Parahorics

Benedict H. Gross

June 18, 2012

Contents

1 Introduction 1

2 Roots and affine roots 3

3 Z and Z/dZ-gradings 5

4 The filtration of parabolics and parahorics (split case) 6

5 A cohomological classification 8

6 The affine diagram and its automorphism 12

7 From cohomological invariants to affine data 14

1 Introduction

This is an expository paper on the structure of parahoric subgroups. We will also treat the simpler caseof parabolic subgroups to motivate the exposition. All of the difficult results on the determination ofthese subgroups up to conjugacy are stated without proof. A full treatment can be found in the papersof Chevalley [5], Borel and Tits [3], and Bruhat and Tits [4]. We will be concerned with the internalstructure of these subgroups, once representatives for thedistinct conjugacy classes have been chosen.

LetA be a complete discrete valuation ring with quotient fieldk. Letπ be a uniformizing parameter inA, and assume that the residue fieldf = A/πA is finite of characteristicp. Let G be a split reductivegroup of rankℓ overA, which is absolutely simple and simply-connected. The additive group ofk hasa locally compact topology coming from the valuation, and the groupG(k) inherits a locally compact

1

topology for which the subgroupG(A) is compact. In fact,G(A) is a maximal compact subgroup ofG(k).

Unlike the case of real Lie groups, the maximal compact subgroups ofG(k) do not lie in a singleconjugacy class. There areℓ further conjugacy classes of maximal compact subgroups – the remainingmaximal parahoric subgroups ofG(k). Following Bruhat and Tits [4], we construct representativesfor the conjugacy classes which stabilize theℓ + 1 distinct vertices in an alcove of the building ofG

overk. Taking the intersections of these maximal compact subgroups, we obtain representatives of the2ℓ+1 − 1 conjugacy classes of general parahoric subgroups, which stabilize the distinct facets of thisalcove.

Bruhat and Tits have shown that each parahoric subgroupP can be identified with theA-valued pointsof a smooth group schemeGP overA with general fiberG over k [4]. HenceP has a descendingfiltrationP ⊲P1⊲P2⊲ · · · wherePm is the subgroup which is the kernel of the reduction mapGP (A) →GP (A/πmA). Moy and Prasad [12] have defined a refinement of this filtration

P ⊲ P1/d ⊲ P2/d ⊲ · · · .

The integerd ≥ 1 is the denominator of the coordinates of the barycenterx of the facet fixed byP ,andd = 1 for P = G(A). The quotient(P/P1/d) = L(f) is the group of points of a split reductivegroupL of rank ℓ over the residue fieldf, and fora ≥ 1 the quotientsPa/d/P(a+1)/d = Va(f) are thegroup of points of unipotent vector groupsVa over f. These afford algebraic linear representations ofL over f which are periodic with periodd: Va ∼= Va+d. In the notation of Moy and Prasad, we havePa/d = G(k)x,a/d.

By the internal structure ofP we mean the structure of the reductive quotientL over f and the alge-braic representationsVa of L which arise from the subquotients of the Moy-Prasad filtration. Reederand Yu have recently elucidated this structure using Vinberg’s theory of torsion automorphisms [15].As a result, when the residual characteristicp of f is a good prime forG which does not divided,they determine the parahoricsP whereVa has semi-stable orbits for the action ofL, in the sense ofgeometric invariant theory, as well as the parahorics wherethe dual representationV ∗

1 has stable orbits.The existence of stable orbits onV ∗

1 allows them to construct interesting families of supercuspidal rep-resentationsG(k), which are compactly induced from complex characters ofP1/d which are trivial onP2/d.

We begin this paper with a review of roots and affine roots [2, Ch VI]. We then review what is knownaboutZ andZ/dZ gradings of the Lie algebrag of G. Most of this theory is due to Richardson [16]Vinberg [23]. After this preparation, we define representatives of the conjugacy classes of of parabolicand parahoric subgroups, and discuss their internal structure, when the simple groupG is split overk. In the last three sections, we use results of Kneser and Tatein Galois cohomology to classifythe isomorphism classes of simple, simply connected groupsG which are split by a tamely ramifiedextension ofk. Following Bruhat and Tits [4] we then associate to each such group overk an affinediagrameR with an automorphismF , from which the internal structure of the parahoric subgroups ofG(k) can be determined. We end with a tabulation of the possible pairs (eR, F ) and the simple groupsG to which they are associated.

I want to thank Mark Reeder and Jiu-Kang Yu for teaching me this material. I also want to thank JKfor preparing the Dynkin diagrams in this paper.

2

2 Roots and affine roots

In this section, we will review the theory of root systems andaffine root systems as presented inBourbaki [2, Ch VI], giving the relation with the simply-connected, absolutely simple, split groupGoverA.

Let S be a maximal split torus inG, let X = Hom(Gm,S) be the cocharacter group ofS and letY = Hom(S,Gm) be its character group. There is a non-degenerate pairing

〈x, y〉 : X × Y → Hom(Gm,Gm) = Z

LetR be the set of roots ofG. This is the finite subset of non-zero elementsα in Y which occur in therepresentation ofS on the Lie algebrag of G. We have the Cartan decomposition of the Lie algebra(cf. [7]), as a representation of the torusS:

g = s ⊕∑

gα

where the Lie algebras of S gives ℓ copies of the trivial character, and each root spacegα is onedimensional, just as in the complex case [18].

LetZ(G) ⊂ S be the center ofG andS∗ = S/Z(G) be the corresponding maximal torus in the adjointgroupG∗ = G/Z(G). The character groupY ∗ of S∗ is theZ-submodule ofY spanned by the roots,and the cocharacter groupX∗ of S∗ is the subgroup ofη in X ⊗ Q which take integral values onR.The quotient groupY/Y ∗ is the Cartier dual of the centerZ(G), which is of multiplicative type, andwe have a duality of finite abelian groups

X∗/X × Y/Y ∗ → Q/Z.

There is also a set of corootsR∨ inside ofX, defined using the theory ofSL2, which is in bijectionwith the set of roots. Each rootα gives a simple reflectionssα of X and a simple reflectionsα∨ of Y ,defined by

sα(x) = x− 〈α, x〉α∨ sα∨(y) = y − 〈α∨, y〉α

These reflections preserve the sets of co-roots and roots respectively. The Weyl groupW = W (R) isthe subgroup ofAut(R) that they generate. The pairingX×Y → Z isW -invariant, and by inspectionthe groupW acts trivially on the quotientsX∗/X andY/Y ∗.

