PART II: THE WONDERFUL COMPACTIFICATION

Transcription

1 PART II: THE WONDERFUL COMPACTIFICATION Contents 1. Introduction 1 2. Construction of the compactification 3 3. The big cell 4 4. Smoothness of the compactification 8 5. The G G-orbits on the compactification 9 6. The structure of the orbits and their closures Independence of regular dominant weight Compactifications in more general spaces The Lie algebra realization of the compactification Log-homogeneous varieties The logarithmic cotangent bundle of G Cohomology of the wonderful compactification The Picard group The total coordinate ring 42 References Introduction Let K be any algebraic group, let τ : K K be an involution of K, and let H = K τ be its fixed point set. The homogeneous space K/H is called a symmetric space. Any algebraic group G is naturally a symmetric space under the action of K = G G by leftand right-multiplication, by the involution τ : G G G G (g, h) (h 1, g 1 ). The fixed point set is and there is an isomorphism H = G = {(g, g 1 ) G G} G = (G G)/G. 1

2 PART II: THE WONDERFUL COMPACTIFICATION 2 In the 1980s, DeConcini and Procesi [DP] showed that any semisimple symmetric space X has a wonderful compactification X a variety satisfying the following properties: (1) X is smooth and complete (2) X X is an open dense subset, and the boundary X\ X = X 1... X l is a union of smooth prime divisors with normal crossings. (3) The closures of the G-orbits on X are the partial intersections X i, for I {1,..., l}. i I In a more general framework, studying equivariant compactifications of homogeneous spaces, Luna and Vust [LV] showed that any homogeneous space X that has a wonderful compactification is necessarily spherical a Borel subgroup acts on X with an open dense orbit. For example, any reductive algebraic group G is a spherical homogeneous space under the two-sided action of G G, and the open orbit of the Borel subgroup B B G G is the open dense Bruhat cell. There are two distinguished classes of equivariant compactifications of spherical homogeneous spaces. The first is the class of toroidal compactifications these are generalizations of toric varieties, and their boundary structure is described combinatorially by fans. Every compactification X of X is dominated by a toroidal compactification X, in the sense that there is a proper birational G-equivariant morphism X X that restricts to the identity along the open locus X. The second class is the class of simple compactifications, which are compactifications on which G acts with a unique closed orbit. Brion and Pauer gave in [BP] a necessary and sufficient criterion for a spherical variety X to have simple compactifications. When such compactifications exist, there is a unique one that is also toroidal. This compactification X has the universal property that for any toroidal compactification X, and any simple compactification X, there are unique morphisms X X X that restrict to the identity along X. If X is smooth, it is the wonderful compactification of X and it has the properties described by DeConcini and Procesi. We will construct the wonderful compactification of a semisimple algebraic group of adjoint type G, following mostly the well-known survey of Evens and Jones [EJ]. Then we will describe two other realizations of the wonderful compactification, one as a variety of Lagrangian subalgebras of g g, and one as a GIT quotient of the Vinberg monoid.

3 PART II: THE WONDERFUL COMPACTIFICATION 3 2. Construction of the compactification From now on, let G be a semisimple connected complex algebraic group of adjoint type that is, with trivial center. Let G be its simply-connected cover, and choose a maximal torus and a Borel subgroup T B G corresponding to T B G. Let U B be the unipotent radical. Because the morphism G G is a central quotient, it is an isomorphism on unipotent subgroups, and we can identify U with its image in B. Let X ( T ) be the character lattice of the torus T, Φ the set of nonzero roots, Φ + the set of positive roots relative to B, and = {α 1,..., α l } the set of simple roots, where l = dim T is the rank of G. Let W = N G( T )/ T be the corresponding Weyl group. There is a standard ordering on X ( T ) given by λ µ λ µ = l n i α i, n i Z 0. i=1 Definition 2.1. A weight λ X ( T ) is dominant if λ, ˇα 0 for every positive coroot ˇα ˇΦ +. It is regular if λ, ˇα > 0 for every positive coroot ˇα ˇΦ +. The dominant weights form a cone the dominant Weyl chamber and the regular dominant weights are exactly the ones that fall in the interior of this cone. This is dual to the notion of a regular semisimple element in the Lie algebra of G. The following lemma, whose proof is left as an exercise, will be useful. Lemma 2.2. Let λ be a dominant weight and let V an irreducible representation of G of highest weight λ. Let v λ be a highest weight vector of V. Then the following are equivalent: (1) λ is regular. (2) The stabilizer of the highest weight space Cv λ in G is B. (3) The stabilizer of λ in the Weyl group W is trivial. From now on let V be an irreducible G-representation of regular highest weight λ. In the diagram (2.1) G End V \{0} ψ G P(End V ), the top arrow is the representation map, the left arrow is a quotient by the center, and the right arrow is a quotient by scalars. All these maps are G G-equivariant, and the representation map

