Диссертация (1136188), страница 15
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These orbits are in one-to-one correspondence with theorbits of the Weyl group acting on the faces of the polytope Pπ . This correspondencepreserves incidence relations. i. e. if F1 , F2 are faces such that F1 ⊂ F2 , then theorbit corresponding to F1 is contained in the closure of the orbit corresponding to F2 .(2) Let σ be another representation of G. The normalizations of subvarieties Xπand Xσ are isomorphic if and only if the normal fans corresponding to the polytopesXπ and Xσ coincide.
If the first fan is a subdivision of the second, then there existsa G × G-equivariant map from the normalization of Xπ to Xσ , and vice versa.The second part of Theorem 2.1 follows from the general theory of sphericalvarieties (see [13], Theorem 5.1) combined with the description of compactificationsXπ via colored fans (see [14], Sections 7, 8).494V. KIRITCHENKOIn what follows, we will only consider regular compactifications of G. The simplest example of a regular compactification is the wonderful compactification constructed by De Concini and Procesi. Suppose that the group G is of adjoint type,i. e. the center of G is trivial. Take any irreducible representation π with a strictlydominant highest weight. It is proved in [6] that the corresponding compactificationXπ of the group G is always smooth and, hence, does not depend on the choiceof a highest weight.
Indeed, the normal fan of the weight polytope Pπ coincideswith the fan of the Weyl chambers and their faces, so the second part of Theorem2.1 applies. This compactification is called the wonderful compactification and isdenoted by Xcan .Other regular compactifications of G can be characterized as follows. The normalization X of Xπ is regular if first, it is smooth, and second, there is a (G × G)equivariant map from X to Xcan . These two conditions can be reformulated interms of the weight polytope Pπ .
Namely, the first condition implies that Pπ isintegrally simple (see [14], Theorem 9), i. e. it is simple and the primitive vectorson the edges meeting at each vertex form a basis of LT . The second conditionimplies that none of the vertices of Pπ lies on the walls of the Weyl chambers, i. e.the normal fan of Pπ subdivides the fan of the Weyl chambers and their faces.A regular compactification X has the following nice properties (see [5] for details), which we will use in the sequel.
The boundary divisor X \ G is a divisorwith normal crossings. The G × G-orbits of codimension s correspond to the facesof Pπ of codimension s and have rank (k − s). Recall that each face F ⊂ Pπ isthe transverse intersection of several facets of Pπ (since Pπ is simple). Then theclosure of the orbit corresponding to F is the transverse intersection of the closuresof the codimension one orbits that correspond to these facets. Each closed orbitof X (such orbits correspond to the vertices of Pπ ) is isomorphic to the product oftwo flag varieties G/B × G/B.2.3. Picard group of compactifications. Let X be the normalization of thecompactification Xπ of G.
We assume that X is regular, and hence smooth. Thenthe second cohomology group H 2 (X) is isomorphic to the Picard group of X (see[2]). There is a description of the Picard group of a regular complete symmetricspace due to Bifet (see [2], Theorem 2.4, see also [3], Proposition 3.2). In our case,this description can be reformulated as follows (such a reformulation is well-knownin the toric case, and in the reductive case it was suggested by K.
Kiumars). Denoteby V (π) the group of all integer virtual polytopes analogous to the weight polytopePπ and invariant under the action of the Weyl group.Proposition 2.2. The Picard group Pic(X) of X is canonically isomorphic to thequotient group of V (π) modulo parallel translations. The isomorphism takes thehyperplane section corresponding to a representation σ to the weight polytope of σand extends to the other divisors by linearity.In particular, if G is semisimple, then Pic(X) = V (π) (the only parallel translation taking a W -invariant polytope to a W -invariant polytope is the trivial one).Let us identify divisors in X with the corresponding polytopes using this isomorphism.ON INTERSECTION INDICES OF SUBVARIETIES IN REDUCTIVE GROUPS495¯¯1 , .
. . , O¯¯l , which areThe variety X has l distinguished boundary divisors Othe closures of codimension one orbits. Let us describe the corresponding virtualpolytopes. Choose l facets Γ1 , . . . , Γl of Pπ so that each orbit of the Weyl groupacting on the facets of Pπ contains exactly one Γi . For example, one may take allfacets that intersect the fundamental Weyl chamber.
Choose the support functionsh1 , . . . , hl corresponding to these facets so that hi (Pπ ) is equal to the integraldistance (with respect to the weight lattice LT ) from the origin to the facet Γi .¯ i of codimension one orbit corresponds to the virtualLemma 2.3. The closure Ōpolytope whose i-th support number is 1 and the other support numbers are 0.Proof. Let σ be any representation of G whose weight polytope P is analogous toPπ .