In Bourbaki, root systems are discussed independent of algebraic groups. Their notation is a bit differ-ent than ours. Here is the translation (cf. [2, Ch VI 1.9]):

Q(R) = Y ∗ P (R) = Y Q(R∨) = X P (R∨) = X∗.

HenceP/Q = Y/Y ∗ P∨/Q∨ = X∗/X.

Since our groupG is assumed to be absolutely simple and split,R is an irreducible (reduced) rootsystem of typeAn n ≥ 1, Bn n ≥ 2, Cn n ≥ 3, Dn n ≥ 4, G2, F4, E6, E7, orE8.

3

Let B be a fixed Borel subgroup ofG containingS. ThenB determines a subset of positive rootsR+.The positive roots are those which occur in the representation of S on the Lie algebrab of B. Thechoice ofB also determines a root basis∆ = {α1, α2, . . . , αℓ}, and every positive root has a basisexpansionα =

∑mi(α).αi with all coefficientsmi ≥ 0. The Weyl group is a finite Coxeter group,

generated by the simple reflectionssαi.

The finite groupAut(R)/W (R) = Aut(R,∆) is isomorphic to the group of automorphisms of theDynkin diagram ofR, which is a symmetric groupS1, S2, S3, with the latter case occurring only whenR is of typeD4. This group acts faithfully on the finite abelian quotient groupsX∗/X andY/Y ∗, andthe natural pairingX∗/X×Y/Y ∗ → Q/Z is non-degenerate andAut(R,∆)-invariant. The full groupAut(R) is isomorphic to the semi-direct productW.Aut(R,∆), and the choice of a Borel gives thesplitting.

The rootsα give linear functionals on the real vector spaceX ⊗ R. A fundamental domain for theaction ofW onX ⊗ R is given by the closed Weyl chamber whereαi(x) ≥ 0 for i = 1, 2, . . . , ℓ. WeequipX ⊗R with the sup norm|x| = Max{|αi(x)|} and letC be the compact spherical alcove whichis the intersection of the closed Weyl chamber with the unit sphere{x : |x| = 1}. A facetF of Cconsists of the pointsx where a fixed proper subset of theαi vanish, and the remaining subsetΣ takepositive values. The barycenterx of F is the point where the remaining subsetΣ of the basic rootsall satisfyαi(x) = 1. Since all roots take integral values on the barycenter,x is a cocharacter of theadjoint torusS∗, and gives a homomorphism

x = η : Gm → S∗ → G∗ → Aut(G).

For example, ifx is the barycenter of the interior of the alcoveC, thenαi(x) = 1 for all simple roots,andx = η is the co-characterρ∨ of S∗ given by half the sum of the positive co-roots.

Letβ =

∑miαi

be the highest root. This is by definition the highest weight for S in the adjoint representation ofG.The multiplicitiesmi = mi(β) in the basis expansion satisfymi ≥ 1 for all i, and

∑mi = h − 1,

whereh is the Coxeter number ofG. These integers, which are critical in what follows, are tabulatedfor the different root systems in [2]. If α is any positive root, then the multiplicitymi(α) of αi in thebasis expansion ofα satisfies0 ≤ mi(α) ≤ mi(β).

We now define the affine root system associated to the split groupG. Consider the set of affine linearfunctionalsψ = ψ(α, n) = α + n onX ⊗ R, with α in R andn in Z. We say thatα is the gradientof the affine rootψ(α, n). The affine rootsψi = ψ(αi, 0) = αi together withψ0 = ψ(−β, 1) = 1 − βform a basis for the affine root system, and the reflections in theℓ+ 1 affine hyperplanes whereψi = 0generate an affine Coxeter group. Puttingm0 = 1, we have the relation

∑miψi = 1

The (closure of the) fundamental alcove inX ⊗R is the compact regionC where theℓ+1 basic affineroots satisfyψi(x) ≥ 0. Since the basic rootsαi = ψi take non-negative values onC, the same is truefor all of the positive roots. Since the highest rootβ satisfies1 − β(x) = ψ0(x) ≥ 0 onC, all positiveroots take values0 ≤ α(x) ≤ 1 onC.

4

A facetF of C consists of the points where a proper subset of the basic affine roots take the value0and the non-empty complementary subset of basic affine rootstake non-zero values. The barycenterx of the facetF is the point ofF where the remaining basic affine roots in the complementary subsetΣ of {0, 1, . . . , ℓ} take the same (rational) value. This value is completely determined by the relation∑miψi = 1. It is equal to1/d with d = d(F ) is the sum of the multiplicitiesmi over the subsetΣ.

For example, if the complementΣ consists of a single affine rootψj, then the facet is a vertexx ofC andψj(x) = 1/mj. At the other extreme, if the facetF is the interior ofC, so the complementΣcontains of all the basic affine roots, then the common valueψi(x) at the barycenter is equal to1/h. Ingeneral, we have1 ≤ d ≤ h.

For a general barycenterx, the elementd.x of X ⊗ R takes integral values on all of the roots, so is acocharacter of the adjoint torus and gives a homomorphism

d.x = η : Gm → S∗ → G∗ → Aut(G).

3 Z and Z/dZ-gradings

In this section, we consider the gradings of the Lie algebra which are induced by a homomorphism

η : Gm → S∗ → G∗ → Aut(G)

as well as by the restriction ofη to the subgroupµd of d-torsion inGm.

TheGm action gives aZ-grading of the Lie algebra

g =∑

a∈Z

g(a),

whereg(a) is the subspace wheret in Gm acts by multiplication byta. The reductive subgroupG(0)of G which is fixed byGm containsS as a maximal split torus. It acts linearly on each subspaceg(a).For example,g(a) = g(0) is the adjoint representation ofG(0).

Whena is non-zero, the subspaceg(a) decomposes as a representation ofS as the direct sum of onedimensional root spaces:

g(a) =∑

〈η,α〉=a

gα.

In particular, whena 6= 0, the eigenspaceg(a) consists entirely of nilpotent elements. Whena = 0,and one must add theℓ dimensional spaces.

Proposition 3.1 If a 6= 0, then everyG(0) invariant polynomial ong(a) is a constant.

This was proved by Richardson[16] and Vinberg[23]. It follows from the fact that over an algebraicallyclosed field,G0 has an open dense orbit ong(a).

5

Now consider the restriction of the homomorphismη to µd. This gives rise to aZ/dZ-grading ofg

g =∑

a∈Z/dZ

g(a)

whereg(a) is the subspace ofg whereζ in µd acts by multiplication byζa [17]. The reductive subgroupG0 of G which is fixed byµd containsS as a maximal split torus and acts on each subspaceg(a). Forexample,g(0) is the adjoint representation ofG0.

