A]/ (X_c-1 A^c-1,  A - X_c-1∑_i=0^c-2 h_c-1-i (-A)^i )) _(X_c-1, A).
  
		Note that if h_c-1 = 0, then the last two rows of the matrix 
    λ I_d - [ H_1 H_2;   0 H_4 ]
 are both zero. This contradicts with the fact that the matrix [ H_1 H_2;   0 H_4 ], which represents g̅ on Φ_d, has in its Jordan normal form a unique Jordan block of eigenvalue λ (cf. <ref>).
		Hence h_c-1≠  0, and ∑_i=0^c-2 h_c-1-i (-A)^i is a unit in k[X_c-1, A] _(X_c-1, A). It follows that
		
    ℛ≅(k[X]/ (X^c  )) _(X) = k[X]/X^c,
 as desired.





 §.§ The intersection length formula


We are now ready to determine the structure of the complete local ring of S_Λ^g̅ at a k-point of it, when p is large enough. It is a consequence of Lemma <ref>, Proposition <ref>, and some commutative algebra. 

	Let x_0 ∈ S_Λ ^g̅(k). Let λ and c be as in Definition <ref>. Assume p>c. Then the complete local ring of S_Λ^g̅ at x_0 is isomorphic to k[X]/ X^c. 


Since S_Λ is smooth of dimension d-1 (cf. <ref>), the complete local ring of S_Λ^g̅ at x_0 is of the form 
    𝒮̂= k[[X_1,⋯, X_d-1]]/I
 for a proper ideal I of k[[X_1,⋯, X_d-1]].[We use this notation because previously we used the notation 𝒮 to denote the local ring of S_Λ^g̅ at x_0.] Let 𝔪 be the maximal ideal of k [[X_1,⋯, X_d-1]] and let 𝔪̅ be the maximal ideal of 𝒮̂. By Lemma <ref> and Proposition <ref>, there is an isomorphism

    β: 𝒮̂/𝔪̅^p  k[X]/ X^c.


We first notice that if R_1 is any quotient ring of k[[X_1,⋯, X_d-1]] with its maximal ideal 𝔪_1 satisfying 𝔪_1 = 𝔪_1^2 (i.e. R_1 has zero cotangent space), then R_1 =k. In fact, R_1 is noetherian and we have 𝔪_1^l = 𝔪_1 for all l∈_≥ 1, so by Krull's intersection theorem we conclude that 𝔪_1 = 0 and R_1 =k. 

Assume c=1. Then 𝒮̂/ 𝔪̅ ^p ≅ k, so 𝒮̂ has zero cotangent space and thus 𝒮̂= k. Next we treat the case c≥ 2. Let α be the composite 
 
    α: k [[ X_1,⋯, X_d-1]] →𝒮̂/𝔪̅^p    k[X]/X^c.
 Let J = α. It suffices to prove that I= J. Note that because β is an isomorphism we have 
    I +  𝔪^p = J.

 In the following we prove 𝔪^p ⊂ I, which will imply I= J and hence the theorem. The argument is a variant of <cit.>. 
 
 Let Y ∈ k [[ X_1,⋯, X_d-1]] be such that α(Y) =X. Since X generates the maximal ideal in k[X]/X^c, we have 
 
    𝔪 =  J +(Y).
 Then by (<ref>) and (<ref>) we have 𝔪 = I + (Y) + 𝔪 ^p,
 and so the local ring k[[X_1,⋯, X_d-1]]/ (I+(Y)) has zero cotangent space. We have observed that the cotangent space being zero implies that the ring has to be k, or equivalently
  
    𝔪 = I + (Y)

 
 Now we start to show 𝔪^p ⊂ I. By (<ref>) we have 𝔪^p ⊂ I+ (Y^p), so we only need to prove Y^p∈  I. We will show the stronger statement that Y^c ∈ I. By Krull's intersection theorem, it suffices to show that Y^c ∈ I + 𝔪 ^pl for all l≥ 1. In the rest we show this by induction on l.
 
  Assume l=1. Note that α (Y^c) =0, so by (<ref> ) we have 
    Y^c ∈ J=  I + 𝔪^p.
 Suppose Y^ c ∈  I + 𝔪^pl for an integer l ≥ 1. Write
   
    Y^c = i +  m,  i ∈ I,  m ∈𝔪^pl.
   By (<ref>) we know 
    𝔪^pl⊂ ( J + (Y)) ^pl⊂∑ _s=0^pl J^s (Y) ^pl-s.
 Thus we can decompose m∈𝔪 ^pl into a sum 
   
    m = ∑_s=0 ^pl j_s Y^pl-s,  j_s ∈ J^s.

