Affine Grassmannians for Triality Groups

arXiv:2107.07372v2 [math.RT] 29 Jul 2021 hc stefntro h aeoyof category the on functor the is which Then cee aldtelo ru soitdto associated group loop the called scheme, erfr[] 2 o motn eut ntesrcueo opgrou loop of r structure of theory the [12]. the [4], on to [3], results applications varieties, have Shimura important results for These [2] Grassmannians. Gr [1], fpqc-sheaf The refer We Grassmannian. affine the definition where ru vrSpec( over group ffieGasanascnb endb opgop.Let groups. loop by defined be can Grassmannians Affine Introduction 1 theorem main the of Proof groups 5 triality for Grassmannians Affine triality 4 and groups orthogonal Special 3 algebras composition Twisted 2 Introduction 1 Contents 3 o hs raiygop sfntr lsiyn suitable classifying functors as descrip groups give triality we these algebras, for composition alge these composition Using twisted algebra. certain of automorphisms as given L D R + 4 (( esuyan rsmnin o aie raiygop.Th groups. triality ramified for Grassmannians affine study We G ote r om fteotooa rtesi rusi var 8 in groups spin the or orthogonal the of forms are they so , t )i h arn oe eiswt variable with series power Laurent the is )) 0 ⊂ ffieGasanasfrTilt Groups Triality for Grassmannians Affine LG k .W osdrtefunctor the consider We ). 0 sasbru uco,adtefq-utetGr fpqc-quotient the and functor, subgroup a is R R 7→ 7→ L k LG hhoZhao Zhihao -algebras, + G 0 Abstract ( 0 R ( LG G R = ) 1 0 = ) ecnie h oiielo group loop positive the consider We . 0 ntectgr of category the on G G 0 ( 0 R ( t G hsfntri ersne ya ind- an by represented is functor This . R (( 0 atcsi xdspace. fixed a in lattices [[ k t t sas ersne ya ind-scheme. an by represented also is ))) ]]) eafil,adlet and field, a be in ftean Grassmannians affine the of tions rsotie rmteoctonion the from obtained bras , . dcinadlclmdl of models local and eduction s rusaeo type of are groups ese k -algebras, als hycnbe can They iables. sadascae affine associated and ps G 0 = G LG 0 ea algebraic an be 0 /L + G 0 L sby is + G 14 11 0 8 4 1 , In [11], Pappas and Rapoport developed a similar theory of twisted loop groups and of their associated affine Grassmannians: Let G be a linear algebraic group over k((t)). Consider the twisted loop group LG, which is also the ind-scheme representing the functor:

LG : R G(R((t))) → for any k-algebra. Since R((t)) is a k((t))-algebra, the definition makes sense. Notice that when G = G k((t)), we recover the previous definition in the untwisted case. To define the positive 0 ⊗k loop group, we assume G is reductive, and choose a parahoric subgroup G of G. This is a smooth group scheme over k[[t]] with G k((t)) = G. Then the positive loop group L+G is an ⊗k[[t]] infinite-dimensional affine group scheme L+G representing the functor:

L+G : R G (R[[t]]) →

The fpqc-quotient GrG = LG/L+G is represented by an ind-scheme over k, which we call the affine Grassmannian associated to G. In this paper, we take the twisted loop group to be a ramified triality group and study the corresponding affine Grassmannian. Our main goal (see Theorem 1.1 below) is an explicit description of the triality affine Grassmannian in terms of lattices with extra structure. Lusztig [10] first showed that affine Grassmannians for simple Lie algebras can be described in terms of certain orders, which are lattices closed under the Lie bracket. Here we aim for an explicit description in terms of lattices in the standard representation which is more in line with such descriptions known for classical groups. For example, Pappas and Rapoport gave such descriptions for affine Grassmannians and affine flag varieties for unitary groups in [11] using lattices (or lattice chains) which are self-dual for a hermitian form. See also work of Görtz [4] and of Smithling [14] for the symplectic and the split orthogonal groups, respectively. It turns out that the case of the ramified triality group, which we consider here, is considerably more complicated. What are triality groups? Let G0 be an adjoint Chevalley group of type D4 over a field F0. Consider the Dynkin diagram:

•

D : 4 ••

•

The Dynkin diagram of type D4 has a symmetry not shared by other Dynkin diagrams: it admits automorphisms of order 3. Since the automorphism of the Dynkin diagram of type D4 is isomorphic to the symmetric group S3, there is a split exact sequence of algebraic groups:

f 1 G Aut(G ) S 1. → 0 → 0 → 3 →

Thus, G0 admits outer automorphisms of order 3, which we call trialitarian automorphisms. The ﬁxed elements of G0 under such an outer automorphism, deﬁne groups of type G2:

G ⇚ 2 • •

2 1 1 Consider the Galois cohomology set H (F0, Aut(G0)) := H (Γ0, Aut(G0)), where Γ0 is the abso- lute Galois group Gal(F0,sep/F0). Adjoint algebraic groups of type D4 over F0 are classiﬁed by 1 1 1 H (F, Aut(G0)) (29.B, [8]), and the map induced by f in cohomology f : H (F0, Aut(G0)) 1 → H (F0,S3) associates to G0 of type D4 the isomorphism class of a cubic ´etale F0-algebra F , see [8]. The possibilities of F are summarized as follows:

F type G0 1 F0 F0 F0 D4 × × 2 F0 ∆ D4 × 3 Galois ﬁeld ext. D4 6 non-Galois ﬁeld ext. D4

1 2 The group G0 is said to be of type D4 if F is split, of type D4 if F ∼= F0 ∆ for some quadratic 3 × separable field extension ∆/F0, of type D4 if F is a cyclic field extension over F0, and of type 6 3 D4 if F is a non-cyclic field extension. In our paper, we consider the D4 case and call the corresponding G0 the triality group. These triality groups are often studied by composition algebras. By composition algebras, we mean algebras (not necessarily associative) with a nonsingular quadratic form q such that q(x y) = q(x)q(y) for all x, y in this algebra. We give a review of symmetric composition · algebras and normal twisted composition algebras in §2. Composition algebras can be used to describe exceptional groups. For example, Springer shows the automorphism of an octonion algebra is of type G2 (§2.3, [15]). Here the octonion algebra is an 8-dimensional composition algebra. We can view this automorphism group as the fixed subgroup of a spin group of an octonion algebra under outer automorphisms (Proposition 35.9, [8]). In §3, we extend this result and show that the subgroup of a spin group of a normal twisted composition algebra, which is 3 fixed under outer automorphisms, is of type D4. This will be our main tool to study affine Grassmannians for triality groups. Let k be a field with characteristic char(k) =2, 3. We assume the cubic primitive root ξ is in 6 k. We set F0 = k((t)) (resp. F = k((u))) the ring of Laurent power series, with ring of integers k[[t]] (resp. k[[u]]). Set u3 = t so that F/F is a cubic Galois extension with Gal(F/F ) = ρ = A , 0 0 h i ∼ 3 where ρ acts on u by ρ(u) = ξu. In §2, we define the normal twisted composition algebra (V, ) over F . Here (V, ) is a 8-dimensional vector space with an F -bilinear product and ∗ ∗ 0 ∗ a nonsingular quadratic form q satisfying certain properties (see Definition 2.1). We also fix a finitely generated projective k[[u]]-module L in V , which we call the standard lattice in V . Denote by , the bilinear form associated to q. We show that the spin group Spin(V, ) over F has h i ∗ an outer automorphism, and the subgroup of Res Spin(V, ), which is fixed under the outer F/F0 ∗ automorphism, is the triality group G we are interested in, i.e.,

G = Res Spin(V, )A3 . F/F0 ∗ We now choose the parahoric group scheme G over Spec(k[[t]]) given by the lattice L. This is a “special” parahoric subgroup in the sense of Bruhat-Tits theory. Recall the generic fiber Gη of + G is equal to G. The quotient fpqc sheaf LGη/L G is by definition the affine Grassmannian for the triality group over Spec(k). Our main theorem is:

3 Theorem 1.1. There is an LGη-equivariant isomorphism

LG /L+G F η ≃ where the functor F sends a k-algebra R to the set of finitely generated projective R[[u]]-modules L (i.e., R[[u]]-lattices) of V R = R((u))8, such that ⊗k ∼ (1) L is self dual under the bilinear form , , i.e., L Hom (L, R[[u]]). h i ≃ R[[u]] (2) L is closed under multiplication, L L L. ∗ ⊂ (3) There exists a L, such that q(a)=0, a a,a =1. ∈ h ∗ i (4) For a as in (3), let e = a + a a. Then, we have e x = x¯ = x e for any x¯ satisfying ∗ ∗ − ∗ x,¯ e¯ =0. (Here, x¯ is the image of x under the canonical map L L/uL.) h i → In particular, it gives a bijection between k-points in the affine Grassmannian for the triality group and a certain set of k[[u]]-lattices in V that satisfy some special conditions. The proof of this theorem is inspired by the construction of twisted composition algebras by Springer in [15, §4.5]: Observe that every normal twisted composition algebra (with isotropic quadratic form) (V, ) ∗ contains a special hyperplane V := Ff Ff , where the sum f + f is the para-unit. Set 0 1 ⊕ 2 1 2 V := V f and V := V f . We can decompose V as V = V V V . In §5, we use similar 1 0 ∗ 1 2 0 ∗ 2 0 ⊕ 1 ⊕ 2 ideas and extend Springer’s results to suitable lattices: To decompose a lattice L in V , we need L is closed under multiplication, self dual, and contains the elements a,a a that play similar roles ∗ as f1,f2 in V (see Theorem 1.1 for details). When L satisfies the conditions (1)-(4) in Theorem 1.1, we can decompose L as L = R[[u]]a R[[u]](a a) L L . Furthermore, there exist a basis ⊕ ∗ ⊕ 1 ⊕ 2 of L1 (resp. L2) such that the multiplication table of L is the same as the standard lattice L. Thus, we can define a morphism g in LG such that L = g(L). The triality group G we consider in this paper is simple and simply connected as a form of the spin group. We can also consider variants G′ with the same adjoint group G′ G and ad ≃ ad use similar ideas. These variants and their associated (global) affine Grassmannians are useful for describing corresponding local models, in the sense of [12]. Indeed, by fixing a coweight µ of G′, we can obtain a description of the corresponding local model as classifying lattices whose distance from the standard lattice is “bounded by µ”. We will describe this in another work. The results in this paper are part of my thesis at Michigan State University. I would like to thank my advisor G. Pappas for his useful suggestions and patient help.

2 Twisted composition algebras

Twisted composition algebras were introduced by Springer in his 1963 lecture notes [15], to get a new description of Albert algebras. We recall the definition from [15] and [9]. Let F0 be a field with char(F0) =2, 3, and let F be a separable cubic field extension of F0. The normal closure of 6 ′ F over F0 is F = F (d), where d satisfies a separable quadratic equation over F0 (see Theorem 4.13, [5]). We can take d = √D, the square root of the discriminant D of F over F0. We set ′ F0 = F0(d). So either F is the Galois extension of F0 with cyclic Galois group of order 3, and ′ ′ ′ ′ ′ then F = F, F0 = F0; or F and F0 are quadratic extensions of F and F0, respectively, and F ′ is the Galois extension of F0. We will focus on the case that the separable cubic extension F/F0 is also normal, and call algebras of this type “normal twisted composition algebras”.

4 Let F/F0 be a cubic Galois extension. We set Γ = Gal(F/F0), with ρ the generator of Γ. Set θ = ρ2, then Γ= 1,ρ,θ . { } Deﬁnition 2.1. A normal twisted composition algebra (of dimension 8) is a 5-tuple (A, F, q, ρ, ), ∗ where A is a vector space of dimension 8 over F with a nonsingular quadratic form q, and associated bilinear form , . We have an F -bilinear product : A A A on F with the h i 0 ∗ × → following properties: (1) The product x y is ρ-linear in x and θ-linear in y, that is: ∗ (λx) y = ρ(λ)(x y), x (λy)= θ(λ)(x y), ∗ ∗ ∗ ∗

(2) We have q(x y)= ρ(q(x))θ(q(y)), ∗ (3) We have x y,z = ρ( y z, x )= θ( z x, y ) h ∗ i h ∗ i h ∗ i for all x,y,z A,and λ F . ∈ ∈ Let A′ = (A′,F,q′,ρ′, ′) be another normal twisted composition algebra. A similitude A ∗ → A′ is deﬁned to be an F -linear isomorphism g : A A′, for which there exists λ F ∗, such that → ∈ q′(g(x)) = ρ(λ)θ(λ)q(x), g(x) ′ g(y)= λg(x y), ∗ ∗ for all x, y A. We denote by A′ = A . The scalar λ is called the multiplier of the similitude. ∈ λ Similitudes with multiplier 1 are called isometries. We will use the following lemmas which their proofs can be found in [15, Lemma 4.1.2, Lemma 4.1.3]. Lemma 2.2. We have (1) x z,y z = ρ( x, y )θ(q(z)), h ∗ ∗ i h i (2) z x, z y = θ( x, y )ρ(q(z)), h ∗ ∗ i h i (3) x z,y w, + x w,y z = ρ( x, y )θ( z, w ), h ∗ ∗ i h ∗ ∗ i h i h i for all x,y,z,w A. ∈ Lemma 2.3. We have (1) x (y x)= ρ(q(x))y, (x y) x = θ(q(x))y, ∗ ∗ ∗ ∗ (2) x (y z)+ z (y x)= ρ( x, z )y, (x y) z + (z y) x = θ( x, z )y, ∗ ∗ ∗ ∗ h i ∗ ∗ ∗ ∗ h i (3) (x x) (x x)= T (x)x q(x)(x x), where T (x) := x x, x F , ∗ ∗ ∗ − ∗ h ∗ i ∈ 0 for all x,y,z A. ∈ Remark 2.4. View a normal twisted composition algebra (A, F, q, ρ, ) as an 8-dim quadratic ∗ space. We can discuss isotropic subspaces in A. An element x A is called isotropic if q(x) = 0. ∈ A subspace W of A is said to be isotropic if q(x) = 0 for all x W . A maximal isotropic ∈ subspace is an isotropic subspace with the maximal dimension. All maximal isotropic subspaces have the same dimension, which is called the Witt index of q. The index is at most equal to half dimension of the vector space, which in our case, is 4.