Whena is non-zero (modulod), the subspaceg(a) decomposes as a representation ofS as the directsum of one dimensional root spaces:

g(a) =∑

〈η,α〉≡a

gα.

Whena = 0, and one must add theℓ dimensional spaces.

These representations were studied overf = C by Kostant and Rallis whend = 2, [9], and by Vinberg[23] and his school [13] whend > 2. Levy [11] considered the general situation where the character-istic of f is a good prime which does not divided. Under these assumptions, one can use the existenceof Cartan subspaces and the theory of complex reflection groups to prove the following generalizationof Chevalley’s theorem [6] (which is the cased = 1).

Proposition 3.2 TheG0-invariant polynomials ong(a) form a polynomial ring, withr(a) independentgenerators.

As a corollary, the geometric quotientg(a)/G0 is affine space of dimensionr(a) over f. The dimen-sions0 ≤ r(a) ≤ ℓ are interesting invariants of theµd action; we haver(a) > 0 if and only if g(a)contains semi-simple elements.

4 The filtration of parabolics and parahorics (split case)

We first define the parabolic subgroups ofG, which are algebraic subgroupsP overA with G/Pprojective. More precisely, we define the2ℓ − 1 parabolic subgroups which contain the fixed BorelB,as these represent the distinct conjugacy classes. They correspond bijectively to the barycentersx offacetsF of the compact spherical alcoveC. We recall thatC is the intersection of the closed Weylchamber defined by the inequalitiesαi(x) ≥ 0 with the unit sphere{x : |x| = 1}, and a barycenterxsatisfiesαi(x) = 1 for a non-empty subsetΣ of ∆ andαj(x) = 0 for the basic roots in∆ − Σ. Henceall rootsα take integral values onx andx = η is a cocharacter of the adjoint torus. Henceη gives ahomomorphism

x = η : Gm → S∗ → G∗ → Aut(G)

and by results of the previous section, an integral gradingg =∑

a∈Zg(a).

For each rootα, we let Uα be the corresponding root group (isomorphic toGa) in G. Then theparabolicP = Px is generated by the torusS and the root groupsUα, for those rootsα which satisfy〈η, α〉 ≥ 0. We define a terminating descending filtration

P ⊲P1 ⊲P2 ⊲ · · ·Pm ⊲ 1

6

where fora ≥ 1, Pa is the unipotent subgroup ofP generated the root groupsUα with 〈η, α〉 ≥ a.The integerm = 〈η, β〉, whereβ is the highest root. The subgroupP1 is the unipotent radical ofP,and fora ≥ 1 the quotientVa = Pa/Pa+1 is a vector group, which is a direct sum of the root spacesgα for those roots with〈η, α〉 = a. HenceVa is isomorphic tog(a) as a representation ofS.

The groupP/P1 = L is the Levi quotient ofP, which is reductive of rankℓ with maximal split torusS. From the definition, it is easy to see that the elements of∆ − Σ give a root basis forL. HenceLhas the same root datum as the subgroupG0 coming from the grading, and the two reductive groupsare isomorphic. One can then use the exponential map (cf. [7]) to prove the following

Proposition 4.1 The representationVa of L is isomorphic to the representationg(a) of G0.

It follows that L has no polynomial invariants onVa, other than constants. The open orbit over aseparably closed field shows that eachVa is a prehomogeneous vector space. Since the weights ofVaare all roots ofG, the representations which occur are very restricted. In particular, whenG is simplylaced, the abelianizationV1 of the unipotent radicalP1 is the direct sum of#Σ irreducible minusculerepresentations ofL, with lowest weights the roots inΣ and distinct central characters.[1].

We now give the analogous definition and results for the parahoric subgroups which contain a fixedIwahori (= the elements ofG(A) which reduce to the fixed Borel inG(f)). Equivalently, we will definethe parahorics subgroups which contain a fixed maximal pro-p-subgroup ofG(k) (= the elements ofG(A) which reduce to the unipotent radical of a fixed Borel inG(f)). Recall the closed alcoveC inX ⊗ R which is defined by the inequalitiesψi(x) ≥ 0. Let x be the barycenter of a facetF of C. Wedefine the parahoric subgroupP = Px fixing the facetF , as well as the Moy-Prasad filtrationPx,a/dassociated to the barycenterx of F with denominatord, as follows. Fix a Chevalley structure onGoverA, consisting of the split maximal torusS and an isomorphismeα : Ga → Uα overA for everyroot α, whereUα is the corresponding root group. Associated to each affine root ψ(α, n) we definethe subgroupUψ = eα(π

nA) of Uα(k). ThenP = Px is the subgroup ofG(k) which is generated byS(A) and the subgroupsUψ, for the affine roots which satisfyψ(x) ≥ 0.

For example, ifx is the vertex ofC whereψ0 = 1 − β takes the value1, thenα(x) = 0 for all roots ofG, andP = G(A). At the other extreme, ifx is the barycenter of the interior ofC, then0 < α(x) < 1for all positive roots. HenceP is the subgroup ofG(A) generated byS(A), Uα(A) for positive roots,andUα(πA) for negative roots. This is just the Iwahori subgroup which reduces to the fixed Borelsubgroup ofG moduloπ. Finally, if x is a barycenter whereψ0(x) = 1−β(x) = 0, then the parahoricPx contains the unipotent subgroupU−β(π

−1A). In particular,P is not contained inG(A).

To define the Moy-Prasad filtrationPa/d = G(k)x,a/d whered is the denominator ofx, we letSn be thesubgroup of the torusS(A) which is the kernel of the reduction mapS(A) → S(A/πnA). ThenPa/dis generated bySn with n ≥ a/d and the subgroupsUψ, for the affine roots which satisfyψ(x) ≥ a/d.The quotientL = P/P1/d is reductive and contains the maximal torusS overA/πA = f. A basis forits root system is given by the gradients of the basic affine rootsψi which vanish atx.

Whena ≥ 1 is not divisible byd, the subquotientVa = Pa/d/P(a+1)/d is the direct sum of one dimen-sional root spacesgα overf whereα(x) + n = a/d for some integern. This determines its structure asa representation ofS, and constrains its structure as a representation ofL. Whena is divisible byd, soa/d = n is an integer, one has to add theℓ trivial root spaces coming from the Lie algebra ofS overf.

7

Lemma 4.2 The root spaces which occur in the representation ofS on the subquotientVa of P = Pxare precisely those which occur in the componentg(a) of theZ/dZ-grading ofg over f, which comesfrom the restriction toµd of the co-characterη = d.x : Gm → Sad. Hence these representations ofS

are isomorphic.

