An application to the discriminant algebra of an algebra with an involution of the second kind is given

Maximal Indexes of Tits Algebras

A. S. Merkurjev¹

Received: April 16, 1996 Communicated by Ulf Rehmann

Abstract. LetGbe a split simply connected semisimple algebraic group over a eld F and letC be the center of G. It is proved that the maximal index of the Tits algebras of all inner forms of GL over all eld extensions L=F corresponding to a given characterofC equals the greatest common divisor of the dimensions of all representations of Gwhich are given by the multiplication bybeing restricted toC. An application to the discriminant algebra of an algebra with an involution of the second kind is given.

1991 Mathematics Subject Classication: Primary 20G15.

LetGbe an adjoint semisimple algebraic group dened over a eldF, let:G^e!G be the universal covering and letC= ker() denote the center ofG^e. In [13] Tits has constructed a homomorphism

:C(F)^!Br(F)

whereC(F) is the group of characters ofC dened overF and Br(F) is the Brauer group of F. For any character ²C(F) one can choose a central simple algebra A(called the Tits algebra), representing the class()²Br(F), in such a way that there is a group homomorphism

Ge^!GL1(A)

restricting to the character on the center C and inducing an irreducible represen- tation over a separable closureF^sep of the eldF. It follows from the representation theory of semisimple algebraic groups that the index ind(A) of the algebraAdivides the dimension of any irreducible representation:G^{eq !GL}(V) of a quasisplit inner form G^eq of G^e such that the restriction of to the center C^q of G^eq is given by the multiplication by (we identify the Galois modules of the character groupsC and C^q). Therefore, if we denote byn(G) the greatest common divisor of the dimensions of all such representations, then ind(A) dividesn(G). The numbersn(G) depend only on the class of the inner forms ofG, i.e. on the Dynkin diagramD= Dyn(G^sep), and the action of the absolute Galois group ofF on Aut(D). In particular, ifGis of

1I would like to thank the Universite de Franche-Comte at Besancon and the Alexander von Humboldt-Stiftung for nancial support.

inner type, then the numbersn(G) depend only on the isomorphism class ofGover F^sepand were computed in [5].

It was proved in [5], case by case, that, for a groupGof inner type, the maximal possible index of the Tits algebra A corresponding to reaches its upper bound n(G). More precisely, there is a eld extension E=F and an inner form G⁰ of the group GF E over E such that for any character of the center of the universal covering of G⁰, dened over E, the index of the Tits algebraA corresponding to equalsn(G) =n(G⁰).

We give here a uniform proof of this statement for all adjoint semisimple algebraic groupsG(not necessarily of inner type). The eldE appears as a function eld of a

\classifying variety"Y for the corresponding adjoint quasisplit groupG^q.

The universal property of the variety Y asserts that any inner form of G over an arbitrary eld extension L=F arises from some L-point of Y. Hence, the Tits algebras over the function eld E = F(Y) are generic ones, and, therefore, are of maximal index. It follows that, if the index of the Tits algebraAcorresponding to reaches the upper boundn(G) over some eld extension, then it does so overF(Y).

In the rst part of the paper we dene, for a group scheme ^G, the dual group scheme^G0 with respect to a ^G-torsor. This construction is a slight generalization of the corollary of Prop. 34 in [10]. For an adjoint semisimple algebraic groupGover a eldF we construct a classifying variety Y overF such that the scheme^G0, dual to GFY with respect to a certain torsor, represents the algebraic family of all inner forms ofG.

In section 4 we dene Tits algebras and give a list of all Tits algebras for all absolutely simple groups of classical types.

The main result is formulated in section 5. The rest of the paper is devoted to the proof of the theorem. In the last section we give an application of the theorem in the case of groups of outer typeA²_n ¹ which was not covered in [5].

All the group schemes considered in the paper are assumed to be at ane of nite type over a Noetherian separated base schemeY.

For a eldF we denote byF^sep a separable closure and by the absolute Galois group Gal(F^sep=F). The split 1-dimensional torus SpecF[t;t ¹] is denoted by^Gm.

1. Dual group scheme with respect to a torsor

Let^Gbe a group scheme over a schemeY, and let :X^!Y be a (left)^G-torsor [7].

Denote by Aut^G(X) the group of all ^G-automorphisms of X over Y. IfX =^G is a trivial torsor, then the map^G(Y)^!Aut^G(X) given by the ruleg ^7!(g^{0 7!}g⁰g ¹) is clearly a group isomorphism.

Consider the sheaf of groups in the at topologyY onY: S(Z) = Aut^GYZ(XY Z):

Proposition1.1. The sheaf S is represented by a group scheme over Y.

Proof. Since :X ^!Y is faithfully at, it is sucient to prove that the restriction ofS onX is represented by a group scheme (by faithfully at descent, [7, Th.2.23]).

But overX the torsor_Y id :X_Y X ^!X is trivial, hence for any schemeZ over X we have a canonical isomorphismS(Z)^{! G}(Z) and therefore the restriction ofS onX is represented by^GY X.