4 descends to the G G-equivariant morphism PART II: THE WONDERFUL COMPACTIFICATION 4 ψ : G P(End V ). The map ψ is an injection this is guaranteed by adjointness if G is simple, and also by the regularity of λ if it is not. Definition 2.3. The wonderful compactification of G is X = ψ(g) P(End V ). Example 2.4. Let G = P GL 2 with G = SL 2. Then all nonzero weights are regular, and we can take V = C 2 to be the standard representation. In this case is the embedding with image and the closure of this image is The boundary of X is X = ψ(g) = {[ a and it is a single smooth prime divisor. c ψ : G P(M 2 2 ) {[ a c ] } b d ad bc 0, X = P(M 2 2 ) = P 3. ] } b d ad bc = 0 = P 1 P 1, Remark 2.5. Example 2.4 does not generalize. For n 3, the standard representation of SL n is not regular, because it is a fundamental representation and it generates one of the edges of the dominant Weyl chamber. In general, the wonderful compactification of P GL n is not simply the projective space P n The big cell Choose a basis of weight vectors of descending weight v 0,..., v n for V, such that v i is in the weight space V λi v 0 V λ of weight λ i, and with the properties i = 1,..., l v i V λ αi λ i > λ j i < j Let B be the opposite Borel to B, let B be its image in G, and let U be their common unipotent radical. Then U v i v i + j>i V λj, and so U stabilizes the affine space {[ ] P 0 (V ) = ai v i a 0 0} = C l.

5 PART II: THE WONDERFUL COMPACTIFICATION 5 Let v0,..., v n be a dual basis for the dual space V, so that each vi has weight λ i. Then U stabilizes the affine space {[ ] P 0 (V ) = ai vi a 0 0} = C l. The following lemma is clear from Lemma 2.2, and from the fact that the unipotent groups U and U act on the affine spaces P 0 (V ) and P 0 (V ) with closed orbits. Lemma 3.1. The action maps U U [v0] P 0 (V ) and U U [v 0 ] P 0 (V ) are isomorphisms, and their images are closed. We use the usual G G-equivariant identification V V End V (v f) (w f(w)v). Then the set {v i vj } is a basis for End V. The affine space {[ ] } P 0 = aij v i vj a 00 0 P(End V ) is U T U-stable, by the observations before Lemma 3.1. Define X 0 = X P 0. This intersection is called the big cell of the wonderful compactification. Proposition 3.2. The intersection of the big cell with the open dense locus ψ(g) is the image of the open Bruhat cell of G: X 0 ψ(g) = ψ(u T U). Proof. One containment is clear: ψ(e) X 0, X 0 is U T U-stable, and ψ is G G-equivariant, so it follows that ψ(u T U) X 0. For the other, choose a representative ẇ N G( T ) for each w W. Then by the Bruhat decomposition, G = U T ẇu. w W If w 1, then ẇv 0 is a weight vector of weight wλ, and wλ λ by Lemma 2.2. It follows that ψ(ẇ) = ẇψ(e) [ = ẇ vi vi [ ] = (ẇvi ) vi / P 0, ]