To prove this, we first observe thatη = d.x is a co-character ofSad, as it takes the value0 or 1 oneach basic rootαi and hence takes integral values on all the roots. To identifythe root spaces whichoccur ing(a), we note that whenever the identityα(x) + n = a/d holds for some integern, thenα(d.x) = 〈η, α〉 ≡ a modulod. Hence the root spacesgα which lie in Va are precisely those whichcontribute to the eigenspaceg(a), and theseS modules are isomorphic overf. The coincidence of theroot spaces in the lemma led Reeder and Yu [15] to a proof of the following stronger result.

Proposition 4.3 Let x be the barycenter of the facetF of C fixed byP , so that the basic affine rootstake the value0 or 1/d on x. Let η = d.x be the associated cocharacter ofSad. Then the subgroupG0 of G over f which is fixed byη(µd) is isomorphic to the reductive quotientL of P . Moreover, foreverya ≥ 1 the subquotientVa = Pa/d/P(a+1)/d is isomorphic as a representation ofL ∼= G0 to thesubmoduleg(a) in the associatedZ/dZ- grading ofg overf.

Since the weights in the representationsVa = g(a) of L are all roots ofG, one can show that therepresentations themselves are very restricted. For example, assume thatG is simply laced and thatd > 1. Then the representationV1 is the direct sum of#Σ irreducible minuscule representations ofL, which are distinguished by their central characters. The lowest weights for these representations, ascharacters ofS, are the gradients of the simple affine roots inΣ.

We now sketch their proof of the proposition, in the simple case where the local fieldk contains thedth roots of unity. LetE be the totally ramified Galois extensionk(Π) of k, whereΠd = π. TheGalois groupGal(E/k) is then isomorphic toµd(k) via its action onΠ. The barycenterx becomeshyperspecial in the building ofG over the extensionE, where there are more affine rootsψ = α+n/dto consider. In particular, the laticeg(E)x,a/d is the scalar multipleΠag(E)x,0 and the reductive quotientof G(E)x,0 is isomorphic toG overf. An argument of Serre and Tate [20] shows that the Galois groupGal(E/k) acts algebraically on the reductive quotient, and a short computation shows that this actionis the inverse of the adjoint action given byη : µd → T → Aut(G).

Since the reductive quotient ofG overk is the subgroup of Galois invariants in the reductive quotientoverE, this gives the isomorphismL ∼= G0 over f. Similarly, the quotientgx,a/d/gx,(a+1)/d is thevector space of Galois invariants inΠag(E)x,0/Π

a+1g(E)x,0. It then follows from our calculation ofthe Galois action ong andΠa that this quotient space is isomorphic to the representation g(a) in theZ/dZ- grading ofg overf.

5 A cohomological classification

In the previous section, we considered the internal structure of the parahoric subgroups of a split,simply-connected, simple groupG over a local fieldk with finite residue field. We now turn to the

8

general case of a simply-connected, simple groupG overk, assuming only thatG splits over a tamelyramified extension ofk.

A remarkable fact, which was discovered by Bruhat and Tits [4], is that the geometry of the alcove inthe building ofG overk, as well as the internal structure of its parabolic subgroups, depends only ontwo invariants of a combinatorial nature. The first is a simple affine diagrameR, which is a connectedaffine Coxeter graph of the type listed in Bourbaki [2] [Ch VI, Thm 4] together with an orientationchosen for each multiple edge. The second is a conjugacy class F in the finite groupAut(eR) ofautomorphisms of this affine diagram. More precisely, the affine diagrameR determines the structureof parahorics inG over the maximal unramified extensionK of k, and the automorphismF of theaffine diagram determines the descent of this structure tok. We will show how one can compute theaffine invariants(eR, F ) from the cohomological data describing the isomorphism class ofG overk,and will describe which invariants occur for the different isomorphism classes.

In this section, we recall the cohomological classificationof the simple, simply-connected groupsG

overk. Much of this theory works over a general field. The first invariant is the split groupG0 overk which becomes isomorphic toG over the separable closureks. The simply-connected groupG0 isdetermined up to isomorphism by its root system. LetS0 be a maximal split torus inG0, letX be thecocharacter group ofS0 andY be the character group. We recall that the set of rootsR is the finitesubset ofY of the non-trivial characters ofS0 which occur in the adjoint representation. SinceG0 issimple, the root system is reduced and irreducible of typeAn n ≥ 1, Bn n ≥ 2, Cn n ≥ 3, Dn n ≥4, G2, F4, E6, E7, orE8.

Fix an isomorphismf : G0 → G over the separable closureks. For an elementσ in the Galois groupof ks overk, we haveσf = f · aσ, whereaσ is a one cocycle with values inAut(G0)(k

s). The class ofaσ in the pointed cohomology setH1(k,Aut(G0)) determines the isomorphism class ofG overk. Webreak the description of this cohomology class into two parts.

The first is the imageqσ of aσ, as a cocycle with values in the quotient groupOut(G0)(ks). If we

fix a pinning (B0, T0, {Xi}) of G0 over k, the finite group of pinned automorphisms overk mapsisomorphically onto the groupOut(G0)(k

s). Since the group of pinned automorphisms is isomorphicto the constantetale groupAut(R,∆) the class ofqσ gives a homomorphism

q : Gal(ks/k) → Aut(R,∆)

up to conjugation. When viewed as a map to the pinned automorphisms, it determines the quasi-split in-ner formGq of G overk. SinceAut(G0) is the semi-direct product of the adjoint quotientG0/Z(G0)with the subgroup of pinned automorphisms, standard twisting arguments in non-commutative coho-mology [19] show that the fibre overq in the surjective mapH1(k,Aut(G0)) → H1(k,Out(G0)) canbe identified with theOut(Gq)(k) orbits on the set of classesc in H1(k,Gq/Z(Gq)), whereZ(Gq) isthe center of the quasi-split groupGq. The orbit of the classc is the second cohomological invariant.

All of this is true over an arbitrary fieldk. In our case, whenk is local andG splits over a tamelyramified extension, the homomorphismq is tamely ramified. Moreover, Kneser’s theorem [10] showsthat the coboundary induces an isomorphism of pointed sets

9

H1(k,Gq/Z(Gq)) → H2(k,Z(Gq)).