We denote by ^G0 the group scheme overY representingS and call^G0 the group scheme dual to^Gwith respect to the^G-torsor :X ^!Y. It follows from the proof of proposition 1.1 that the group schemes^G0Y X and^GY X are isomorphic overX.

By denition the group scheme ^G0 acts onX overY. Proposition1.2. The morphism:X^!Y is a^G0-torsor.

Proof. By faithfully at descent we may assume that X = ^G is a trivial ^G-torsor.

Then the action of^{G0 ! G} onX clearly leads to the structure of a trivial ^G0-torsor onX.

There is a natural bijection of the set of isomorphism classes of ^G-torsors : X^!Y overY and the setH¹(Y;^G) (see [7]).

Letf :^G!G1 be a morphism of group schemes overY, and let :X ^!Y be a^G-torsor. A^G1-torsor¹ :X^{1 !}Y representing the image of the class of under the map

H¹(Y;^G)^!H¹(Y;^G0)

is called the image of the ^G-torsor : X ^! Y under f. Let ^G0 (resp. ^G10) be the group scheme dual to ^G (resp. ^G1) with respect to the torsor : X ^! Y (resp.

¹:X^1!Y). The natural group homomorphism Aut^G(X)^!Aut^G1(X¹)

induces a group scheme homomorphismf⁰ :^G0!G10 overY of the dual group schemes.

2. ^PGL-torsors

Let p : ^{V !} Y be a vector bundle over Y and ^E = EndY(^V) (viewed as a vector bundle overY). Consider the group scheme ^G =^PGL(^V) overY. Let :X ^!Y be a^G-torsor. The group scheme^G acts on ^E and onX over Y, hence on^EY X. Denote by Sec^G(^E) the (Y;^O_Y)-algebra of^G-invariant sectionsX ^!E_Y X of the vector bundle^EY X ^!X. Consider the sheafT of algebras onY:

T(Z) = Sec^GYZ(^EY Z):

Proposition2.1. The sheafT is represented by the total space of an Azumaya alge- bra overY.

Proof. By faithfully at descent we may assume thatX =^Gis a trivial torsor. Then for any scheme Z over Y we have T(Z) = MorY(Z;^E), hence T is represented by

E which is the total space of the associated locally free sheaf^EndY(^V) of Azumaya algebras.

We call an Azumaya algebra^AoverY whose total space representsT the algebra associated to the ^G-torsor : X ^! Y. It follows from the proof of proposition 2.1 that the^OX-algebra^Ais isomorphic to(^EndY(^V)).

Consider the sheaf of sets onY:

U(Z) = Iso^OZ alg(^A;^EndY(^V))

for any:Z^!Y. The group^G(Z) acts naturally onU(Z) makingU a^G-torsor.

Proposition2.2. The sheaf U is represented by the^G-torsor :X ^!Y.

Proof. A morphism : Z ^! X over Y denes a trivialization of the torsor X Y Z ^! Z and, hence, an isomorphism of ^OZ-algebras ()^A and ()(^EndY(^V)). Therefore, we get a map X(Z) ^! U(Z) which gives rise to a map of sheaves

MorY(;X)^!U:

To prove that this map is a bijection, by faithfully at descent one may assume that X is a trivial torsor. By the Skolem-Noether theorem in this case the statement is clear.

Remark2.3. Proposition 2.2 shows how to reconstruct the ^G-torsor : X ^! Y out of the algebra ^A. Thus, we have a bijection between the set of isomorphism classes of Azumaya algebras^AoverY such that^A! (^EndY(^V)) and the set of isomorphism classes of^PGL(^V)-torsors overY.

The group scheme ^PGL1(^A) over Y acts naturally on the sheaf U, and the action commutes with that of ^G. Hence, we have a group scheme homomorphism

PGL

1(^A)^!G0 where^G0 is the group scheme dual to^G with respect to the^G-torsor :X ^!Y. To prove that this homomorphism is an isomorphism, by faithfully at descent, one may consider the split situation in which our statement is clear. Hence,

G

0=^PGL(^V)⁰=^PGL1(^A):

Now let ^G be an arbitrary group scheme overY, let :X ^!Y a^G-torsor and

let f :^G!PGL(^V)

be a projective representation overY, where ^V is a vector bundle over Y. Denote by^Athe Azumaya algebra onY associated to the^PGL(^V)-torsor, which is equal to the image of under f. We call^A the algebra associated to the^G-torsor and the projective representationf. There is a natural group scheme homomorphism

f⁰:^G0!PGL1(^A) where^G0 is the group scheme dual to^G with respect to.

3. Inner forms

LetG be a semisimple algebraic algebraic group dened over a eld F with center Z(G). Denote byGthe corresponding adjoint groupG=Z(G). An algebraic groupG⁰ overF is called a twisted form of GifG^0sep'G^sep. The set of isomorphism classes of twisted forms ofGis in 1{1 correspondence with the setH¹(F;Aut(G^sep)) ([10]).