5 It turns out that a normal twisted composition can be obtained by scalar extension from a symmetric composition algebra. Recall from [8], §34 that a symmetric composition algebra (of dimension 8) is a triple (S,⋆,q), where (S, q) is an 8-dimensional F0-quadratic space (with nondegenerate bilinear form , ) and ⋆ : S S S is a bilinear map such that for all x,y,z S, h i × → ∈ q(x⋆y)= q(x)q(y), x⋆y,z = x,y⋆z . h i h i By [8, Lemma 34.1], the above deﬁnition is equivalent to

x⋆ (y ⋆ x)= q(x)y = (x⋆y) ⋆ x, for all x, y S. ∈

Given a symmetric composition (S,⋆,q) over F0. We can get a normal twisted composition algebra S˜ = S (F,ρ) as follows: ⊗ S (F,ρ) := (S F,F,q , ρ, ) ⊗ ⊗F0 F ∗ where q is the scalar extension of q to F and is deﬁned by extending ⋆ linearly to S F F ∗ ⊗F0 and setting x y = (id ρ)(x) ⋆ (id θ)(y) for all x, y S F ∗ S ⊗ S ⊗ ∈ ⊗F0 (see [9]). A normal twisted composition algebra A over F is said to be reduced if there exist a symmetric composition algebra S over F and λ F ∗ such that A is isomorphic to S˜ . 0 ∈ λ Example 2.5. The main tool that we use in this paper is the normal twisted composition algebra obtained from the para-Cayley algebra. Consider the Cayley (octonion) algebra (C, ) over F ⋄ 0 with the bilinear form , and the conjugacy map: h i r(x) := x, e e x, for x A. h i − ∈ Consider the new multiplication ⋆ : x⋆y := r(x) r(y). This new multiplication yields x⋆y,z = ⋄ h i x,y⋆z . So (C,⋆) is a symmetric composition algebra, which is called the para-Cayley algebra h i (see [8, 34.A]). The multiplication table of (C,⋆) is given by Table 1 (we write “ ” instead of · 0 for clarity); and denote by (V, ) the normal twisted composition algebra obtained from the ∗ para-Cayley algebra.

y ⋆ e1 e2 e3 e4 e5 e6 e7 e8 e e e e e 1 · · ·− 1 · − 2 3 − 4 e e e e e 2 ·· 1 · − 2 · − 5 − 6 e e e e e 3 · − 1 ·· − 3 − 5 · 7 e e e e e 4 · − 2 − 3 5 · · ·− 8 x e e e e e 5 − 1 ··· 4 − 6 − 7 · e e e e e 6 2 · − 4 − 6 · · − 8 · e e e e e 7 − 3 − 4 · − 7 · 8 ·· e e e e e 8 − 5 6 − 7 · − 8 ··· Table 1: The split para-Cayley algebra multiplication x⋆y

6 Remark 2.6. Let (A, F, q, ρ, ) be a normal twisted composition algebra. Consider the extended ∗ algebra A′ = A F . We claim this extension algebra A′ is also a twisted composition algebra. ⊗F0 In fact, we have a nice description of A′. Consider an isomorphism of F -algebras

∼ ν : F F F F F given by r r (r r ,ρ(r )r ,θ(r )r ). ⊗F0 → × × 1 ⊗ 2 7→ 1 2 1 2 1 2 Note that ρ id is identiﬁed with the map deﬁned byρ ˜(r , r , r ) = (r , r , r ) to make the ⊗ F 1 2 3 2 3 1 following diagram commutative:

ρ⊗id F FF F F ⊗F0 ⊗F0

ρ˜ F F F F F F. × × × × We deﬁne the twisted vector spaces ρA and θA:

ρA = ρx x A , θA = θx x A , { | ∈ } { | ∈ } with the operations: ρ(rx)= ρ(r)ρx, ρ(x + y)= ρx + ρy, and θ(rx)= θ(r)θx, θ(x + y)= θx + θy, ∼ for all x, y A, r F . Then there is an F -isomorphism A F A ρA θA given by: ∈ ∈ ⊗F0 → × × x r (rx, r(ρx), r(θ x)) ⊗ 7→ (see [9, Remark 2.3]). To describe the multiplication in A F and A ρA θA, we need to ⊗F0 × × consider F -bilinear maps:

: ρA θA A, : θA A ρA, : A ρA θA, ∗id × → ∗ρ × → ∗θ × → given by ρx θy = x y, θx y = ρ(x y), x ρy = θ(x y), ∗id ∗ ∗ρ ∗ ∗θ ∗ for all x, y A. Then the product : (A ρA θA) (A ρA θA) A ρA θA given by ∈ ⋄ × × × × × → × × (x, ρx, θx) (y, ρy, θy) = (ρx θy,θ x y, x ρy), ⋄ ∗id ∗ρ ∗θ will make the following diagram commutative:

∗⊗id (A F ) (A F ) F A F ⊗F0 × ⊗F0 ⊗F0

(A ρA θA) (A ρA θA) ⋄ A ρA θA. × × × × × × ×

Finally, deﬁne quadratic forms ρq : ρA F and θq : θA F by → → ρq(ρx)= ρ(q(x)), θq(θx)= θ(q(x)).

7 We have an isomorphism:

ρ θ ρ θ (A F0 F, F F0 F, qF ,ρ idF , idF ) A A A, F F F, q q q, ρ,˜ . ⊗ ⊗ ⊗ ∗⊗ ≃ × × × × × × ⋄ 3 Special orthogonal groups and triality

In this section, we recall special orthogonal groups and spin groups for twisted composition algebras. Let (V, q) be a vector space with a nonsingular quadratic form q over a field F , with char(F ) different from 2. Denote by , the bilinear form corresponding to q. For any h i f End (V ), there exists an element σ (f) End (V ) such that x, f(y) = σ (f)(x),y . We ∈ F q ∈ F h i h b i can see this using matrices: If b GL(V ) denotes the Gram matrix of , with respect to a fixed ∈ h i basis, then x, y = xtby. Let σ (f) = b−1f tb. Then x, f(y) = xtbf(y) = σ (f)(x),y . It is h i q h i h b i easy to see that σ : End (V ) End (V ) given by f σ (f) is an involution of End (V ), and q F → F 7→ q F we call σq the involution corresponding to the quadratic form q. The special orthogonal group SO(V, q) is the subgroup of isomorphism group Isom(V, q) that preserve the form , and have h i determinant equal to 1:

SO(V, q) := g Isom(V, q) g(x),g(y) = x, y . { ∈ | h i h i} Elements g SO(V, q) are called proper isometries (Improper isometries are elements in Isom(V, q) ∈ that preserve the form with determinant equal to 1). − ⊗n The Cliﬀord algebra C(V, q) is the quotient of the tensor algebra T (V ) := ≥ V by the ⊕n 0 ideal I(q) generated by all the elements of the form v v q(v) 1 for v V . Since T (V ) is a ⊗ − · ∈ graded algebra, T (V )= T (V ) T (V ), where T (V )= T (V V ) and T (V )= V T (V ). This 0 ⊕ 1 0 ⊗ 1 ⊗ 0 induces a Z/2Z-grading of C(V, q):

C(V, q)= C (V, q) C (V, q). 0 ⊕ 1

We call C0(V, q) the even Clifford algebra and C1(V, q) the odd Clifford algebra. When dim V = n, n n−1 we have dim C(V, q)=2 , and dim C0(V, q)=2 (see [7, Chapter IV]). For every quadratic space (V, q), the identity map on V extends to an involution on the tensor algebra T (V ) which preserve the ideal I(q): (v v )t := v v for v ,...v V . It is therefore inducing 1 ⊗···⊗ r r ⊗···⊗ 1 1 r ∈ a canonical involution of the Clifford algebra τ : C(V, q) C(V, q) given by τ(v v ) = → 1 ··· d v v . By using the even Clifford algebra C (V, q), now we can consider the universal covering d ··· 1 0 of SO(V, q), which is the spin group Spin(V, q) defined by:

Spin(V, q)= c C (V, q)∗ cV c−1 = V, τ(c)c =1 . { ∈ 0 | } For any c Spin(V, q), we have a linear map χ : x cxc−1. This is an element in SO(V, q) since ∈ c 7→ q(χ (x)) = cxc−1cxc−1 = q(x), and we can show that Spin(V, q) SO(V, q) given by c χ is c → 7→ c surjective. We have an exact sequence:

1 Z/2Z Spin(V, q) SO(V, q) 1. → → → → The special orthogonal group scheme SO(V, q) and the spin group scheme Spin(V, q) over F are

8 deﬁned by:

SO(V, q)(R) := g Isom(V , q) g(x),g(y) = x, y , det g =1 , { ∈ R | h i h i } Spin(V, q)(R) := c C (V , q)∗ cV c−1 = V , τ(c)c =1 , { ∈ 0 R | R R } for any F -algebra R, where V := V R. In particular, when the quadratic space (V, q) is the R ⊗F normal twisted composition algebra (V, ), we denote by SO(V, ) (resp. Spin(V, )) the special ∗ ∗ ∗ orthogonal group (resp. spin group) for (split) normal twisted composition algebras. The Cliﬀord algebra for (V, ) has a special structure. Consider the twisted vector spaces ρV, ∗ θV in Remark 2.6. For any x V , we have two F -linear maps ∈ l : ρV θV, r : θV ρV x → x → given by l (ρy)= θ(x y) and r (θz)= ρ(z x). x ∗ x ∗ (see [9, §3]). By Lemma 2.3, it follows that the F -linear map:

ρ θ 0 rx α : (V, ) EndF ( V V ), given by x ∗ → ⊕ 7→ lx 0 satisﬁes α(x)2 = q(x)id. Hence we can extend this map to: α : C(V, ) End (ρV θV ) by ∗ → F ⊕ the universal property of Cliﬀord algebra. In fact, this map is an isomorphism of algebras with involution (see [8, Proposition (36.16)]):

∼ ρ θ α : (C(V, ), τ) (End ( V V ), σρ θ ), ∗ → F ⊕ q⊥ q

If we restrict this isomorphism to the even Cliﬀord algebra C0(V, q), we get

∼ ρ θ α : (C (V, ), τ) (End ( V ), σρ ) (End ( V ), σθ ), 0 ∗ → F q × F q ρ θ where σρq, σθ q are involutions corresponding to quadratic forms q, q (we still use α to denote the isomorphism for simplicity).

Remark 3.1. For any g SO(V, )(F ), there exist g ,g SO(V, )(F ) such hat: 1 ∈ ∗ 2 3 ∈ ∗ g (x y)= g (x) g (y), i =1, 2, 3 (mod 3) i ∗ i+1 ∗ i+2

(see [8, Proposition (36.17)]). It is easy to see that when gi satisfy the above equation, the following diagram D(gi,gi+1,gi+2) commutes:

C (V, q)α End (ρV ) End (θV ) 0 F × F ρ θ C0(gi) Int( gi+1)×Int( gi+2) C (V, q)α End (ρV ) End (θV ). 0 F × F

9 Here the automorphism C (g ): C (V, ) C (V, ) of the even Cliﬀord algebra given by 0 i 0 ∗ → 0 ∗ C (g )(v v )= g (v ) g (v ). 0 i 1 ··· 2r i 1 ··· i 2r Conversely, for any g ,g ,g SO(V, )(F ), if the diagram D(g ,g ,g ) commutes, we will 1 2 3 ∈ ∗ i i+1 i+2 get g (x y)= g (x) g (y),i =1, 2, 3 (mod 3). These two equivalent statements are called i ∗ i+1 ∗ i+2 “the principle of triality”.

By using this isomorphism α, we have the following result for normal twisted composition algebras. Similar results for symmetric composition algebras can be found in [8, Proposition (35.7)]. Let us set Spin(V, ) := Spin(V, )(F ). ∗ ∗ Theorem 3.2. There is an isomorphism

Spin(V, ) = (g ,g ,g ) SO(V, q)×3 g (x y)= g (x) g (y), for any x, y V ∗ ∼ { 1 2 3 ∈ | i ∗ i+1 ∗ i+2 ∈ } Proof. Let c C (V )∗. Using the isomorphism with involution α, we obtain ρg End (ρV ) ∈ 0 2 ∈ F and θg End (θV ) such that 3 ∈ F ρ g2 0 ρ θ α(c)= θ EndF ( V ) EndF ( V ). 0 g3 ∈ ×

We have ρ ρ σq(g2) 0 g2 0 α(τ(c)c)= θ θ = I, 0 σq(g3) 0 g3

−1 which implies σq(g2)g2 =1, σq(g3)g3 = 1, i.e., g2,g3 are isometries. Consider χc(x)= cxc V . −1 ∈ By applying α on both sides, we have α(χc(x)) = α(c)α(x)α(c ), which gives us:

0 r 0 ρg r θσ (g ) χc(x) = 2 · x · q 3 . l 0 θg l ρσ (g ) 0 χc(x) 3 · x · q 2 ρ θ θ ρ This is equivalent to g2rx = rχc(x) g3, g3lx = lχc(x) g2, i.e.,

g (x y)= g (x) χ (y), g (x y)= χ (x) g (y). (3.3) 2 ∗ 3 ∗ c 3 ∗ c ∗ 2 −1 −1 Finally, χc is an isometry since q(χc(x)) = cxc cxc = q(x). By Remark 3.1, Equation (3.3) yields the diagram D(χc,g2,g3) commuting, which shows that C0(χc) is the identity on the center of C (V, ). By [8, Proposition (13.2)], the isometry χ is proper. Similarly, g ,g are 0 ∗ c 2 3 also proper isometries. Thus, let g1 = χc. We get related equations as above. We now send c (g ,g ,g ) that gives as above. This giving map is an injective group homomorphism since 7→ 1 2 3 α is an isomorphism. It is also surjective, since, given any (g1,g2,g3) satisfying gi(x y) = ρ ∗ g2 0 gi+1(x) gi+2(y), there exist c C0(V ) such that α(c)= θ . ∗ ∈ 0 g3

From Theorem 3.2, we have an isomorphism between group schemes over F0:

Res (Spin(V, ))(R) = (g ,g ,g ) Res (SO(V, )(R)×3 g (x y)= g (x) g (y) F/F0 ∗ ∼ { 1 2 3 ∈ F/F0 ∗ | i ∗ i+1 ∗ i+2 }

10 for any F -algebra R. The transformationρ ˜ : (g ,g ,g ) (g ,g ,g ) is an outer automorphism 0 1 2 3 7→ 2 3 1 of Res (Spin(V, )) satisfyingρ ˜3 =1. Hereρ ˜ generates a subgroup of Aut(Res (Spin(V, ))), F/F0 ∗ F/F0 ∗ which is isomorphic to A . Consider the ﬁxed points of Res (Spin(V, )) under A = ρ˜ . 3 F/F0 ∗ 3 h i We obtain the triality group for the special orthogonal groups G:

G(R) := Res (Spin(V, ))A3 (R) F/F0 ∗ = g SO(V, )(R F ) g(x y)= g(x) g(y) for all x, y V R . ∼ { ∈ ∗ ⊗F0 | ∗ ∗ ∈ ⊗F0 } for any F0-algebra R.

4 Aﬃne Grassmannians for triality groups

In this section we give the definition of affine Grassmannians for triality groups. Recall that the affine Grassmannian for general groups is representable by an ind-scheme and is a quotient of loop groups. Let k be a field. We consider the field K = k((t)) of Laurent power series with indeterminate t and coefficients in k. Let = k[[t]] be the discretely valued ring of power series with coefficients OK in k. For a k-algebra R, we set D = Spec(R[[t]]), resp. D∗ = D t =0 = Spec(R((t))), which R R R \{ } we picture as an R-family of discs, resp., an R-family of punctured discs. We recall some functors from [11, §1]: Let X be a scheme over K. We consider the functor LX from the category of k-algebras to that of sets given by

R LX(R) := X(R((t))). 7→ If is a scheme over , we denote by L+ the functor from the category of k-algebras to that X OK X of sets given by R L+ (R) := (R[[t]]). 7→ X X The functors LX,L+ give sheaves of sets for the fpqc topology on k-algebras. In what follows, X we will call such functors “k-spaces” for simplicity.