Finally, we can compute the finite abelian groupH2(k,Z(Gq)) from the root systemR of G using Tateduality [19]. Let Mq be the Cartier dual ofZ(Gq), which is a finite, commutative,etale group schemeoverk. Over the separable closure,Mq(k

s) = P/Q = Y/Y ∗ is the quotient of the weight lattice bythe root lattice. The Galois group acts via the homomorphismq with imageJq in Aut(R,∆). Tate’stheorem then gives canonical isomorphisms

H2(k,Z(Gq)) = Hom(H0(k,Mq),Q/Z) = Hom((Y/Y ∗)Jq ,Q/Z) = (X∗/X)Jq

.

The groupOut(Gq)(k) is isomorphic to the centralizer ofJq in Aut(R,∆), and this centralizer actsnaturally on the classc of G in (X∗/X)Jq . Summarizing our results, we have proved the following.

Proposition 5.1 The isomorphism class of the tamely ramified groupG over the local fieldk is com-pletely determined by the following data:

• The irreducible root systemR

• The tamely ramified homomorphismq : Gal(ks/k) → Aut(R,∆) with imageJq

• The orbit of the cohomology classc in the finite abelian group(X∗/X)Jq under the action of thecentralizer ofJq in Aut(R,∆).

Since the groupsAut(R,∆) andX∗/X are so small, we can tabulate all of the possibilities whichoccur. The imageJ = Jq of the homomorphismq is a subgroup ofS1, S2, S3, so is determined by itsorder, which we tabulate as#J .

10

R #J (X∗/X)J Dynkin Diagram

An−1 1 Z/nZ ◦ ◦ · · · ◦ ◦

A2n+1 2 Z/2Z◦ · · · ◦

◦uuuuIII

I l

◦ · · · ◦

A2n 2 1◦ · · · ◦ ◦

l

◦ · · · ◦ ◦

Bn 1 Z/2Z ◦ ◦ ◦ · · · ◦ +3◦

Cn 1 Z/2Z ◦ ◦ ◦ · · · ◦ks ◦

D2n 1 (Z/2Z)2 ◦sss

◦ ◦ ◦ · · · ◦ ◦◦

KKKD2n+1 1 Z/4Z

Dn 2 Z/2Z◦

ttt◦ ◦ ◦ · · · ◦ ◦ l

◦JJJ

D4 3, 6 1◦

rrr��◦66

◦◦

LLL

\\

G2 1 1 ◦_jt ◦

F4 1 1 ◦ ◦ +3◦ ◦

E6 1 Z/3Z◦ ◦ ◦ ◦ ◦

◦

E6 2 1◦ ◦

◦ ◦tttJJJ l

◦ ◦

E7 1 Z/2Z◦ ◦ ◦ ◦ ◦ ◦

◦

E8 1 1◦ ◦ ◦ ◦ ◦ ◦ ◦

◦

To illustrate, we will describe the tamely ramified simply-connected groupsG with root system of typeE6. There is the split groupG0 of rank6, and for each quadratic field extensionL of k which is eitherunramified or tamely ramified there is the quasi-split groupGq of rank4 which is split byL. Finally,whenq is trivial, the groupG0 has a inner formG of rank2, corresponding to a non-trivial classc inX∗/X = Z/3Z. Both non-trivial classes inX∗/X lie in the sameAut(R,∆) = Z/2Z orbit. (As aconsequence,whenG has rank2, the mapAut(G)(k) → Out(G)(k) = Z/2Z is not surjective.)

11

6 The affine diagram and its automorphism

We recall thatK is the maximal unramified extension ofk. LetE be the unique cyclic, tamely ramifiedextension ofK which splits the groupG, and lete be the degree ofE overK. SinceG is split overE,an alcove in the building ofG overE is a simplex whose facets correspond to the non-empty subsetsof the nodes of the extended Dynkin diagramR associated to the based root system(R,∆).

We can determine the structure of an alcove in the building ofG overK using the techniques of tamedescent. Since the cyclic groupGal(E/K) acts via an element of ordere in the groupAut(R,∆) ofpinned outer automorphisms ofG, the fixed affine root system has been computed in [14]. The alcoveof G overK is again a simplex, whose facets correspond to the non-emptysubsets of the nodes of thetwisted affine diagrameR.

The affine diagrams which arise are tabulated below. Note that the removal of any non-empty subsetof nodes (with the edges attached to those nodes) results in an ordinary Dynkin diagram. Whene = 1the positive integers attached to the nodes give the multiplicities of the simple rootsαi in the highestroot β. Whene > 1, they givee times the multiplicities of the simple roots in the highest short root,or twice the highest short root of the group fixed by the pinnedouter automorphism. Whene = 1 wewrite eR simply asR.

An

◦1

OOOOOOOOO

ooooooooo

◦1

◦1

· · · ◦1

◦1

Bn

◦1 KKK

◦2

◦2

· · · ◦2

+3◦2◦

1

sss

Cn ◦1

+3◦2

◦2

· · · ◦2

◦2

◦1

ks

Dn

◦1 KKK ◦

1sss◦2

◦2

· · · ◦2

◦2◦

1

sss ◦1

KKK

E6

◦1

◦2

◦3◦2

◦1

◦2

◦1

E7

◦1

◦2

◦3

◦4◦3

◦2

◦1

◦2

E8

◦2

◦4

◦6◦5

◦4

◦3

◦2

◦1

◦3

F4 ◦1

◦2

◦3

+3◦4

◦2

G2 ◦3

◦2

_jt ◦1

2A2n+1

◦2 KKK

◦4

◦4

· · · ◦4

◦2

ks◦2

sss

12

2A2n ◦2

+3◦4

◦4

· · · ◦4

◦4

+3◦4

2Dn ◦2

◦2

ks ◦2

· · · ◦2

◦2

+3◦2

2E6 ◦2

◦4

◦6

◦4

ks ◦2

3D4 ◦3

_*4◦6

◦3

Finally, we obtain the alcove ofG overk by unramified descent, which gives a homomorphism

Gal(K/k) → Aut(eR).

The imageF of Frobenius is a well-defined conjugacy class in the finite groupAut(eR). The facets ofthe alcove overk correspond to non-emptyF -stable subsets of the nodes ofeR.

To enumerate the pairs(eR, F ) which occur, we need to know the structure of the finite group ofautomorphisms for each affine diagram. Whene = 1 the groupAut(R) is the semi-direct product(X∗/X) ⋊ Aut(R,∆), and whene > 1 the groupAut(eR) is the abelian quotient(X∗/X)Gal(E/K).These finite groups are tabulated below, where we useT2n to denote the dihedral group of order2n andSn to denote the symmetric group onn letters. We give the semi-direct product decomposition whene = 1: the normal subgroupX∗/X acts simply-transitively on the nodes with multiplicity1 and thecanonical splitting is given by the subgroup fixing one such node.