The natural homomorphism

G(F^sep)^!Aut(G^sep); g^7!(g^07!gg⁰g ¹) induces the map

:H¹(F;G(F^sep))^!H¹(F;Aut(G^sep)):

A twisted form G⁰ of the group G is called an inner form of G if the cocycle corresponding toG⁰ belongs to the image of. The groupGis called of inner type if Gis an inner form of a split group.

Assume now that Gis an adjoint group, i.e. G=G. LetX be a G-torsor over F. It corresponds to some element ²H¹(F;G(F^sep)) ([10]). It is straightforward

to check that the groupG⁰, dual to Gwith respect to the torsor X, corresponds to ()²H¹(F;Aut(G^sep)).

We have proved

Proposition3.1. LetG andG⁰ be adjoint semisimple algebraic groups over a eld F. Then G⁰ is an inner form of G i there is a G-torsor X overF such that G⁰ is the dual group with respect to theG-torsorX.

Remark3.2. The second condition of proposition 3.1 can be taken as the denition of an inner form of an adjoint group (in order to avoid referring to cocycles).

4. Tits algebras

LetGbe an adjoint semisimple algebraic group dened over a eldF, letG^e!Gbe the universal covering, andC the kernel of the covering. It is known that C, being the center of G^e, is a closed subscheme of G^e of multiplicative type (not necessarily reduced) ([2],[12]). Denote byC the nite -module Hom(C^sep;^Gm) of characters.

The group Gis an inner form of some quasisplit group dened over F ([1],[12]).

By proposition 3.1, there exists aG-torsorX over F such that the group G⁰, dual to Gwith respect to X, is quasisplit. The choice of a point of X overF^sep denes an isomorphismG^sep! G^0sep which is uniquely determined up to conjugation. This isomorphism extends uniquely to an isomorphismG^esep! G^e0sep whereG^e0 is the uni- versal covering ofG⁰ overF ([12]). Hence, we obtain an isomorphism of the centers ':C^sep! C^sep0 . One can easily see that this isomorphism is dened overF (hence, induces an isomorphism of -modules':C^! C⁰) and depends only on the choice of theG-torsorX (which is not unique in general) but not on the point of X over F^sep.

Denote byB a Borel subgroup inG^e0 dened overF, byT a maximal torus inB dened overF and by the subgroup in T generated by roots ofG^e relative toT. The restriction map induces the natural isomorphism of -modules

T=^! C:

There is a partial ordering onT: we write > for, ²T if is a sum of roots ofB. In each coset ofT^e= there is a unique minimal element with respect to this ordering called the minimal weight.

Choose a character ² C dened overF and put ⁰ ='()²C⁰. By the representation theory of quasisplit semisimple groups (see [13]) there is an irreducible representation ~ : G^{e0 ! GL}(V) such that the restriction of ~ to C⁰ is given by multiplication by ⁰. Consider a central simple F-algebra A associated to the G- torsorX and the projective representation:G^{0 !PGL}(V) induced by ~ (section 2). The algebra A is called the Tits algebra of the group G corresponding to the representation. Its class in Br(F) depends only on the choice of character²C ([13]) and is called the Tits class of the groupGcorresponding to . By construction, the index ofAdivides dimV. Denote byn(G) the greatest common divisor of the numbers dimV for all representations ~:G^e0!GL(V) such that the restriction of ~ onC⁰ is given by multiplication by⁰. We have observed that indA divides n(G) (see [5]). If= 0, then n(G) = 1.

Let²C(F)^'(T=) and²T be the minimal weight in the coset. Let ~:G^{e0 !GL}(V) be a representation (unique up to an isomorphism) with the highest weight called the minimal representation. The Tits algebra corresponding to is called the minimal Tits algebra of G and is denoted byA. The algebraA is the canonical representative of the Tits class corresponding to. For example, if= 0, thenA=F. Any Tits algebra is Brauer equivalent to a minimal one.

Remark4.1. The isomorphism ' : C^{sep !} C^sep0 depends on the choice of the G- torsorX. Another choice ofX changes'by an automorphism ofC⁰ induced by an (outer) automorphism ofG⁰, but clearly does not change the numbersn(G).

Remark4.2. By denition, the numbersn(G) depend only on the quasisplit inner form ofGand hence do not change if we replaceGby any inner form of it. In turn, the class of inner forms ofGis uniquely determined by the isomorphism class of G overF^sepand the action of on the group of outer automorphisms

Out(G^sep) = Aut(G^sep)=Int(G^sep) = Aut(Dyn(G^sep))

of the groupG^sep. If we changeF by a eld extensionE=F such thatF is separably closed inE, the numbers n(G) do not change. If G is a group of inner type (i.e.

G⁰ is a split group, or equivalently, acts trivially on Out(G^sep)) then the numbers n(G) depend only on the isomorphism class ofGoverF^sepand are computed in [5].