6 PART II: THE WONDERFUL COMPACTIFICATION 6 and therefore ψ(u T ẇu) X 0 =. So the only Bruhat cell whose image intersects X 0 is the open cell U T U. Remark 3.3. Since U T U is dense in G, its image ψ(u T U) is dense in X 0, and X 0 = ψ(u T U) P 0. Proposition 3.4. Let Z be the closure of ψ(t ) in P 0. Then Z = C l. Proof. Let t T and choose a preimage t T. Then [ ] ψ(t) = t vi vi [ ] = ( tv i ) vi [ ] = λi ( t)v i vi [ = v 0 v0 + ] λ i ( t) λ( t) v i vi. Since λ i λ, λ λ i = n ij α j, n ij Z 0. Then the image of t becomes [ ψ(t) = v 0 v0 + ] 1 αj (t) n v i v ij i [ l = v 0 v0 1 + α i=1 i (t) v i vi + ] 1 αj (t) n v i v ij i i>l Define a map F : C l ψ(t ) [ (z 1,..., z l ) It is clear that F is an isomorphism. v 0 v 0 + l z i v i vi + ( i>l i=1 Πz n ij j ] ) v i vi Consider the action map χ : U U Z X 0 (u, v, z) uzv 1. Theorem 3.5. The morphism χ is an isomorphism, and therefore X 0 = C dim G is smooth. Lemma 3.6. There is a U U-equivariant morphism β : X 0 U U such that β(χ(u, v, z)) = (u, v).

7 PART II: THE WONDERFUL COMPACTIFICATION 7 Proof. The morphism β 1 : P 0 P 0 (V ) [A] [Av o ] is well-defined. Moreover, for any (u, v, t) U U T, β 1 (ψ(u, v, t)) = utv[v 0 ] = u[v 0 ], so the image β 1 (ψ(u T U)) is the closed set U [v 0 ] = U by Lemma 3.1. Extending to the closure X 0, β 1 gives a surjection β 1 : X 0 U. Dually, define β 2 : P 0 P 0 (V ) [A] [v0 A 1 ] Once again this induces β 2 : X 0 U. Define β : X 0 U U x (β 1 (x), β 2 (x)). Lemma 3.7. Let A be an algebraic group acting on a variety Y. Suppose that there is an A- equivariant morphism β : Y A, where A is viewed as a left A-module. Then Y = A f 1 (e). Proof. Consider the maps f : A β 1 (e) Y (a, y) a y and g : Y A β 1 (e) y (β(y), β(y) 1 y). They are inverses of one another. Proof of Theorem 3.5. In view of the morphism β from Lemma 3.6, Lemma 3.7 implies that X 0 = U U β 1 (e, e). It is clear from the construction of β that ψ(t ) β 1 (e, e), and since the fiber β 1 (e, e) is closed, Z β 1 (e, e).

8 PART II: THE WONDERFUL COMPACTIFICATION 8 But X 0 is irreducible of dimension dim G, so the fiber β 1 (e, e) is irreducible of dimension dim T, and the inclusion Z β 1 (e, e) is actually an equality. 4. Smoothness of the compactification We will show that X is smooth by showing that it is a union of copies of the big cell X 0. To this end, we will need the following lemmas. Lemma 4.1. Let A be a semisimple group acting on an irreducible representation V with highest weight vector v 0. Then A [v 0 ] is the unique closed orbit of the action of A on P(V ). Proof. An orbit A [v] is closed if and only if it is projective, which is the case if and only if the stabilizer of [v] is parabolic. Up to conjugation we may assume this parabolic is a standard parabolic, and then [v] is stabilized by the Borel consisting of the positive roots, so it is a highest weight vector. Since V is irreducible it has a unique highest weight space, so [v] = [v 0 ]. Lemma 4.2. Suppose A is an algebraic group acting on an irreducible variety Y with a unique closed orbit Z. If U Y is an open subset that intersects Z, then Y = au. a A Proof. The set AU = a A au is open, so its complement W = Y \AU is closed and A-stable. Then W contains a closed A-orbit, which by uniqueness must be the closed orbit Z. But then Z W, so Z U = a contradiction. Proposition 4.3. Suppose that W X is a closed G G-stable subvariety. Then W = a(w X 0 ). a G G Proof. The tensor product V V is an irreducible representation of G G, so by Lemma 4.1 the action of G G on P(V V ) has the unique closed orbit (G G)[v 0 v0]. If W is closed and G G-stable, it contains a closed orbit, so by uniqueness (G G)[v 0 v0] W. But since W X 0 is open in W, and since [v 0 v0 ] X 0, it follows by Lemma 4.2 that W = a(w X 0 ). The following theorem is immediate: a G G