13

eR Aut(eR) Affine Dynkin Diagram

An−1 T2n = Z/nZ ⋊ S2

◦

OOOOOOOOO

ooooooooo

◦ ◦ · · · ◦ ◦

Bn Z/2Z◦

KKK◦ ◦ · · · ◦ +3◦

◦sss

Cn Z/2Z ◦ +3◦ ◦ · · · ◦ ◦ ◦ks

D2n+1 T8 = (Z/4Z) ⋊ S2

◦KKK ◦

sss◦ ◦ · · · ◦ ◦

◦sss ◦

KKK

D2n T8 = (Z/2Z)2⋊ S2

◦KKK ◦

sss◦ ◦ · · · ◦ ◦

◦sss ◦

KKK

D4 S4 = (Z/2Z)2⋊ S3

◦ LLL ◦rrr◦

◦rrr ◦

LLL

E6 S3 = (Z/3Z) ⋊ S2

◦ ◦ ◦ ◦ ◦

◦

◦

E7 Z/2Z◦ ◦ ◦ ◦ ◦ ◦ ◦

◦

E8 1◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦

◦

G2 1 ◦ ◦_jt ◦

F4 1 ◦ ◦ ◦ +3◦ ◦

2A2n+1 Z/2Z◦

KKK◦ ◦ · · · ◦ ◦ks

◦sss

2A2n 1 ◦ +3◦ ◦ · · · ◦ ◦ +3◦

2Dn Z/2Z ◦ ◦ks ◦ · · · ◦ ◦ +3◦

2E6 1 ◦ ◦ ◦ ◦ks ◦

3D4 1 ◦ _*4◦ ◦

7 From cohomological invariants to affine data

To identify the parahoric structure ofG, we need a recipe to pass from the cohomological invariants(R, q, c) to the affine data(eR, F ). The recipe for the parahoric subgroups ofG over k is then asfollows. Suppose that there areℓ+ 1 orbits of the automorphismF on the nodes of the affine diagrameR . Then the groupG has rankℓ overk, and the parahoric subgroupsP of G(k) which contain a fixedIwahori correspond bijectively to the non-emptyF -stable subsetsΣ of the nodes ofeR. The integerd = d(P ) in the Moy-Prasad filtration

P ⊲ P1/d ⊲ P2/d ⊲ · · ·

is the sum, taken over the subsetΣ, of the integral labels of the respective nodes. The reductive quotientL(f) = (P/P1/d) is quasi-split overf. Over the separable closure,L contains the maximal split torus

14

S and its root basis is given by the complement ofΣ in the basic affine roots. The action ofF on thiscomplement determines the descent ofL to f.

The affine diagrameR is obtained from the root systemR and the restriction of the homomorphismqto the inertia subgroup ofGal(ks/k). If the imageIq of (tame) inertia inAut(R,∆) is cyclic of ordere, then the affine diagram has typeeR. Note that non-isomorphic quasi-split groupsGq can have thesame affine diagram, as the latter only depends on the ordere of the tame inertia subgroup, whereas theisomorphism class of the group depends on the tamely ramifiedextension ofk which splitsG. Thusnon-isomorphic groups can have the same internal parahoricstructure.

To determine the automorphismF of eR, we first assume thate = 1. Then the homomorphismq factorsthroughGal(K/k) and the image of Frobenius gives a conjugacy classs in the groupAut(R,∆).

If s = 1 the isomorphism class ofG is determined by theAut(R,∆) orbit of the cohomology classcin X∗/X. This gives a well-defined conjugacy classF = c × 1 in the groupAut(R) = (X∗/X) ⋊

Aut(R,∆).

If s 6= 1 the groupG is determined by the〈s〉 orbit of the cohomology classc in the quotient group(X∗/X)〈s〉. If we lift c to a classc∗ in X∗/X, then the productF = c∗ × s gives a well-definedconjugacy class inAut(R).

Whene > 1, F is equal to the cohomology classc in (X∗/X)I = (X∗/X)Jq = Aut(eR).

With this recipe, we can easily identify the different isomorphism classes of groups associated to eachpair (eR, F ). We give the affine diagram with its automorphism marked, then describe the simplegroup overk to which it belongs. This description is also tabulated (with a bit more information) in thesurvey article of Tits [22]. WhenF = 1, the reductive quotientsL of the parahoric subgroups ofG areall split over the residue fieldf. Tits calls such a groupG residually split.

◦ 244

4

◦ 1+++

i7→i+ai∈Z/nZ

◦ 0

◦n−1

���

◦n−2

���

The affine diagram isAn−1. This is ann-gon, andF is a rotation by±a units inT2n. Writea/n = b/min lowest terms. There arem orbits on the nodes, and the rank ofG overk ism− 1.

LetD be the division algebra of degreem2 overk with invariantb/m. ThenG = SLn/m(D). This isthe split formG0 = SLn whenF = 1 and hencea ≡ 0.

15

◦ · · · ◦◦

uuuuIII

I l

◦ · · · ◦

The affine diagram isAn−1 with n = 2m+ 1. This is ann-gon andF is a reflection fixing a vertex inT2n. There arem+ 1 orbits on the nodes, and the rank ofG overk ism.

Let L be the unramified quadratic extension ofk and letW be a non-degenerate Hermitian space ofodd rankn = 2m+ 1 overL. ThenG = SU(W ). This is the unramified quasi-split formGq.

◦ · · · ◦III

I

◦uuuuIII

I l ◦uuu

u◦ · · · ◦

The affine diagram isAn−1 with n = 2m. This is ann-gon andF is a reflection fixing a vertex inT2n

(hence fixing the opposite vertex). There arem+ 1 orbits on the nodes, and the rank ofG overk ism.

Let L be the unramified quadratic extension ofk and letW be a non-degenerate Hermitian space ofeven rankn = 2m overL which contains an isotropic subspace of dimensionm. ThenG = SU(W ).This is the unramified quasi-split formGq.

◦ ◦ · · · ◦ ◦l

◦ ◦ · · · ◦ ◦

The affine diagram isAn−1 with n = 2m. This is ann-gon andF is a reflection through the midpointof an edge inT2n. There arem orbits on the nodes, and the rank ofG overk ism− 1.

Let L be the unramified quadratic extension ofk and letW be a non-degenerate Hermitian spaceof even rankn = 2m overL which does not contain an isotropic subspace of dimensionm. ThenG = SU(W ).

◦KKK◦ ◦ · · · ◦ +3◦

◦sss

The affine diagram isBn andF is trivial. There aren+ 1 nodes and the rank ofG overk is n.