Example 4.1. Ar + If = OK is the affine space of dimension r over K , then L is the infinite X ∞ O X dimensional affine space L+ = Ar, via: X iQ=0 ∞ ∞ L+ (R) = Hom (k[[t]][T , ..., T ], R[[t]]) = R[[t]]r = Rr = Ar(R). X k[[t]] 1 r iY=0 iY=0

r Let be the closed subscheme of A deﬁned by the vanishing of polynomials f1, ..., fn X OK in k[[t]][T , ..., T ]. Then L+ (R) is the subset of L+Ar(R) of k[[t]]-algebra homomorphisms 1 r X k[[t]][T , ..., T ] R[[t]] which factor through k[[t]][T , ..., T ]/(f , ..., f ). If X is an aﬃne K- 1 r → 1 r 1 n scheme, LX is represented by a strict ind-scheme.

Let X be a linear algebraic group over K. The loop group associated to X is the ind-scheme LX over Spec(k). We list some properties of loop groups:

(1) L(X Y )= LX LY ; ×k ×k

11 (2) If k′ is a k -ﬁeld extension, then we have an isomorphism of ind-schemes over k′

LX Spec(k′) L(X Spec(k′((t)))) ×k ≃ ×k((t))

′ ′ (3) Assume that K /K is a finite extension of K, where K = k((u)). If X = ResK′/K H for some linear algebraic group H over K′, then we have an isomorphism of ind-schemes over k: LX LH, ≃ by (LX)(R)= X(R((t))) = H(R((t)) k((u))) = H((R((u))) = LH(R). ⊗k((t)) Now let be a flat affine group scheme of finite type over k[[t]]. Let X = denote the generic X Xη fiber of . This is a group scheme over k((t)). We consider the quotient sheaf over Spec(k): X + X := L /L . F Xη X This is the fpqc sheaf associated to the presheaf which to a k -algebra R associates the quotient (R((t)))/ (R[[t]]). Generally, the affine Grassmannian for is the functor GrX : Alg Sets X X X k → which associates to a k-algebra R the isomorphism classes of pairs ( , α) where D is a left E E → R fppf -torsor and α (D∗ ) is a section. X ∈E R Here a pair ( , α) is isomorphic to ( ′, α′) if there exists a morphism of -torsors π : ′ E E X E→E such that π α = α′. The datum of a section α (D∗ ) is equivalent to the datum of an ◦ ∈ E R isomorphism of -torsors X ≃ 0 D∗ D∗ , g g α, E | R −→ E| R 7→ · where := is viewed as the trivial -torsor. The loop group LX acts on the affine Grass- E0 X X mannian via g [( , α)] = [( , gα)]. · E E Proposition 4.2. If Spec(k[[t]]) is a smooth affine group scheme, then the map LX GrX X → → given by g [( ,g)] induces an isomorphism of fpqc quotients: 7→ E0

X = GrX . F ∼ Proof. See [16, Proposition 1.3.6].

Here are a few observations:

(1) If ρ : H is a map of group schemes which are ﬂat of ﬁnite presentation over k[[t]], then X → there is a map of functors:

GrX Gr , ( , α) (ρ∗ ,ρ∗α), → H E 7→ E X X D∗ where ρ∗ = H denotes the push out of torsors, and ρ∗α = (id, α) : (H 0) R X E × E × E | → (H ) D∗ in this description. × E | R (2) If k′ is a k-ﬁeld extension, then we have:

′ GrX Spec(k ) Gr ′ . ×k ≃ X×k[[t]]Spec(k [[t]])

12 Remark 4.3. When X = GL , a G -bundle on D is canonically given by a rank n vector n E → R bundle, i.e., a rank n locally free R[[t]]-module L. The trivialization α induces an isomorphism of R((t))-modules L[t−1] R((t))n. By taking the image of L L[[t]] under this isomorphism, we ≃ ⊂ obtain a well defined finite locally free R[[t]] -module Λ = Λ R((t))n such that Λ[t−1] = (E,α) ⊂ R((t))n. Note that Λ depends only on the class of ( , α). E Now we can define affine Grassmannians for triality groups. In what follows, let k be a field with char(k) =2, 3. Suppose that the cubic primitive root ξ is in k. We set F = k((u)), F0 = k((t)) 3 6 with u = t. Thus F/F0 = k((u))/k((t)) is a cubic Galois field extension. Set Γ = Gal(F/F0) with generator ρ with ρ(u)= ξu. Then k[[t]] (resp. k[[u]]) is the ring of integers of F0 (resp. F ). Recall that (V, ) is a normal twisted composition algebra obtained from the para-Cayley ∗ algebra over F , i.e., there is a basis e , ..., e of (V, ) in the Table 1, with the multiplication { 1 8} ∗ x y = (id ρ)(x) ⋆ (id θ)(y) for all x, y C F, ∗ C ⊗ C ⊗ ∈ ⊗F0 where (C,⋆) is the split para-Cayley algebra. The quadratic form of (V, ) is determined by the ∗ multiplication from Lemma 2.3. Denote by , the bilinear form: , : V V F corresponding h i h i ⊗ → to the quadratic form. Let R be an F -algebra. Notice that the base change V R is isomorphic 0 ⊗F0 to R((u))8. A finitely generated projective submodule in V R is called a lattice in V R. ⊗F0 ⊗F0 We set L = 8 R[[u]]e , and call it the standard lattice in V R. ⊕i=1 i ⊗F0 In §3, we defined the triality group for special orthogonal groups over F0 = k((t)):

G(R) = Res (Spin(V, ))A3 (R) F/F0 ∗ = g SO(V, q)(R k((u))) g(x y)= g(x) g(y) for all x, y V k((u)) . ∼ { ∈ ⊗k((t)) | ∗ ∗ ∈ ⊗k((t)) } for any k((t))-algebra R. Let G be the aﬃne group scheme over k[[t]] that represents the functor from k[[t]]-algebras to groups that sends R to

G (R) := g SO (k[[u]] R) g(x y)= g(x) g(y) for all x, y L . { ∈ 8 ⊗k[[t]] | ∗ ∗ ∈ }

Here G is the parahoric subgroup given by L by Proposition 1.3.9, [6]. We denote by LGη (resp. + L G ) the functor from the category of k-algebras to groups given by LGη(R)= Gη(R((t))) (resp. + + L G (R)= G (R[[t]])). The quotient fpqc sheaf LGη/L G is by deﬁnition the aﬃne Grassmannian for the triality group. Our main theorem is:

Theorem 4.4. There is an LGη-equivariant isomorphism

LG /L+G F η ≃ where the functor F sends a k-algebra R to the set of ﬁnitely generated projective R[[u]]-modules L (i.e., R[[u]]-lattices) of V R = R((u))8 such that ⊗k ∼ (1) L is self dual under the bilinear form , , i.e., L Hom (L, R[[u]]). h i ≃ R[[u]] (2) L is closed under multiplication, L L L. ∗ ⊂ (3) There exists a L, such that q(a)=0, a a,a =1. ∈ h ∗ i (4) For a as in (3), let e = a + a a. Then, we have e x = x¯ = x e for any x¯ satisfying ∗ ∗ − ∗ x,¯ e¯ =0. (Here, x¯ is the image of x under the canonical map L L/uL.) h i →

13 This theorem is proven in the next section. It gives a bijection between k-points in the aﬃne Grassmannian for the triality group and a certain set of k[[u]]-lattices in V that satisfy some special conditions.