We would like to classify the Tits classes of all adjoint semisimple algebraic groups. A Tits algebra of the product of adjoint semisimple groups is the tensor product of the Tits algebras of factors. Since any adjoint semisimple group is the product of the groupsG¹=R_L=F(G) whereGis an absolutely simple adjoint group over a nite separable eld extensionL=F ([12]), it suces to describe the Tits alge- bras ofG¹. IfG^e!Gis the universal covering ofGwith kernelC, then

Ge¹=R_L=F(G^e)^!R_L=F(G) =G¹ is the universal covering ofG¹ with kernelC¹=RL=F(C).

LetF LF^sep, ⁰= Gal(F^sep=L) . We have a canonical isomorphism :C(L) = (C) ^0! (C¹) =C¹(F);

and for any⁰²C(L) the Tits algebraA with=(⁰) for the groupG¹ equals the corestriction in the extensionL=F of the Tits algebraA⁰ ofG([13]). Hence, it is sucient to classify the Tits classes of absolutely simple adjoint groups.

Below is the list of minimal Tits algebras and numbers n(G) for absolutely simple adjoint groups. We use the notation and the computations from [4] and [5].

4.1. Type An. An adjoint simple algebraic group of the typeAn, dened overF, is isomorphic to the projective unitary groupG=^PGU(B;), whereB is an Azumaya algebra of degreen+ 1 over an etale quadratic extensionL=F with an involution of the second kind trivial on F. Its universal covering is the special unitary group Ge=^SU(B;)

Assume rst that Lsplits, i.e. L ^'FF. In this case B ^'AA^op with the switch involution whereAis a central simple algebra of degreen+ 1 overF, where

Ge =^SL1(A) andG=^PGL1(A). ThenC =_n⁺¹, andC =^Z=(n+ 1)^Zwith the trivial -action. For anyi= 0;1;:::;n, consider the natural representation

i:G^e!GL1(ⁱA)

where ⁱA are external powers of A (see [4]). In the split case, i is the i-th ex- ternal power representation known as a minimal representation. Hence, ⁱA for i = 0;1;:::;n are minimal Tits algebras of G. If = i+ (n+ 1)^{Z 2} C, then n= (n+ 1)=gcd(i;n+ 1).

Now letG^e=^SU(B;), whereBis a central simple algebra of degreen+1 with a unitary involution over a quadratic separable eld extensionL=F. The group acts onC=^Z=(n+ 1)^Zbyx^7! x through Gal(L=F). The only non-trivial element in C(F) is=ⁿ⁺¹² + (n+ 1)^Z(whennis odd). There is a natural homomorphism

:G^e!GL1(D(B;));

whereD(B;) is the discriminant algebra (see section 10). In the split caseis the external ⁿ⁺¹² -power representation. Hence, the algebra D(B;) is the minimal Tits algebra for the groupGcorresponding to. The number n equals 2 if (n+ 1) is a 2-power and equals 4 otherwise (see section 10).

4.2. Type Bn. An adjoint simple algebraic group of type Bn, dened over F, is isomorphic to the special orthogonal group G = ^O+(V;q) where (V;q) is a non- degenerate quadratic form of dimension 2n+ 1. Its universal covering is the spinor groupG^e=^Spin(V;q). ThenC=²,C=^Z=2^Z=^f0;^g. The embedding

G ,e ^!GL1(C⁰(V;q));

whereC⁰(V;q) is the even Cliord algebra of (V;q), is, in the split case, the spinor representation known as a minimal representation. Hence, the even Cliord algebra C⁰(V;q) is the minimal Tits algebraA. The numbern equals 2ⁿ.

4.3. Type Cn. An adjoint simple algebraic group of type Cn, dened over F, is isomorphic to the group of projective similitudes G = ^PGSp(A;), where A is a central simple algebra of degree 2n with a symplectic involution . Its universal covering is the symplectic group G^e = ^Sp(A;). Then C = ² and C = ^Z=2^Z=

f0;^g. The embedding

G ,e ^!GL1(A)

is, in the split case, a minimal representation. Hence,Ais the minimal Tits algebra A. The numbern is the largest 2-power which divides 2n.

4.4. TypeDn. An adjoint simple algebraic group of typeDn, dened overF(of non- trialitarian type ifn= 4), is isomorphic to the group of proper projective similitudes G=^PGO+(A;;f) whereAis a central simple, algebra of degree 2nwith an orthog- onal pair (;f) (see [4]). Its universal covering is the spinor groupG^e=^Spin(A;;f).

Then C = ^f0;;⁺; ^g where factors through the special orthogonal group

O

+(A;;f). The composition

Spin(A;;f)^!O+(A;;f),^!GL1(A)

is, in the split case, the standard minimal representation. Hence, A is the minimal Tits algebraA. The numbern equals the largest 2-power which divides 2n.

Assume that the discriminant of is trivial (i.e. the center Z of the Cliord algebraC(A;;f) splits). The group acts trivially onC. The natural compositions

Spin(A;;f),^!GL1(C(A;;f))^!GL1(C(A;;f))

where C(A;;f) are simple components of C(A;;f), are, in the split case, semispinor minimal representations. Hence, C(A;;f) are minimal Tits algebras A. The numbersn equal 2^{n 1}.