Let W be a non-degenerate orthogonal space of odd dimension2n + 1 over k (the bilinear pairingis even and the associated quadratic formQ is non-singular) which contains an isotropic subspace ofdimensionn. ThenG = Spin(W ). This is the split formG0.

16

◦JJJ

l ◦ ◦ · · · ◦ +3◦◦

ttt

The affine diagram isBn andF is the non-trivial involution. There aren orbits on the nodes and therank ofG overk is n− 1.

Let W be a non-degenerate orthogonal space of odd dimension2n + 1 overk (the bilinear pairing iseven and the associated quadratic formQ is non-singular) which does not contain an isotropic subspaceof dimensionn. ThenG = Spin(W ).

◦ +3◦ ◦ · · · ◦ ◦ ◦ks

The affine diagram isCn andF is trivial. There aren+ 1 nodes and the rank ofG overk is n.

LetW be a non-degenerate symplectic space of even dimension2n overk. ThenG = Sp(W ). This isthe split formG0.

◦ · · · ◦ ◦ks

◦uuuuIII

I l

◦ · · · ◦ ◦ks

The affine diagram isCn with n = 2m andF is the non-trivial involution. There arem + 1 orbits ofthe nodes and the rank ofG overk ism.

LetD be the quaternion division algebra overk and letW be a non-degenerate Hermitian space overD of dimensionn. ThenG = U(W ).

◦ · · · ◦ ◦ks

l

◦ · · · ◦ ◦ks

The affine diagram isCn with n = 2m+ 1 andF is the non-trivial involution. There arem+ 1 orbitsof the nodes and the rank ofG overk ism.

LetD be the quaternion division algebra overk and letW be a non-degenerate Hermitian space overD of dimensionn. ThenG = U(W ).

17

◦KKK ◦

sss◦ ◦ · · · ◦ ◦

◦sss ◦

KKK

The affine diagram isDn andF is trivial. There aren+ 1 nodes and the rank ofG overk is n.

LetW be a non-degenerate orthogonal space of even dimension2nwhich contains a maximal isotropicsubspace of dimensionn. ThenG = Spin(W ) and the center of the Clifford algebra is the splitetalequadratic extensionL = k + k. This is the split formG0.

◦;;

;;◦

����

l ◦ ◦ · · · ◦ ◦ l

◦

���� ◦

;;;;

The affine diagram isDn andF is the central element of order2 in T8. There aren − 1 orbits on thenodes and the rank ofG overk is n− 2.

Let W be a non-degenerate even orthogonal space of even dimension2n which does not contain anisotropic subspace of dimensionn − 1. ThenG = Spin(W ) and the center of the Clifford algebra isthe splitetale quadratic extensionL = k + k.

◦

��

◦ · · · ◦sssKKK◦

��l

◦

ff

◦ · · · ◦sssKKK◦

ff

The affine diagram isDn with n = 2m+ 1 andF is an element of order4 in T8. There arem orbits onthe nodes and the rank ofG overk ism− 1.

LetD be the quaternion division algebra overk and letW be an anti-Hermitian space of dimensionnoverD, such that the center of the Clifford algebra (in the sense ofTits [21] and Jacobson[8]) is thesplit etale quadratic algebraL = k + k. ThenG = U(W ).

18

◦YY

��

◦ · · · ◦sssKKK◦WW

��◦

�����

8888

8 l

◦◦ · · · ◦

sssKKK◦

The affine diagram isDn with n = 2m andF is an element of order2 in P∨/Q∨ which is not centralin T8. There arem+ 1 orbits on the nodes and the rank ofG overk ism.

LetD be the quaternion division algebra overk and letW be an anti-Hermitian space of dimensionnoverD, such that the center of the Clifford algebra is the splitetale quadratic algebraL = k+ k. ThenG = U(W ).

◦��

��◦ MMM<<◦ ◦ · · · ◦ ◦ l

◦qqq<<

◦

====

The affine diagram isDn andF has order2 in Aut(R,∆). There aren orbits on the nodes and the rankof G overk is n− 1.

Let W be a non-degenerate orthogonal space of even dimension2n where the center of the Cliffordalgebra is the unramified quadratic extensionL of k. ThenG = Spin(W ). This is the unramifiedquasi-split formGq.

◦YY

��

◦ · · · ◦sssKKK◦WW

��l

◦◦ · · · ◦

sssKKK◦

The affine diagram isDn with n = 2m + 1 andF is the remaining class of order2 in T8. There arem+ 1 orbits on the nodes and the rank ofG overk ism.

LetD be the quaternion division algebra overk and letW be an anti-Hermitian space of dimensionnoverD, such that the center of the Clifford algebra is the unramified quadratic extensionL of k. ThenG = U(W ).

19

◦

��

◦ · · · ◦sssKKK◦

��◦

�����

8888

8 l

◦

ff

◦ · · · ◦sssKKK◦

ff

The affine diagram isDn with n = 2m andF has order4 in T8. There arem orbits on the nodes andthe rank ofG overk ism− 1.

LetD be the quaternion division algebra overk and letW be an anti-Hermitian space of dimensionnoverD, such that the center of the Clifford algebra is the unramified quadratic extensionL of k. ThenG = U(W ).

◦��◦��

◦

<<<<

����

◦

◦

>>

The affine diagram isD4 andF has order3 in S4. There are3 orbits on the nodes and the rank is2.

This is the unramified quasi-split inner formGq, split by a cubic extension.

◦ ◦ ◦ ◦ ◦

◦

◦

The affine diagram isE6 andF is trivial. There are7 nodes and the rank is6.

This is the split formG0.

◦ ◦◦ ◦ ◦

tttJJJ l

◦ ◦

The affine diagram isE6 andF is a non-trivial involution inS3. There are5 orbits on the nodes and therank is4.

This is the unramified quasi-split formGq.

20

◦

,,

◦~~

◦LLL

++

◦rrr||

◦LLL rrr

◦

RR

◦

SS

The affine diagram isE6 andF is an element of order3 in S3. There are3 orbits on the nodes and therank is2.

◦ ◦ ◦ ◦ ◦ ◦ ◦

◦

The affine diagram isE7 andF is trivial. There are8 nodes and the rank is7.

This is the split formG0.

◦ ◦ ◦◦ ◦

tttJJJ l

◦ ◦ ◦

The affine diagram isE7 andF is the non-trivial involution. There are5 orbits on the nodes and therank is4.

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦

◦

The affine diagram isE8 andF is trivial. There are9 nodes and the rank is8.

This is the split formG0.