Then X is isomorphic to the normalization of the compactification Xσ . Thusa generic linear functional f on Xσ can also be regarded as a rational functionon X. Let us find the zero and the pole divisors of f . The zero divisor D is thedivisor corresponding to the weight polytope of σ. The pole divisor is a linear¯¯1 , . . . , O¯¯l . It is not hard to show that the coefficientscombination of the divisors Oare the support numbers h1 (Pσ ), .
. . , hl (Pσ ), i. e. the integral distances from theorigin to the facets of Pσ corresponding to Γ1 , . . . , Γl . Indeed, for toric varieties,this statement is well-known (see [8], Section 3.4). In particular, this holds for¯¯ in X of the maximal torus T ⊂ G. Note that the toric variety T¯¯the closure T¯¯corresponds to the same polytope Pσ and the codimension one orbits of T are the¯¯. Hence, in order to have the right coefficients inirreducible components of Oi ∩ T¯¯¯1 ∩T¯¯, . . .
, Ō¯l ∩T¯¯ in T¯¯, wethe decomposition of D ∩ T along the hypersurfaces Ōmust have¯ 1 + · · · + hl (Pσ )Ō¯ l.D = h1 (Pσ )Ō¯ j ) = 0, unless i = j.It follows that hi (ŌAnother useful collection of divisors consists of the closures in X of codimensionone Bruhat cells in G. Denote these divisors by D1 , . . . , Dk . They can also bedescribed as follows. Denote by ω1 , . .
. , ωk the fundamental highest weights of G.Let Xi be the compactification corresponding to the irreducible representation πiof G with the highest weights ωi . Then by Theorem 2.1 there is an equivariantmap from the wonderful compactification Xcan to Xi , and hence, there is alsoan equivariant map p : X → Xi . The divisor Di is the preimage under the mapp of the hyperplane section (regarded as a divisor in Xi ) corresponding to therepresentation πi .To each dominant weight λ = m1 ω1 + · · · + mk ωk there corresponds the weightdivisor D(λ) = m1 D1 + · · · + mk Dk .
The polytope of this divisor is the weightpolytope Pλ of the irreducible representation with the highest weight λ. Note thatλ is the only vertex of Pλ inside the fundamental Weyl chamber. Hence, it belongsto all facets of Pλ corresponding to Γ1 , . . .
, Γl (e. g. some of the facets mightdegenerate to the vertex λ). This implies the following lemma.Lemma 2.4. Let D(λ) be the weight divisor corresponding to a weight λ ∈ LT andlet Pλ be its polytope. Then hi (Pλ ) = hi (λ) for any i = 1, . . . , l.496V. KIRITCHENKOCombination of these two lemmas leads to the following result.Corollary 2.5. Let D be the divisor on X corresponding to a polytope P . Weassume that P is analogous to Pπ and identify the respective facets.
Then forany face F ⊂ P of codimension s that intersects the fundamental Weyl chamberD and for any point λ ∈ F ∩ D, the divisor D can be written uniquely as a linear¯¯i such that the corresponding facetscombination of D(λ) and of boundary divisors OΓi do not contain F . Namely, if F = Γi1 ∩ · · · ∩ Γis , thenX¯ j.[hj (P ) − hj (λ)]ŌD = D(λ) +j∈{1,...,l}\{i1 ,...,is }2.4. Ring of conditions. Let Z1 and Z2 be two algebraic subvarieties in G. Onecan define their intersection index Z1 Z2 as the number of points in the intersectiongZ1 ∩Z2 for a generic g ∈ G. Kleiman’s transversality theorem ensures that such anintersection index is well-defined, i.
e. for a generic g ∈ G the intersection gZ1 ∩ Z2is transverse and its cardinality does not depend on the choice of g [12]. We nowconsider the group C ∗ (G) of all formal linear combinations of algebraic subvarietiesin G up to the following equivalence relation. Two subvarieties Z1 , Z2 of the samedimension are equivalent if and only if for any subvariety Y of complementarydimension the intersection indices Z1 Y and Z2 Y coincide.For any two equivalence classes in C ∗ (G) represented by subvarieties Z1 and Z2define their product as the class of the subvariety gZ1 ∩ Z2 for a generic g ∈ G. DeConcini and Procesi showed that this product is well defined in C ∗ (G), i.
e. it doesnot depend on the choice of representatives [7]. The ring C ∗ (G) is called the ringof conditions of G.In what follows, we will also use the relation between the ring of conditions andthe (co)homology ring of a regular compactification. Namely, the subvarieties Z1and Z2 represent the same class in the ring of conditions C ∗ (G) if there exists aregular compactification X of the group G such that the closures of Z1 and Z2 inX represent the same homology class in X and have proper intersections with allG × G-orbits [6].
In particular, we will use regular compactifications to computethe intersection indices of subvarieties in G.2.5. Integration of polynomials. Let f (x1 , . . . , xk ) be a homogeneous polynomial function of degree d defined on a real affine space Rk with coordinates(x1 , .
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