5 Proof of the main theorem

Similar to the proof of [11, Theorem 4.1] in the unitary group case, it suﬃces to check the following two statements:

(i) For any R, g LG (R), L = g(L) satisfies condition (1)-(4). ∈ η (ii) For any L F (R) with (R, ) a local henselian ring with maximal ideal , there exists ∈ M M g LG (R) such that L = g(L). ∈ η Part (i) is easy to prove, since g preserves the bilinear form , and the product . For any h i ∗ x, y L, let x = g(x ),y = g(y ) where x ,y L. Then x y = g(x ) g(y ) = g(x y ) ∈ 0 0 0 0 ∈ ∗ 0 ∗ 0 0 ∗ 0 ∈ L, so (2) satisfied. (1) is obvious via g(x),g(y) = x, y . For (3), let a = g(e ). Then h i h i 4 a a,a = g(e ) g(e ),g(e ) = g(e ),g(e ) = e ,e = 1, and q(a) = q(e ) = 0. For h ∗ i h 4 ∗ 4 4 i h 5 4 i h 4 5i 4 g(e)= g(a)+ g(a a), we have g(e) g(x)+ g(x)= g(x) g(e)+ g(x)= 0 for any g(x) satisfying ∗ ∗ ∗ g(x),g(e) = x, e = 0. h i h i To prove part (ii), the key is to find a basis in L such that the multiplication table under the basis is the same as Table 1, i.e., we need to find a basis f L such that f f = f for { i} ∈ i ∗ j k e e = e in the Table 1. Thus we can define g by g(e )= f , and g is then in LG (R). i ∗ j k i i η We claim that a as in assumption (3) is a primitive element in L (an element in L that extends to a basis of L). Consider the quotient map

R[[u]] R R/ = κ, → → M where κ is the residue ﬁeld of R. There is a base change L L κ, and we still denote → ⊗R[[u]] byx ¯ the image of x L. Considera ¯ L κ. We have a a, a¯ = 1, hencea ¯ = 0. By ∈ ∈ ⊗R[[u]] h ∗ i 6 Nakayama’s lemma, we can extend a to a basis of L. Similarly, we can show that a a is also ∗ a primitive element. Here a,a a are independent by a,a a = 1. Let v , ..., v be any base ∗ h ∗ i 1 6 extension for a,a a. We deﬁne a sublattice L L: ∗ 0 ⊂ L := x L x, a =0, x, a a =0 . 0 { ∈ | h i h ∗ i } 6 For any x L, we can write x as i=1 rivi + r7a + r8(a a) for some ri R[[u]]. Consider ′ ∈ ′∗ ′ ∈ ′ v = vi a, vi a a a a, vi a. ItP is easy to see that v ,a = 0, v ,a a = 0, so v L0. i − h i ∗ − h ∗ i h i i h i ∗ i i ∈ And v′,a,a a are linear independent. We obtain i ∗ 6 6 6 x = r v′ + (r + r v ,a a )a + (r + r v ,a )(a a). i i 7 ih i ∗ i 8 ih i i ∗ Xi=1 Xi=1 Xi=1

Therefore, L = R[[u]]a R[[u]](a a) L , where L is a sublattice of rank 6. ⊕ ∗ ⊕ 0 0 Set f = a,f = a a. Here f ,f play similar roles as for e ,e in the Table 1. By Lemma 1 2 ∗ 1 2 4 5

14 2.2 and Lemma 2.3, we obtain a hyperbolic subspace R[[u]]a R[[u]](a a) with: ⊕ ∗ f f = f , f f = f , 1 ∗ 1 2 2 ∗ 2 1 f f = f f =0, 1 ∗ 2 2 ∗ 1 q(f )= q(f )=0, f ,f =1. 1 2 h 1 2i Lemma 5.1. We have L f L , f L L , 0 ∗ i ⊂ 0 i ∗ 0 ⊂ 0 for i =1, 2.

Proof. For any x L , we have x f ,f = ρ( f f , x ) = 0,and x f ,f = ρ( f f , x )= ∈ 0 h ∗ i ii h i∗ i i h ∗ i i+1i h i∗ i+1 i 0 by Lemma 2.3. Similarly for f x. i ∗ Deﬁne the ρ-linear transformations t : L L , given by t (x) = x f for i = 1, 2. Take i 0 → 0 i ∗ i L = t (L )= L f . Trivially, t (L ) L . Both L are isotropic with rank (L ) 3 since f is i i 0 0 ∗ i i i ⊂ i i i ≤ i an isotropic element. For any x L , we have ∈ 0 (f x) f + (f x) f = θ( f ,f )x = x, 2 ∗ ∗ 1 1 ∗ ∗ 2 h 1 2i by Lemma 2.3. So L = L + L . Since rank(L ) 3, we must have a direct sum composition: 0 1 2 i ≤ L = L L . 0 1 ⊕ 2 Lemma 5.2.

(1) For any x L , t2(x)= f x (i =1, 2 mod 2). ∈ 0 i − i+1 ∗ (2) For any x L , t3(x)= x. ∈ i i − (3) From (2), t is a R[[t]]-isomorphism when restricted at L , more precisely, we have t : L i i i i → L , x x f . The inverse map t−1 = t2 is a θ-linear transformation. i 7→ ∗ i i − i (4) For x L ,y L , we have t (x),t (y) = ρ( x, y ). ∈ 1 ∈ 2 h 1 2 i h i Proof. (1) For x L , we have t2(x) = ((x f ) f )= ((f f ) x)= (f x) by Lemma ∈ 0 1 ∗ 1 ∗ 1 − 1 ∗ 1 ∗ − 2 ∗ 2.3. A similar argument gives t2(x)= f x. 2 − 1 ∗ (2) For any x L , we have t3(x)= ((f x) f ). Consider ∈ 1 1 − 2 ∗ ∗ 1 (f x) f + (f x) f = θ( f ,f )x = x, 2 ∗ ∗ 1 1 ∗ ∗ 2 h 1 2i by Lemma 2.3. Let x = z f L for some z L . Then f x = f (z f )=0by q(f ) = 0. ∗ 1 ∈ 1 ∈ 0 1 ∗ 1 ∗ ∗ 1 1 Hence (f x) f = x, and we obtain t3(x) = x. Similar calculations for y L , and gives 2 ∗ ∗ 1 1 − ∈ 2 t3(y)= y. 2 − Part (3) follows from (2) immediately. For (4), we know that t (x),t (y) = x f ,y f = h 1 2 i h ∗ 1 ∗ 2i ρ( f (y f ), x ), and h 1 ∗ ∗ 2 i f (y f )= t2(y f )= t2 t (y)= t3(y)= y, 1 ∗ ∗ 2 − 2 ∗ 2 − 2 · 2 − 2 by (1) and (2). Hence t (x),t (y) = ρ( x, y ). h 1 2 i h i

15 Remark 5.3. (1) From the proof of above Lemma, we can see that f L = 0, and L f =0 i ∗ i i ∗ i+1 for i =1, 2 mod 2. (2) Since L ,L are isotropic and , restricted to L is nondegenerate, the L are in duality 1 2 h i 0 i by the isomorphism L L∨ given by x x, . Hence L Hom(L , R[[u]]). 1 → 2 7→ h −i 1 ≃ 2 Lemma 5.4. We have

(1) L L R[[u]]f ,L L R[[u]]f , 1 ∗ 2 ⊂ 1 2 ∗ 1 ⊂ 2 (2) L L L (i =1, 2 mod 2). i ∗ i ⊂ i+1 Proof. (1) For any x L ,y L , we write x as x = x f with x L , and y as y = y f ∈ 1 ∈ 2 1 ∗ 1 1 ∈ 1 1 ∗ 2 with y L . Consider 1 ∈ 2 x y = (x f ) (y f )= ((y f ) f ) x + θ( x ,y f )f , ∗ 1 ∗ 1 ∗ 1 ∗ 2 − 1 ∗ 2 ∗ 1 ∗ 1 h 1 1 ∗ 2i 1 by Lemma 2.3. Notice that (y f ) f L f = 0. Thus we have x y = θ( x ,y f )f . 1 ∗ 2 ∗ 1 ∈ 2 ∗ 1 ∗ h 1 1 ∗ 2i 1 Further,