◦ ◦ ◦ +3◦ ◦

The affine diagram isF4 andF is trivial. There are5 nodes and the rank is4.

21

This is the split formG0.

◦ ◦_jt ◦

The affine diagram isG2 andF is trivial. There are3 nodes and the rank is2.

This is the split formG0.

◦KKK◦ ◦ · · · ◦ ◦ks

◦sss

The affine diagram is2A2m+1 andF is trivial. There arem+ 1 nodes and the rank ofG overk ism.

LetL be a tamely ramified quadratic extension ofk and letW be a non-degenerate Hermitian space ofeven rankn = 2m overL which contains an isotropic subspace of dimensionm. ThenG = SU(W ).This is a ramified quasi-split formGq.

◦JJJ

l ◦ ◦ · · · ◦ ◦ks

◦ttt

The affine diagram is2A2m+1 andF is the non-trivial involution. There arem orbits on the nodes andthe rank ofG overk ism− 1.

Let L be a tamely ramified quadratic extension ofk and letW be a non-degenerate Hermitian spaceof even rankn = 2m overL which does not contain an isotropic subspace of dimensionm. ThenG = SU(W ).

◦ +3◦ ◦ · · · ◦ ◦ +3◦

The affine diagram is2A2m andF is trivial. There arem+ 1 nodes and the rank ofG overk ism.

LetL be a tamely ramified quadratic extension ofk and letW be a non-degenerate Hermitian space ofodd rankn = 2m+ 1 overL. ThenG = SU(W ). This is a ramified quasi-split formGq.

22

◦ ◦ks ◦ · · · ◦ ◦ +3◦

The affine diagram is2Dn andF is trivial. There aren nodes and the rank ofG overk is n− 1.

LetW be a non-degenerate even orthogonal space of even dimension2n where the center of the Clif-ford algebra is a tamely ramified quadratic extensionL of k. ThenG = Spin(W ). This is a ramifiedquasi-split formGq.

◦ · · · ◦ +3◦◦

uuuuIII

I l

◦ · · · ◦ +3◦

The affine diagram is2Dn with n = 2m andF is the non-trivial involution. There arem orbits on thenodes and the rank ofG overk ism− 1

Let D be the quaternion division algebra overk and letW be an anti-Hermitian space of dimensionn overD, such that the center of the Clifford algebra is a tamely ramified quadratic extensionL of k.ThenG = U(W )

◦ · · · ◦ +3◦l

◦ · · · ◦ +3◦

The affine diagram is2Dn with n = 2m+ 1 andF is the non-trivial involution. There arem+ 1 orbitson the nodes and the rank ofG overk ism.

Let D be the quaternion division algebra overk and letW be an anti-Hermitian space of dimensionn overD, such that the center of the Clifford algebra is a tamely ramified quadratic extensionL of k.ThenG = U(W ).

◦ ◦ ◦ ◦ks ◦

The affine diagram is2E6 andF is trivial. There are5 nodes and the rank is4.

This is a tamely ramified quasi-split formGq.

23

◦ _*4◦ ◦

The affine diagram is3D4 andF is trivial. There are3 nodes and the rank is2.

This is a tamely ramified quasi-split formGq.

References

[1] H. Azad, M. Barry, G. SeitzOn the structure of parabolic subgroups, Communications in Alge-bra,18 (1990), pp. 551–562.

[2] N. Bourbaki,Lie groups and Lie algebras, Chap. 4-6, Springer-Verlag, 2002.

[3] A. Borel, J. Tits,Groupes reductifs, Publ. Math. IHES27 (1965), pp. 55–150.

[4] F. Bruhat, J. Tits,Groupes reductifs sur un corps locale I,II, Publ. Math. IHES41, 60(1972, 1984), pp. 5 – 251, 197–376.

[5] R. Carter,Finite groups of Lie type. Conjugacy classes and complex characters, Wiley, 1985.

[6] C. Chevalley,Classification des groupes algebriques semi-simples, Springer-Verlag, 2005.

[7] M. DemazureSous groupes paraboliques des groupes reductifs, SGA3, Springer Lecture Notes153(1970), pp. 426–517.

[8] N. Jacobson,Clifford algebras for algebras with involution of type D, J. Algebra,1 (1964),pp. 288–300.

[9] B. Kostant, S. Rallis,Orbits and representations associated with symmetric spaces, Amer. J.Math.93 (1971), pp. 753–809.

[10] M. Kneser,Galois-Kohomologie halbeinfacher algebraischer Gruppenuberp-adischen KorpernI,II , Math. Zeit.88, 89(1965), pp. 40–47, 250–272.

[11] P. LevyVinberg’sθ-groups in positive characteristic and Kostant-Weierstrass slices, Transform.Groups,14, no. 2, (2009), pp. 417–461.

[12] A. Moy, G. Prasad,Unrefined minimal K-types for p-adic groups, Inventiones math.,116(1994),pp. 393–408.

[13] D. Panyushev,On invariant theory ofθ-groups, J. Algebra,283(2005), pp. 655–670.

[14] M. Reeder,Torsion automorphisms of simple Lie algebras, L’Enseignement Math.,56(2) (2010),pp. 3–47.

[15] M. Reeder and J.-K. Yu,Epipelagic representations.

24

[16] R.W. Richardson,Conjugacy classes in parabolic subgroups of semisimple algebraic groups,Bull. London Math. Soc.,6 (1974), pp. 21–24.

[17] J.-P. Serre,Coordonnees de Kac, Oberwolfach Reports,3 (2006), pp. 1787-1790.

[18] J.-P. Serre,Complex semisimple Lie algebras, Springer Monographs in Mathematics (2001).

[19] J.-P. SerreGalois cohomologySpringer Monographs in Mathematics (2002).

[20] J.-P. Serre and J. Tate,Good reduction of abelian varietiesAnnals of math.88 (1968), pp. 492–517.

[21] J. Tits,Formes quadratiques, groupes orthogonaux, et algebres de Clifford, Inventiones math5(1968), pp. 19–41.

[22] J. Tits, Reductive groups overp-adic fields, Proc. Symp. Pure Math.,33, Part 1, Amer. Math.Soc., Providence, RI (1979), pp. 29–69.

[23] E.B. Vinberg,The Weyl group of a graded Lie algebra, Izv. Akad. Nauk SSSR Ser. Mat.,40no. 3(1976), pp. 488-526. English translation: Math. USSR-Izv.10 (1977), pp. 463–495.

[24] E.B. VinbergOn the classification of nilpotent elements of graded Lie algebras, Doklady 16(1975), pp. 1517-1520.

25