9 PART II: THE WONDERFUL COMPACTIFICATION 9 Theorem 4.4. For any G G-orbit O, the closure O has the property O = a(o X 0 ). a G G In particular, X = a G G ax 0, and so X is smooth. 5. The G G-orbits on the compactification First we describe the T -orbits on the closure Z = C l of the torus T from Proposition 3.4. For each I {1,..., l}, define and Z I = {(z 1,..., z l ) z i = 0 if i I} = C l I Z I = {(z 1,..., z l ) Z I z i 0 if i I}. Then it is clear that the ZI are exactly the T -orbits on Z, and each such orbit has a distinguished basepoint Each Z I is the closure of Z I z I = (z 1,..., z l ), z i = 1 if i I. in Z, and the boundary Z\ψ(T ) = is the union of the coordinate hyperplanes in C l in particular, it is a divisor with normal crossings. Now we describe the U T U-orbits on X 0. Using the isomorphism of Theorem 3.5, define and l i=1 Z i χ : U U Z X 0 Σ I = χ(u U Z I ) dim G I = C Σ I = χ(u U Z I ). Then each Σ I is a U T U-orbit on X 0, Σ I is its closure, and the boundary is a divisor with normal crossings. X 0 \ψ(u T U) = The closure of every G G-orbit on X is a union of translations of its intersection with X 0 so, there are at most 2 l such orbit closures. l i=1 Σ i Lemma 5.1. Suppose that W is a projective variety and U W is an open affine subset. Then the boundary W \U is a union of irreducible components of codimension 1.

10 PART II: THE WONDERFUL COMPACTIFICATION 10 Proposition 5.2. Let S i be the closure of Σ i in X. Then l X\ψ(G) = S i. i=1 Proof. Since X is projective and ψ(g) is an affine open subset, X\ψ(G) = S α is a union of irreducible components of codimension 1 by Lemma 5.1. This union is G G-stable, and because G G is connected every component S α is G G-stable. By Proposition 4.3, S α = a(s α X 0 ). a G G Then the intersection S α X 0 is a U T U-stable irreducible hypersurface in X 0, so it is equal to Σ i for some i. It follows that S α = Σ i = S i. Define the intersection S I = i I S i. Then S J S I if and only if J I, and S I X 0 = Σ I, so by Proposition 4.3 S I = aσ I. a G G In particular, S I is smooth. Define SI = S I \ S J. Then SI = aσ I = (G G)Σ I = (G G)(U T U)z I = (G G)z I, a G G and SI is a single G G-orbit. We collect all these results into a single theorem: I J Theorem 5.3. There are exactly 2 l G G-orbits in X, given by S I = (G G)z I, I {1,..., l}. Their closures S I are smooth, and the boundary of X is a divisor with normal crossings. X\ψ(G)