By (<ref>) and (<ref>), we have 

    Y^c = i +  ∑ _s=0^pl j_s Y ^pl-s .

 Splitting the summation ∑_ s = 0 ^pl into two sums ∑_s= 0 ^pl-c and ∑_ s = pl-c +1 ^pl and moving the sum ∑_s= 0 ^pl-c to the left hand side, we obtain
 
    Y^c - ∑ _s=0^pl-c j_s Y ^pl-s =  i + ∑ _s=pl-c+1^pl j_s Y ^pl-s.

 
   Denote 
    A : = ∑_s =0  ^pl-c j_s Y^pl-s-c .
 Then the left hand side of (<ref>) is equal to (1-A) Y^c. Hence we have 
    (1-A)  Y^c = i + ∑_ s= pl-c+1 ^pl j_s Y^pl -s⊂ I + J^pl-c+1(<ref>) I+ (I+𝔪^p) ^pl-c+1 = I + 𝔪 ^p (pl-c+1)⊂ I + 𝔪 ^p(l+1),
 where for the last inclusion we have used c< p. Since 1-A is a unit in k[[X_1,⋯, X_d-1]] (because c<p), we have Y^c ∈ I + 𝔪^p(l+1). By induction, Y^c ∈ I + 𝔪^pl for all l∈_≥ 1, as desired.


Let g ∈ J_b(_p) be regular semisimple and minuscule. Assume ^g ≠∅ and keep the notation of <ref>. Let x_0 ∈ (δ(^♭)∩^g)(k). Let (_d-1, _d) ∈ S_Λ (k) correspond to x_0 via Proposition <ref> and define λ, c as in Definition <ref>. Assume p>c. Then the complete local ring of δ(^♭)∩^g at x_0 is isomorphic to k[X]/ X^c. Moreover, we have c=m(Q(T))+1/2, where Q(T) as in Theorem <ref>. In particular, 1≤ c≤ n/2. 



The first part follows immediately from Proposition <ref> and Theorem <ref>. It remains to show that 
    c=m(Q(T))+1/2.
 Suppose x_0∈_Λ' for some vertex lattice Λ' (not necessarily equal to Λ = L(g)^∨). Let L be the associated special lattice. Then we have (<ref>) 
    (Λ')_W^∨⊆ L⊆Λ'_W,   (Λ')_W^∨⊆Φ(L)⊆Λ'_W.
 Hence the eigenvalue λ of g̅ on Φ(ℒ_d)/ℒ_d-1≅ (L+Φ(L))/L appears among the eigenvalues of g̅ on Λ' /(Λ')^∨, and so the minimal polynomial of g̅ on Λ' /(Λ')^∨ in 𝔽_p[T] is equal to Q(T) by the proof of Theorem <ref>. Notice that the characteristic polynomial of g̅ on Φ(ℒ_d) (in k[T]) divides  R(T)Q(T) (the characteristic polynomial of g̅ on Λ'_W/L(g)) and also is divided by R(T) (the characteristic polynomial of g̅ on (Λ')_W^∨/L(g)). It follows that c, the multiplicity of λ of  g̅ on Φ(ℒ_d), is equal to the multiplicity of λ in R(T)Q(T). The desired formula for c then follows since 
    m(Q(T))+1=2· the multiplicity of  Q(T) in R(T)Q(T).
 Finally, we note that m(Q(T)) is a positive odd integer not greater than the degree of P(T), and the latter, being the type of the vertex lattice Λ = L(g)^∨, is an even integer ≤ t_max (cf. <ref>). The bound for c follows from the value of t_max given in <ref>. 





 §.§ Root computation

Having chosen the ordered basis e_1,⋯, e_d, f_1,⋯, f_d of Ω, we have a corresponding trivialized maximal torus 𝕋 of 𝔾 (i.e. a maximal torus together with an isomorphism to _m^d). Denote its natural characters by 
    ϵ_1,⋯, ϵ_d.
 Then all the roots of (𝕋, 𝔾) are given by 

    α_i,j : = ϵ_i + ϵ_j,   1 ≤ i < j ≤ d
 

    β_i,j : = ϵ_i - ϵ_j,   1 ≤ i < j ≤ d
 

    - α_i,j,    - β_i,j ,   1 ≤ i < j ≤ d.

