Диссертация (1136188), страница 6

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The key ingredient is a descriptionof the tangent bundles of regular compactifications due to Ehlers [11] andBrion [5]. This description is outlined in Subsection 4.2.The remaining problem is to describe the Chern classes [S1 ], .

. . , [Sn−k ]so that their intersection indices with hyperplane sections may be computed explicitly. So far there is such a description for the first and the lastChern classes (see Subsection 3.3). Namely, [S1 ] is the class of a generichyperplane section corresponding to the irreducible representation withthe highest weight 2ρ.

Here ρ is the sum of all fundamental weights ofG. This description follows from a result of A. Rittatore [25] concerningthe first Chern class of reductive group compactifications. The last Chernclass [Sn−k ] is up to a scalar multiple the class of a maximal torus in G.There is a hope that the intersection indices of other Chern classes Si withhyperplane sections can also be computed using a formula similar to theBrion-Kazarnovskii formula.If a complete intersection is a curve, i.e., m = n − 1, then the formulaof Theorem 1.1 involves only the first Chern class [S1 ]. In this case, thecomputation of [S1 ] together with the Brion-Kazarnovskii formula allowsus to compute explicitly the Euler characteristic and the genus of a curvein G in terms of the weight polytopes of π1 , . . . , πm (see Corollaries 4.9 and4.10, Subsection 4.3).

Note that these two numbers completely describe thetopological type of a curve.Most of the constructions and results of this paper can be extendedwithout any change to the case of arbitrary spherical homogeneous spaces.This is discussed in Section 5.I am very grateful to Mikhail Kapranov and Askold Khovanskii for numerous stimulating discussions and suggestions. I would like to thank Kiumars Kaveh for useful discussions and Michel Brion for valuable remarksANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS1229on the first version of this paper.

I am also grateful to the referee for manyuseful remarks and comments.Part of the results of this paper were included into my PhD thesis at theUniversity of Toronto [19].Throughout this paper, whenever a group action is mentioned, it is always assumed that a complex algebraic group acts on a complex algebraicvariety by algebraic automorphisms. In particular, by a homogeneous spacefor a group I will always mean the quotient of the group by some closedalgebraic subgroup.The following remarks concern notations. In this paper, the term equivariant (e.g. equivariant compactification, bundle, etc.) will always meanequivariant under the action of the doubled group G × G, unless otherwise stated. The Lie algebra of G is denoted by g.

I also fix an embeddingG ⊂ GL(W ) for some vector space W . Then for g ∈ G and A ∈ g, notationAg and gA mean the product of linear operators in End(W ).2. Equivariant compactifications and the ring of conditionsThis section contains some well-known notions and theorems, which willbe used in the sequel. First, I define the notion of spherical action anddescribe equivariant compactifications of reductive groups following [9], [15]and [26]. Then I state Kleiman’s transversality theorem [20] and recall thedefinition of the ring of conditions [10, 8].Spherical action. Reductive groups are partial cases of more generalspherical homogeneous spaces.

They are defined as follows. Let G be aconnected complex reductive group, and let M be a homogeneous spaceunder G. The action of G on M is called spherical, if a Borel subgroup ofG has an open dense orbit in M . In this case, the homogeneous space M isalso called spherical. An important and very useful property, which characterizes a spherical homogeneous space M , is that any compactificationof M equivariant under the action of G contains only a finite number oforbits [21].There is a natural action of the group G × G on G by left and right multiplications. Namely, an element (g1 , g2 ) ∈ G × G maps an element g ∈ Gto g1 gg2−1 . This action is spherical as follows from the Bruhat decomposition of G with respect to some Borel subgroup.

Thus the group G can beconsidered as a spherical homogeneous space of the doubled group G × Gwith respect to this action. For any representation π : G → GL(V ) thisTOME 56 (2006), FASCICULE 41230Valentina KIRITCHENKOaction can be extended straightforwardly to the action of π(G) × π(G) onthe whole End(V ) by left and right multiplications. I will call such actionsstandard.Equivariant compactifications.

With any representation π one canassociate the following compactification of π(G). Take the projectivizationP(π(G)) of π(G) (i.e., the set of all lines in End(V ) passing through a pointof π(G) and the origin), and then take its closure in P(End(V )). We obtaina projective variety Xπ ⊂ P(End(V )) with a natural action of G×G comingfrom the standard action of π(G) × π(G) on End(V ). Below I will list someimportant properties of this variety.Assume that P(π(G)) is isomorphic to G. Fix a maximal torus T ⊂ G.Let LT be its character lattice.

Consider all weights of the representation π,i.e., all characters of the maximal torus T occurring in π. Take their convexhull Pπ in LT ⊗ R. Then it is easy to see that Pπ is a polytope invariantunder the action of the Weyl group of G. It is called the weight polytopeof the representation π. The polytope Pπ contains information about thecompactification Xπ .Theorem 2.1.1) ([26], Proposition 8) The subvariety Xπ consists of a finite numberof G × G-orbits.

These orbits are in one-to-one correspondence with theorbits of the Weyl group acting on the faces of the polytope Pπ .2) Let σ be another representation of G. The normalizations of subvarieties Xπ and Xσ are isomorphic if and only if the normal fans correspondingto the polytopes Xπ and Xσ coincide. If the first fan is a subdivision of thesecond, then there exists an equivariant map from the normalization of Xπto Xσ , and vice versa.The second part of Theorem 2.1 follows from the general theory of spherical varieties (see [21], Theorem 5.1) combined with the description of compactifications Xπ via colored fans (see [26], Sections 7, 8).In particular, suppose that the group G is of adjoint type, i.e., the centerof G is trivial.

Let π be an irreducible representation of G with a strictlydominant highest weight. It is proved in [9] that the corresponding compactification Xπ of the group G is always smooth and, hence, does notdepend on the choice of a highest weight.

Indeed, the normal fan of theweight polytope Pπ coincides with the fan of the Weyl chambers and theirfaces, so the second part of Theorem 2.1 applies. This compactification iscalled the wonderful compactification and is denoted by Xcan . It was introduced by De Concini and Procesi [9]. The boundary divisor Xcan r G is aANNALES DE L’INSTITUT FOURIERCHERN CLASSES OF REDUCTIVE GROUPS1231divisor with normal crossings.

There are k orbits O1 , . . . , Ok of codimensionone in Xcan . The other orbits are obtained as the intersections of the closures O1 , . . . , Ok . More precisely, to any subset {i1 , i2 , . . . , im } ⊂ {1, . . . , k}there corresponds an orbit Oi1 ∩ Oi2 ∩ . . . ∩ Oim of codimension m. So thenumber of orbits equals to 2k . There is a unique closed orbit O1 ∩ . .

. ∩ Ok ,which is isomorphic to the product of two flag varieties G/B × G/B. HereB is a Borel subgroup of G.Compactifications of a reductive group arising from its representationsare examples of more general equivariant compactifications of the group.A compact complex algebraic variety with an action of G × G is calledan equivariant compactification of G if it satisfies the following conditions.First, it contains an open dense orbit isomorphic to G.

Second, the actionof G × G on this open orbit coincides with the standard action by left andright multiplications.The ring of conditions. The following theorem gives a tool to definethe intersection index on a noncompact group, or more generally, on ahomogeneous space. Recall that two irreducible algebraic subvarieties Y1and Y2 of an algebraic variety X are said to have proper intersection ifeither their intersection Y1 ∩ Y2 is empty or all irreducible components ofY1 ∩ Y2 have dimension dim Y1 + dim Y2 − dim X.Theorem 2.2 (Kleiman’s transversality theorem, [20]).

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