If the discriminant of is not trivial, then interchanges⁺ and and is the only nontrivial -invariant character.

4.5. Exceptional types.

4.5.1. Trialitarian typeD⁴. The image of the map ^!Aut(C) contains a subgroup of order 3. It implies thatC(F) = 0 and there are no nontrivial characters and Tits algebras.

4.5.2. TypeE⁶. In this caseC^'Z=3^Zand for a nontrivial character²C(F) one hasn= 27.

4.5.3. TypeE⁷. In this caseC^'Z=2^Zand for a nontrivial character²C(F) one hasn= 8.

4.5.4. Types E⁸, F⁴ and G². In these cases C = 0 and there are no nontrivial characters and Tits algebras.

5. The classifying variety of a group

LetG be an adjoint semisimple algebraic group over a eld F and Y be a scheme overF. Consider the group scheme^G =GFY overY, and an arbitrary^G-torsor : X ^! Y. Denote by ^G0 the dual scheme with respect to this torsor. For any rational pointy ²Y(F) the ber^G_y⁰ of^G0 overy is dual to^Gy =Gwith respect to theG-torsor_y :X_y ^!SpecF. Hence, by proposition 3.1, an algebraic group^G_y⁰ is an inner form ofG. So, we can view the scheme ^G0 as the algebraic family of inner forms ofG.

Now we take a specic schemeY. LetG ,^!GLn be any faithful representation overF. Consider the homogeneous variety Y =^GLn=Gand the canonicalG-torsor : ^GLn ^! Y. The variety Y is called the classifying variety of G. The universal property ofY asserts that any inner form of Gis a member of the algebraic family

G

0 over Y:

Proposition5.1. For any inner form G⁰ of G overF there exists a rational point y²Y(F) such that G^{0 'G}_y⁰ overF.

Proof. This follows from Hilbert's Theorem 90 and the exact sequence of pointed sets ([7],[10])

Y(F)^!H¹(F;G(F^sep))^!H¹(F;^GLn(F^sep)) induced by the exact sequence

1^!G(F^sep)^!GLn(F^sep)^!Y(F^sep)^!1:

Now let G¹ be any adjoint semisimple algebraic group over F, and G be its quasisplit inner form. Consider the classifying variety Y = ^GLn=G and the group scheme ^G0 dual to ^G = GF Y with respect to the ^G-torsor : ^GLn ^! Y. By

proposition 5.1, we haveG^1'G_y⁰ for somey²Y(F). Let²Y be the generic point.

The generic ber^G0 is an adjoint semisimple algebraic group over the function eld F(Y). The^G-torsor enables us to identify the character moduleCof the center of the universal coverings of the groupsG¹, Gand^G0.

Now we formulate the main result.

Theorem5.2. For any character²C(F), the index of the Tits class of the group

G

0 corresponding to equalsn(G¹) =n(G) =n(^G0).

Corollary 5.3. For any adjoint semisimple algebraic groupG¹over a eld F there exists a eld extensionE=F and an inner formG²of the groupG¹FE overE such thatF is separably closed inE and for any characterof the center of the universal covering of G², with dened over E, the index of the Tits class of the group G² corresponding toequalsn(G¹) =n(G²).

6. G-modules

Let^G be a group scheme over a scheme Y. Assume that ^G acts on a schemeX over Y. The morphism of the ^G-action onX we denote by

:^GY X ^!X:

A^G-module^F onX is a quasicoherent^OX-module^F together with an isomorphism of^OGYX-modules

':^F! p^2F

(wherep²:^GY X^!X is the projection), satisfying the cocycle condition p²³(')(id)(') = (mid)(')

wherem:^GY ^G!G is the multiplication.

Giving a ^G-module structure on a quasicoherent ^OX-module ^F is equivalent to giving, naturally in Y-schemes Z, a homomorphism of the group ^G(Z) into the automorphism group of the pair (XY Z;^FY Z) ([8],[11]).

Assume that ^G acts on an Azumaya algebra ^B over X, i.e. the structure of

G-module^Bis given by an^OGYX-algebra isomorphism :^B! p^2B:

Denote byM(^G;X;^B) the abelian category of^G-modules^FonX, which are also left^B-modules and coherent^OX-modules, such that the following diagram commutes:

^{BF ! F}

'^{??y ??y}'

p^2Bp^{2F !} p^2F;

where the horizontal maps are given by the action of ^B on ^F. Morphisms in the category are morphisms of^B- and^G-modules.

If the algebra ^Bis trivial, i.e. ^B=^OX, then the category is simply denoted by M(^G;X).