We denote by 
    A_i,j :  =  (𝔾) _α_i,j
    𝔄_i,j: =  (𝔾) _-α_i,j
    B_i,j: =  (𝔾) _β_i,j
    𝔅_i,j : =  (𝔾) _-β_i,j.


We use the symbols A_i,j, etc. to denote a chosen basis vector of A_i,j, etc. 
We have 

    [A_i,j, 𝔄_i,j ] ≡ [B_i,j, 𝔅_i,j] ≡ 0 𝕋.

Now P^+ is the span of 

    𝕋∪A_i,j, B_i,j, 𝔅_i,j |  1≤ i<j ≤ d .

P^- is the span of 

    𝕋∪A_i,j, B_i,j  |  1≤ i < j ≤ d ∪𝔅_i,j  |   1≤ i < j <d∪𝔄_i,d |  1≤ i < d.

P_0 is the span of 

    𝕋∪A_i,j, B_i,j  |  1≤ i < j ≤ d ∪𝔅_i,j  |   1≤ i < j <d .

(We see that indeed Φ switches P^+ and P^- and we have P^+ ∩ P^- =  P_0.)

We may identify 𝒯_x_0 S_Λ ^+ =  P^-/ P_0 with the span of 

    𝔄_i,d |   1≤ i < d.

From this we see that _k 𝒯_x_0 S_Λ ^+ is indeed d-1, as expected.

Now we investigate the vector space 𝒯_x_0 (S_Λ ^+) ^g̅. Recall that this is the same as 

    ( P^- /  P_0)^ h_0 = X∈ P^- /  P_0 | ( h _0) X ≡ X  P_0.


 We first study the equation 

    [ϖ , X] ≡ 0  P_0
 for ϖ∈ P_0 and X ∈ P^- /  P_0.

Now suppose we have an element ϖ of P_0, of expansion 

    ϖ = τ + ∑ _i< j <d( a_i,j A_i,j + b_i,j B_i,j + b̅_i,j𝔅_i,j) + ∑_i<d (a_i,d A_i,d + b_i,d B_i,d),
 where τ∈𝕋.
Suppose we are also given 
    X = ∑ _1≤ i < d X^i 𝔄_i,d ∈span_k 𝔄_i,d  |   1≤ i <d =  P^- /  P_0.

The following relations are automatic:

    [A_i,j , X]  ≡ 0   P_0,  ∀ 1≤ i< j ≤ d-1.


    [A_i,d , X]  ≡ [B_i,d , X] ≡ 0   P_0,  ∀ 1≤ i< d.

Hence [ϖ , X]  P_0 depends only on the component 

    τ+ ∑_ i< j <d b_i,j B_i,j + b̅_i,j𝔅_i,j
 of ϖ.
In the following i,j,i',j' are indices with 1≤ i< j <d, 1≤ i' <d, 1≤ j' <d. Write ∼ for colinearity. We have 

    [𝔅_i,j,𝔄_j',d ] ∼δ _j,j'𝔄_i,d
    
     [B_i,j, 𝔄_i',d] ∼δ_i,i'𝔄_j,d

We may rescale the B_i,j's and 𝔅_i,j's to arrange

    [𝔅_i,j,𝔄_j,d ] = 𝔄_i,d
    
    [B_i,j, 𝔄_i,d] = 𝔄_j,d


We have, modulo P_0,

    [ϖ , X] ≡ [ τ + ∑_ i< j <d b_i,j B_i,j + b̅_i,j𝔅_i,j ,  X]  = [τ ,X] + ∑_ i< j <d b_i,j X^i  𝔄_j,d  + ∑_ i< j <db̅_i,j X^j 𝔄_i,d


    =[τ, X] + ∑_1≤ l <d(∑_i<l  b_i,l X^i + ∑ _ l<j<db̅_l,j X^j ) 𝔄_l,d.

Also 

    [τ,X] = ∑_1≤ l <d X^l  · (- α_l,d )(τ) 𝔄_l,d

Write τ^l: = (-α _l,d) (τ).
Hence [ϖ , X] ≡ 0  P_0 if and only if 

    τ^l X^l + ∑_i<l  b_i,l X^i + ∑ _ l<j<db̅_l,j X^j = 0, ∀ 1≤ l <d.

We view this as a system of d-1 linear equations in the unknowns 
X^1,⋯, X^d-1. When ϖ satisfies a suitable regularity condition, this system should have nullity 0 or 1
.




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