x ,y f = θ( t (x ),t (y f ) ) h 1 1 ∗ 2i h 1 1 2 2 ∗ 2 i = θ( x, t (y) ), h 2 i by Lemma 5.2 (4). Hence x y = ρ( x, t (y) )f . Similarly, we have y x = ρ( t (x),y )f . ∗ h 2 i 1 ∗ h 1 i 2 (2) For any x , x L , we ﬁrst claim that x x L . Consider x x ,f = θ( f 1 2 ∈ 1 1 ∗ 2 ∈ 0 h 1 ∗ 2 1i h 1 ∗ x , x )=0by f L = 0, and x x ,f = ρ( x f , x )=0by L f = 0. Using Lemma 1 2i 1 ∗ 1 h 1 ∗ 2 2i h 2 ∗ 2 1i 1 ∗ 2 2.3, we ﬁnd that

t (x ) t (x ) = (x f ) (x f ) 1 1 ∗ 1 2 1 ∗ 1 ∗ 2 ∗ 1 = f (x (x f )) − 1 ∗ 2 ∗ 1 ∗ 1 = f (f (x x )) 1 ∗ 1 ∗ 1 ∗ 2 by x f ,f = 0 and f , x = 0. We also have f (f (x x )) = f ( t2(x x )) = h 1 ∗ 1 1i h 1 2i 1 ∗ 1 ∗ 1 ∗ 2 1 ∗ − 2 1 ∗ 2 t4(x x )= t (x x ). Therefore, 2 1 ∗ 2 − 2 1 ∗ 2 t (x ) t (x )= t (x x ). 1 1 ∗ 1 2 − 2 1 ∗ 2 Since x x L , we obtain that t (x x ) L . Hence L L L . Similarly, L L L . 1∗ 2 ∈ 0 2 1∗ 2 ∈ 2 1∗ 1 ⊂ 2 2∗ 2 ⊂ 1 We now prove that L has the same multiplication table as the Table 1: We want to ﬁnd a basis x , x , x of L (resp. y ,y ,y of L ) such that t (x ) = id (resp. t (y ) = id). { 1 2 3} 1 { 1 2 3} 2 1 i − 2 i − Consider the quotient map R[[u]] κ = R[[u]]/( ,u). We set L¯ = L κ, L¯ = L κ → M ⊗R[[u]] i i ⊗R[[u]] with multiplicationx⋆ ¯ y¯ = x y, and ∗ t¯ : L¯ L¯ , given by t¯ (¯x)=¯x⋆ f¯ , i i → i i i for i =0, 1, 2.

Proposition 5.5. Given (L, , , ) satisfying (1)-(4) as above. Then (L,⋆¯ ) is isomorphic to the ∗ h i split para-Cayley algebra.

16 Proof. It is easy to see q(¯x⋆ y¯) = q(¯x)q(¯y), and x⋆¯ y,¯ z¯ = y¯ ⋆ z,¯ x¯ , so L¯ is a symmetric h i h i composition algebra. By [8, Lemma (34.8)], a symmetric algebra is a para-Cayley algebra if and only if it admits a para-unit, i.e., there exist an elemente ¯ L¯, such that ∈ e⋆¯ e¯ =¯e, e⋆¯ x¯ =x⋆ ¯ e¯ = x,¯ − for allx ¯ L¯ satisfying e,¯ x¯ = 0. Set e = f +f in our case. We can see that e is an idempotent ∈ h i 1 2 element by e⋆e = (f + f ) (f + f )= f + f = e. By condition (4), we gete⋆ ¯ x¯ =x⋆ ¯ e¯ = x¯, 1 2 ∗ 1 2 1 2 − for allx ¯ L¯ satisfying e,¯ x¯ =0. Thuse ¯ is a para-unit in L¯, and L¯ is a para-Cayley algebra. It ∈ h i is split since q is an isotropic norm. Lemma 5.6. For t¯ : L¯ L¯ , we have t¯ (¯x)=¯x⋆ f¯ = x¯ for any x¯ L¯ . Then L¯ = L¯ ⋆ f¯ = i i → i i i − ∈ i i 0 i x¯ L¯ x⋆¯ f¯ = x¯ , i =1, 2. { ∈ 0 | i − } Proof. By [8, Lemma (34.8)], we can deﬁnex ¯ y¯ = (¯e⋆ x¯) ⋆ (¯y ⋆ e¯) as a unital composition ⋄ algebra with the identity elemente ¯. We havex⋆ ¯ y¯ = r(¯x) r(¯y), where r(¯x)= e,¯ x¯ e¯ x¯ is the ⋄ h i − conjugation ofx ¯. By [15, Proposition 1.2.3],

x¯ y¯ +¯y x¯ x,¯ e¯ y¯ y,¯ e¯ x¯ + x,¯ y¯ e¯ =0. ⋄ ⋄ − h i − h i h i Usingx⋆ ¯ y¯ = r(¯x) r(¯y) and r(¯x), r(¯y) = x,¯ y¯ , we obtain ⋄ h i h i x⋆¯ y¯ +¯y⋆ x¯ = e,¯ x¯ r(¯y)+ e,¯ y¯ r(¯x) x,¯ y¯ e.¯ h i h i − h i Lety ¯ = f¯. We getx⋆ ¯ f¯ + f¯ ⋆ x¯ = r(¯x). Therefore, ifx ¯ L¯ f¯, we have f¯ ⋆ x¯ =0by q(f¯) = 0, i i i ∈ 0 ∗ i i i and x⋆¯ f¯ =x⋆ ¯ f¯ + f¯ ⋆ x¯ = e,¯ x¯ e¯ x¯ = x.¯ i i i h i − − This implies L¯ ⋆ f¯ x¯ L¯ x⋆¯ f¯ = x¯ . It is obvious that x¯ L¯ x⋆¯ f¯ = x¯ L¯ ⋆ f¯. 0 i ⊂{ ∈ 0 | i − } { ∈ 0 | i − }⊂ 0 i Hence we get L¯ = L¯ ⋆ f¯ = x¯ L¯ x⋆¯ f¯ = x¯ , i 0 i { ∈ 0 | i − } and t¯ = id. i − So far we know t : L L is a ρ-linear isomorphism with t3 = id, and t¯ = id. We i i → i i − i − will use non-abelian Galois cohomology to prove that t and id are the same up to ρ-conjugacy. i − More precisely, if we ﬁx a basis for L = R[[u]]3 and let A GL (R[[u]]) represent t , we can ﬁnd i ∼ i ∈ 3 i a new basis for L with transition matrix b GL (R[[u]]), such that i ∈ 3 I = b−1A ρ(b). − i Let Γ = 1,ρ,θ be the cyclic group. Set B = Aut(L ) = GL (R[[u]]). Consider the quotient { } 1 3 map R[[u]] κ. Since (R[[u]], (u)), (R, ) are henselian pairs, we obtain the exact sequence: → M 1 U GL (R[[u]]) GL (κ) 1 → → 3 → 3 → where U is the kernel of GL (R[[u]]) GL (κ). Here Γ acts on GL (R[[u]]) by ρ(u)= uξ, and Γ 3 → 3 3 acts trivially on GL3(κ). We obtain the exact sequence of pointed sets:

1 U Γ GL (R[[u]])Γ GL (κ)Γ H1(Γ,U) H1(Γ, GL (R[[u]]) H1(Γ, GL (κ)). → → 3 → 3 → → 3 → 3

17 by [13, Proposition 38]. Since U is a unipotent group over k[[u]] with char(k) = 3, we have 1 1 6 H (Γ,U) = 1. Hence the only element mapped to the base point of H (Γ, GL3(k)) is the base point of H1(Γ, GL (R[[u]]), i.e., for any [a ] H1(Γ, GL (R[[u]])) satisfying [¯a ] = 1, we have 3 s ∈ 3 s [as] = 1. Consider t : L L . The subgroup of GL (R[[u]])generated by t is 1,t2, id, t , t2, id 1 1 → 1 3 1 { 1 − − 1 − 1 } given by t3 = id. If we ﬁx the basis and use A to represent t , we get t2 = A ρ(A ), 1 − 1 1 1 1 1 t3 = A ρ(A )θ(A )= I. Deﬁne a map: 1 1 1 1 − a :Γ GL (R[[u]]) → 3 given by ρ a = A . Using a = a sa , we get θ a = a ρ(a ) = A ρ(A ), and 7→ ρ − 1 st s t 7→ θ ρ ρ 1 1 1 a = I. Hence the image of Γ = ρ, θ, 1 is the subgroup t4 = t ,t8 = t2,t12 = id t . 7→ 1 { } { 1 − 1 1 1 1 } ⊂ h i This is a 1-cocycle. Consider the image [¯a] of [a] under the map