Let ^Abe an Azumaya algebra on Y. Consider the Azumaya algebra^B =^A on X, where : X ^! Y is the structure morphism, and the category M(Y;^A) of left^A-modules which are coherent^OY-modules. For^M2M(Y;^A) the^OX-module

F=^Mhas a natural structure of a^B-module. Since=p², it follows that we also have a natural^G-module structure on^F given by the isomorphisms

':^{F '}()^M= (p²)^M'p^2F: Thus, we have obtained a functor

:M(Y;^A)^!M(^G;X;^B); ^M7!(^M;'):

Proposition6.1. If:X ^!Y is a^G-torsor thenis an equivalence of categories.

Proof. Under the isomorphisms

GY X^! X_Y X; (g;x)^7!(gx;x)

GY ^GY X ^! XY XY X; (g¹;g²;x)^7!(g¹g²x;g²x;x)

the action morphism is identied with the rst projectionp¹ :XY X ^!X and morphismsmid, idare identied with the projectionsp¹³;p¹²:XYXYX^! XY X. Hence, the isomorphism'giving a^G-module structure on an^OX-module

F can be identied with descent data, i.e. with an isomorphism :p^{1F !} p^2F

of^OX^YX-modules satisfying the usual cocycle condition (p²³ )(p¹² ) =p¹³ :

The statement follows now by faithfully at descent ([7, Prop.2.22]).

7. Modules under groups of multiplicative type

LetC be a diagonalizable group scheme over a eld F, and let C = Hom(C;^Gm) be the character group. It is known thatC = SpecF[C], whereF[C] is the group algebra ofCoverF, and the comorphism

m:F[C]^!F[C]FF[C] of the multiplication is given by the formulam() =([2]).

To introduce an action of Con an ane schemeX = SpecAoverF is the same as to give aC-graded structure on theF-algebraA([3]):

A= ^a

²CA: The comorphism of the action ofC onX,

:A^!F[C]FA is given by the formula

(^X

²Ca) = ^X

²C(a):

The trivial action corresponds to the trivial graded structure: A= 0 for ⁶= 0.

Let M be an A-module. A C-module structure of the associated ^OX-module

F=M^fis given by an isomorphism ofF[C]F A-modules

': (F[C]FA)_A;M^! (F[C]F A)A;p²M;

satisfying the cocycle condition. Let '(11m) = ^P2_C(1m), where m;m²M. Since (eid)'= id ([11]), wheree: SpecF ^!Gis the group unit, it follows that

m= ^X

²Cm: ()

It is easy to check that the cocycle condition implies that (m)equalsmif=and equals 0 if⁶=. Hence, the equality () gives rise to the direct sum decomposition

M= ^a

²CM

makingM aC-gradedA-module. Therefore, the categoryM(C;X) is equivalent to the category of nitely generatedC-graded modules.

Let an algebraic group Gover a eld F act on an ane scheme X over F and on an Azumaya algebra ^B on X. Assume that a closed central group subscheme CGof multiplicative type acts trivially onX and^B. Denote byC the -module of characters Hom(C^sep;^Gm). Since the groupC^sep is diagonalizable, it follows that for any^F2M(G;X;^B) we have a decomposition

F

sep= ^a

²C(^Fsep) () into a direct sum ofG^sep-submodules (^Fsep) on X^sep (since C is central and acts trivially onX and^B).

Choose any -invariant character ² C (dened over F). Clearly, (^Fsep) and its direct complement in () are dened over F, hence we have a canonical decomposition

F=^FF

into a direct sum of G-submodules on X. In other words, these submodules are uniquely determined by the property thatc (c) is trivial on^F and invertible on

F for allc inC.

Consider the full subcategories M(G;X;^B) and M(G;X;^B) in M(G;X;^B) consisting of allG-modules^F such that^F=^F and^F =^F respectively. It is clear that M(G;X;^B)^'M(G;X;^B)M(G;X;^B):

If = 0 is the trivial character then the categoryM(G;X;^B) is equivalent to the categoryM(G=C;X;^B).

8. Equivariant algebraic K-theory

TheK-groups of the categoryM(^G;X;^B) (see section 6) we denote byK(^G;X;^B).

These groups are clearly contravariant with respect to at ^G-morphisms in X. If

B=^OX is the trivial algebra we simply writeK(^G;X).

Let G be an algebraic group over F acting on a scheme X over F. We will need the following particular cases of the localization theorem [11, th. 2.7] and the homotopy invariance theorem [11, cor. 4.2] in equivariant algebraicK-theory.

Proposition8.1. Let U X be an open G-equivariant subscheme. Then the re- striction homomorphismK⁰(G;X)^!K⁰(G;U) is surjective.

Proposition8.2. Assume thatGacts linearly on an ane space ^A_nF overF. Then the structure morphismp:^A_nF ^!SpecF induces an isomorphism

p:K(G;SpecF)^! K(G;^A_nF):

The category M(G;SpecF) is equivalent to the category of nite dimensional representations ofGoverF. The groupK⁰(G;SpecF) we denote byR(G).

Assume that G acts on an Azumaya algebra ^B over X and contains a closed central subschemeC overF of multiplicative type, acting trivially onX and^B. For ² C(F) the K-groups of the category M(G;X;^B) we denote by K(G;X;^B).

SinceK(G;X;^B) is a canonical direct summand ofK(G;X;^B) (section 7), it follows that the statements of propositions 8.1 and 8.2 still hold if we replaceK byK.

The group K⁰(G;SpecF) we simply denote by R(G). It is generated by the classes of all representations: G ^!GL(V) such that the restriction of to C is given by.

9. Proof of the theorem

LetG¹ be an adjoint semisimple group over a eld F, let Gbe the quasisplit inner form ofG¹with universal coveringG^e!G, and letCbe the kernel of the covering.

Choose a faithful representation G ,^!GL_n overF and consider the classifying varietyY =^GLn=G over F and the group scheme ^G0 over Y dual to ^G =G_FY with respect to the ^G-torsor : ^GLn ^! Y. Let be the generic point of Y. The

G-torsor enables us to identify the character modulesC andC⁰, whereC⁰ is the kernel of the universal covering of^G0. Choose a character⁰²C⁰ dened overF(Y) and denote by²C the corresponding character overF.

Consider a representation ~:G^e!GL(V) such that the restriction of ~toC is given by. Consider also the Azumaya algebra ^A on Y associated to the ^G-torsor and the projective representation:G^!PGL(V) induced by ~(section 2). We know that there is an isomorphism ofG-algebras

(^A)^'(^End(V FY))

on^GLn(section 2) and that^A is the Tits algebra corresponding to the character⁰ (section 4). We have to show that ind^A =n(G).

Consider the homomorphism

:K⁰(^Aop )^!Z;

taking anA^op -moduleM to dim_F⁽_Y⁾M. It is easy to see that im() = ind^Adeg^AZ:

Consider also the homomorphism : R(G^e) ^!Z, taking a representation space U to dimFU. It is clear that im() = n(G)^Z. For the proof of the theorem it is sucient to nd a surjective homomorphism

:R(G^e)^!K⁰(^Aop )

such that the compositionequals deg^A= dimV . The homomorphism will be found as a composite of seven epimorphisms¹;²;:::;⁷.

Consider ^GLn as an open subvariety of the ane space ^A = ^An_F² of all n n-matrices over F on which the group G (and hence G^e) acts linearly. The open

embedding^GLn,^!A is clearlyG^e-equivariant. By proposition 8.2 (see also a remark at the end of section 8) the structure morphism^{A !}SpecF induces an isomorphism

¹:R(G^e) =K⁰(G;^e SpecF)^! K⁰(G;^{e A}): By proposition 8.1, the restriction homomorphism

²:K⁰(G;^{e A})^!K⁰(G;^{e GL}n) is surjective.

Denote by^Bthe algebra^End(VFY) =^OGLnFEndV on^GLn. The group Ge clearly acts on ^B. Consider two functors

M(G;^{e GL}n)^u_v M⁰(G;^{e GL}n;^Bop); u(^F) =VF^F; v(^M) =V ^EndV^M;

whereVis theF-vector space dual toV. The canonical isomorphismsV^EndVV^' F and VF V ^' EndV show that uand v are mutually inverse equivalences of categories. Hence, the functoruinduces an isomorphism

³:K⁰(G;^{e GL}n)^! K⁰⁰(G;^{e GL}n;^Bop):

Since the centerC of G^e acts trivially on ^GLn and ^B, it follows that the categories M⁰(G;^{e GL}n;^Bop) and M(G;^GLn;^Bop) are equivalent. Hence, we have an isomor- phism

⁴:K⁰⁰(G;^{e GL}n;^Bop)^! K⁰(G;^GLn;^Bop):

The isomorphismGFX ^'GY X shows that the categoriesM(G;^GLn;^Bop) andM(^G;^GLn;^Bop) are equivalent. Hence, we have an isomorphism

⁵:K⁰(G;^GLn;^Bop)^! K⁰(^G;^GLn;^Bop):

Since :^GLn ^!Y is a^G-torsor and^{B 'A}, it follows from proposition 6.1 that the functor

:M(Y;^Aop)^!M(^G;^GL_n;^Bop) is an equivalence of categories. Hence, induces an isomorphism

⁶:K⁰(^G;^GLn;^Bop)^! K⁰(Y;^Aop): By localization (Proposition 8.1), the functor

M(Y;^Aop)^!M(^Aop ); ^F7!stalk of^F at the generic point induces an epimorphism

⁷:K⁰(Y;^Aop)^!K⁰(^Aop ):

It can be easily checked that the composition=⁷⁶¹takes the class of a representation spaceU of the groupG^eto the generic stalk^F where

^F=VFUF^OGLn

and hence satises the desired condition.

10. Examples

LetL=F be a Galois quadratic eld extension, = Gal(L=F), and letB be a central simple algebra overLof degree 2nwith involution of the second kind trivial on F. Consider the special unitary groupG^e=^SU(B;) overF. The groupG^e(F) ofF- points ofG^econsists of all elementsb²Bsuch that(b)b= 1 and Nrd(b) = 1 where Nrd is the reduced norm homomorphism. The Galois group acts onC ^'Z=2n^Z through its factor group =^f1;^gby(k+ 2n^Z) = k+ 2n^Z(see section 4). The Tits algebra corresponding to the only nontrivial character=n+2n^Z2C(F) can be constructed as follows (see [4],[5]).