1 H1(Γ, GL (R[[u]]) H1(Γ, GL (k)). → 3 → 3 We get [¯a ] = [t¯] = 1 by Lemma 5.6. Therefore [a ] = 1. In matrix language, there exist ρ − ρ b GL (R[[u]]) such that ∈ 3 I = b−1( A )ρ(b), t id. − 1 1 ∼− We have a similar conclusion for t2. Using the above we see that there exist a basis x , x , x for L , and a dual basis y ,y ,y { 1 2 3} 1 { 1 2 3} for L , such that t (x )= x , t (y )= y , x ,y = δ . By Lemma 5.2, we have 2 1 i − i 2 i − i h i ji ij x f = x , f x =0, i ∗ 1 − i 1 ∗ i x f =0, f x = x , i ∗ 2 2 ∗ i − i y f =0, f y = y , i ∗ 1 1 ∗ i − i y f = y , f y =0. i ∗ 2 − i 2 ∗ i By Lemma 5.4, we have x y = δ f , y x = δ f . i ∗ j − ij 1 i ∗ j − ij 2 It remains to calculate the terms in L L . To approach this goal, we define a wedge product i ∗ i : L L L given by ∧ i × i → i+1 u v := t−1(u) t (v), ∧ i ∗ i for any u, v L . Let u L . It is immediate to get ∈ i ∈ 1 u u = t−1(u) t (u) ∧ 1 ∗ 1 = (f u) (u f ) 2 ∗ ∗ ∗ 1 = ((u f )) u) f ∗ 1 ∗ ∗ 2 = f f =0 1 ∗ 2 by f ,u f = 0, q(u) = 0. By linearizing the equation, we find u v = v u for u, v L . h 2 ∗ 1i ∧ − ∧ ∈ 1 A similar argument can be made for u, v L . Now define a trilinear function ,, on L ∈ 2 h i i by u,v,w := u, v w . It is an alternating trilinear function since u,w,v = u, w v = h i h ∧ i h i h ∧ i

18 u, v w = u,v,w , and −h ∧ i −h i v,u,w = v,u w h i h ∧ i = v,t−1(u) t (w) h i ∗ i i = ρ( t (w) v,t−1(u) ) h i ∗ i i = t (t (w) v),u h i+1 i ∗ i = t2(w) t (v),u h i ∗ i i = w v,u = u, v w . h ∧ i −h ∧ i We can now calculate the terms in L L . Consider x x . We have x x , x = x i ∗ i 1 ∗ 2 h 1 ∗ 2 1i −h 1 ∗ x ,t (x ) = x x , x f =0 by x ,f = 0. Similarly x x , x = 0. Hence we have 2 1 1 i −h 1 ∗ 2 1 ∗ 1i h 2 1i h 1 ∗ 2 2i x x = by for some b = x x , x R[[u]]. Multiplying by y on the right side, we obtain 1 ∗ 2 3 h 1 ∗ 2 3i ∈ 1 (x x ) y = (by ) y . Since (x x ) y + (y x ) x = θ( x ,y )x = x , and y x = 0, 1 ∗ 2 ∗ 1 3 ∗ 1 1 ∗ 2 ∗ 1 1 ∗ 2 ∗ 1 h 1 1i 2 2 1 ∗ 2 we have x = ρ(b)(y y ). 2 3 ∗ 1 Therefore b,ρ(b)−1 R[[u]], which implies b R[[u]]∗. Let b = 1 (replace by by y , and also ∈ ∈ − 3 − 3 replace b−1x by x ), We get x x = y . We can perform similar calculations for the other 3 − 3 1 ∗ 2 − 3 x x and y y . By using the alternating trilinear form, we obtain i ∗ j i ∗ j Table 2: x x Table 3: y y i ∗ j i ∗ j x x x y y y ∗ 1 2 3 ∗ 1 2 3 x 0 -y y y 0 x x 1 3 2 1 − 3 2 x y 0 -y y x 0 x 2 3 1 2 3 − 1 x y y 0 y x x 0 3 − 2 1 3 − 2 1 Therefore, we complete the multiplication table of L. By letting g(e4)= f1,g(e5)= f2, and

g(e1)= x1, g(e6)= x2, g(e7)= x3, g(e8)= y1, g(e3)= y2, g(e2)= y3.

We obtain g(e ) g(e )= g(e e ). So, there exist g LG (R) such that L = g(L). i ∗ j i ∗ j ∈

19 References

[1] A. Beauville and Y. Laszlo. Un lemme de descente. C. R. Acad. Sci. Paris S´er. I Math., 320(3):335–340, 1995.

[2] A. Beilinson and V. Drinfeld. Quantization of hitchin’s integrable system and hecke eigen- sheaves. preprint.

[3] U. G¨ortz. On the ﬂatness of models of certain Shimura varieties of PEL-type. Math. Ann., 321(3):689–727, 2001.

[4] U. G¨ortz. On the ﬂatness of local models for the symplectic group. Adv. Math., 176(1):89– 115, 2003.

[5] N. Jacobson. Basic algebra. I. W. H. Freeman and Company, New York, second edition, 1985.

[6] M. Kisin and G. Pappas. Integral models of Shimura varieties with parahoric level structure. Publ. Math. Inst. Hautes Etudes´ Sci., 128:121–218, 2018.

[7] M.-A. Knus. Quadratic and Hermitian forms over rings, volume 294 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1991. With a foreword by I. Bertuccioni.

[8] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol. The book of involutions, volume 44 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits.

3 [9] M.-A. Knus and J.-P. Tignol. Triality and algebraic groups of type D4. Doc. Math., (Extra vol.: Alexander S. Merkurjev’s sixtieth birthday):387–405, 2015.

[10] G. Lusztig. Singularities, character formulas, and a q-analog of weight multiplicities. In Analysis and topology on singular spaces, II, III (Luminy, 1981), volume 101 of Ast´erisque, pages 208–229. Soc. Math. France, Paris, 1983.

[11] G. Pappas and M. Rapoport. Twisted loop groups and their aﬃne ﬂag varieties. Adv. Math., 219(1):118–198, 2008. With an appendix by T. Haines and Rapoport.

[12] G. Pappas and X. Zhu. Local models of Shimura varieties and a conjecture of Kottwitz. Invent. Math., 194(1):147–254, 2013.

[13] J.-P. Serre. Galois cohomology. Springer Monographs in Mathematics. Springer-Verlag, Berlin, english edition, 2002. Translated from the French by Patrick Ion and revised by the author.

[14] B. Smithling. Topological ﬂatness of local models for ramiﬁed unitary groups. II. The even dimensional case. J. Inst. Math. Jussieu, 13(2):303–393, 2014.

[15] T.A. Springer and F.D. Veldkamp. Octonions, Jordan algebras and exceptional groups. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000.

20 [16] X Zhu. An introduction to aﬃne Grassmannians and the geometric Satake equivalence. In Geometry of moduli spaces and representation theory, volume 24 of IAS/Park City Math. Ser., pages 59–154. Amer. Math. Soc., Providence, RI, 2017.

Dept. of Mathematics, Michigan State University, East Lansing, MI, 48824, USA E-mail address: [email protected]