Consider the Severi-Brauer varietyX overLcorresponding to the algebraBand the canonical locally free sheafJ of rank 2nonX, soB= EndX(J) [9]. The canonical nondegenerate bilinear form on then^th-exterior power ofJ

ⁿJⁿJ ^!2nJ^'OX

induces in the usual way an involution of the rst kind on the algebra ⁿB = End^OX(ⁿJ) over L. One can check that the involutions and ⁰ = ⁿ on ⁿB commute. Therefore, the set ^fx ²ⁿB : (x) = ⁰(x)^gis a central simple algebra over F. We denote this algebra by D(B;) and call it the discriminant algebra of (B;) ([4]). It is the Tits algebra corresponding to the character.

The discriminant algebra enjoys the following properties:

1. The degree of D(B;) equals ²_nⁿ.

2. The restriction oftoD(B;) is an involution of the rst kind. In particular, the exponent ofD(B;) divides 2.

3. D(B;)F L ^{' n}B Bⁿ. Since exp(Bⁿ) divides 2, it follows that ind(Bⁿ) also divides 2, and hence indD(B;) divides 4.

Let G^e0 be the quasisplit inner form ofG^e. It is the special unitary group of the hyperbolic hermitian form over the quadratic extension L=F ([12]). Since G^{e0sep '}

SL

2n(F^sep) it follows that

R(G^e0sep)^'Z[t¹;t²;:::;t²n ¹]

whereti is the class of thei^th-exterior power of the standard representation of^SL2n. This ring isC=^Z=2n^Z-graded, the degree oftibeing equal toi(mod2n). The rank mapR(G^e0sep)^!Ztakes t_i to ²_iⁿ. The action of the Galois group onR(G^e0sep) is given by(ti) =t²n i. We have also ([13]):

R(G^e0)^'Z[t¹;t²;:::;t²n ¹]:

Using this description of the ringR(G^e0) and the fact that the image of the map R(G^e0)^!Z, taking a representation spaceU of the groupG^e0to dimFU, equalsn^Z, one can easily compute the numbern(G) for G=G=C^e (see [6]): n(G) is equal to 2 ifnis a 2{power and equals 4 otherwise. Hence, the corollary of the theorem gives in this case the following

Proposition10.1. For any Galois quadratic eld extension L=F andn ^{2 N} there is a eld extension E=F and a central simple algebra B of degree 2n over EF L with involution of the second kind trivial on E such that indD(B;) = 2 if nis a 2-power and equals 4 otherwise.

References

[1] A. Borel, J. Tits. Groupes reductifs. Publ. Math. I. H. E. S.27(1965), 55{151.

[2] M. Demazure, P. Gabriel. Groupes Algebriques Lineares. Masson, Paris, 1970.

[3] R. G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat. Toroidal imbeddings I.

Lecture Notes in Math. 339, Springer-Verlag, Berlin, 1973.

[4] M.-A. Knus, A. S. Merkurjev, M. Rost, J.-P. Tignol. Book of involutions, in preparation.

[5] A. S. Merkurjev, I. A. Panin, A. Wadsworth. Index reduction formulas for twisted ag varieties. Preprint, 1995.

[6] A. S. Merkurjev, I. A. Panin, A. Wadsworth. Index reduction formulas for twisted ag varieties II, Preprint, 1996.

[7] J. S. Milne. Etale Cohomology. Princeton Univ. Press, Princeton, N. J., 1980.

[8] D. Mumford, J. Fogarty. Geometric Invariant Theory. Springer-Verlag, 1982.

[9] D. Quillen. Higher Algebraic K-theory. Lecture Notes in Math. 341, Springer- Verlag, Berlin (1973), 85{147.

[10] J.-P. Serre. Cohomologie Galoisienne, 5^eme edition. Lecture Notes in Math. 5, Springer-Verlag, Berlin, 1994.

[11] R. Thomason. Algebraic K-theory of group scheme action, in: \Algebr. topol.

and algebr. K-theory (ed. W.Browder), Proc. conf. Princeton, oct.24{28, 1983, Princeton, N.J., 1987, 539{563.

[12] J. Tits. Classication of algebraic semisimple groups, in: Algebraic Groups and discontinuous Subgroups, (eds. A.Borel and G.D.Mostow), Proc. Symp. Pure Math., Vol. 9, 1966, 33{62

[13] J. Tits. Representations lineaires irreductibles d'un groupe reductif sur un corps quelconque. J. reine angew. Math. 247(1971), 196{220.

A. S. Merkurjev

Sankt-Petersburg State University andFakultat fur Mathematik

Universitat Bielefeld Postfach 100131 D 33501 Bielefeld Germany

[email protected]