11 PART II: THE WONDERFUL COMPACTIFICATION The structure of the orbits and their closures Let g = Lie G, and consider the root space decomposition g = h + g α. α Φ Fix a subset I {1,..., l}. Let I = {α i i I}, and let Φ I be the set of roots spanned by I. This produces the standard Levi subalgebra l I = h + α Φ I g α, and the parabolic subalgebras p ± I subgroups of G by L I, P ± I, U ± I. = l I + b ± with nilpotent radicals u ± I. Denote the corresponding Let V I be the irreducible representation of L I generated by applying L I to the highest weight vector v 0 that is, V I = Ul I v 0 where Ul I is the universal enveloping algebra of l I. For any x u I, x v 0 = 0. Because u I is normal in p I, it follows that V I is p I -stable. Lemma 6.1. The stabilizer of V I in G is exactly the parabolic subgroup P I. Proof. Let Q be the stabilizer of V I in G. It is already clear that P I Q, so Q is a standard parabolic subgroup and in particular it is connected. It is enough to show that p I = Lie Q. Let t T be such that α i (t) = 1 whenever α i I, and α i (t) 1 otherwise. Then t Z(L I ), and we pick a preimage t T. Since V I is an irreducible representation of L I, t acts on it by the scalar λ( t). Let α i I be a root whose corresponding root space is not contained in p I, and let x g αi be nonzero. Then t acts on xv I by λ( t) α i ( t), and this scalar is distinct from λ(t) because α i (t) 1. It follows that V I xv I = 0. Because λ is regular, xv I 0, so x Lie Q. So the only root spaces contained in Lie Q are the ones also contained in p I. Now let J = j {0,..., n} λ λ j = n i α i, n i Z 0. α i Φ I The set {v j j J}, which consists of weight vectors whose weights can be obtained from λ by subtracting the simple roots in I, is a basis for V I.

12 PART II: THE WONDERFUL COMPACTIFICATION 12 Proposition 6.2. Let pr VI End V denote the projection onto V I. Then z I = [pr VI ]. Proof. Recall that z I = (z 1,..., z l ) C l is the point whose coordinates are 0, i I z i = 1, i I From the isomorphism of Proposition 3.4, it is identified with the following point in X 0 : z I = v 0 v0 + z i v i vi + ( ) n z ij j v i vi = v j v j. i I i>l j J Proposition 6.3. The stabilizer of z I in G G is { (ux, vy) UI L I U I L I xy 1 Z(L I ) }. Proof. Suppose (r, s) G G stabilizes z I = [pr VI ]. Then [rpr VI s 1 ] = [pr VI ], so in particular r stabilizes the image V I of pr VI. By Lemma 6.1, this means r P I. Moreover, if r = ux U I L I, then since the action of U I on V I is trivial. Dually, s P I Moreover, if s = vy U I L I, then Then [rpr VI ] = [xpr VI ], by applying the preceding discussion to V I End V = V V V V = End V. [pr VI s 1 ] = [pr VI y 1 ]. [pr VI ] = [rpr VI s 1 ] = [xpr VI y 1 ], under the isomorphism so xy 1 acts trivially on P(V I ), and so acts by a scalar on the irreducible representation V I of L I. It follows that xy 1 Z(L I ). Remark 6.4. Because the stabilizer of z I is contained in P I P I, there is a surjection SI = (G G)/Stab G G (z I ) G/P I G/P I. The fiber of this surjection is P I P I /Stab G G(z I ) = L I L I /{(x, y) xy 1 Z(L I )} = L I /Z(L I ), which is a semisimple group of adjoint type and smaller rank than G.

13 PART II: THE WONDERFUL COMPACTIFICATION 13 In particular, this gives an isomorphism S {1,...,l} = G/B G/B between the unique closed G G-orbit on X and the product of two copies of the flag variety of G. The natural embedding End V I End V induces a closed embedding of projective varieties P(End V I ) P(End V ), and z K P(End V I ) if and only if I K. Define the map L I P(End V I ) P(End V ) g g v j vj = [gz I ]. j J This descends to an injection G I = L I /Z(L I ) ψ I P(End V I ). Since V I is a regular representation of L I, it is a regular representation of the simply-connected cover of the semisimple adjoint group G I, and we can apply the entire previous discussion to the compactification the wonderful compactification of G I. The quotient P I P I induces an action of P I P I on X I. Theorem 6.5. The map X I = ψ I (G I ) P I /U I Z(L I ) P I /U I Z(L I) = G I G I ϕ : G G PI P X I S I I is an isomorphism of G G-varieties. In particular, S I fibers over the partial flag variety G/P I G/P I with fiber X I. Proof. It is enough to show that ϕ is bijective, because the target S I is smooth. Recall that and I K if and only if z K X I. In this case S I = I K S K ϕ(g G {z K }) = S K. So every G G-orbit is contained in the image of ϕ, and ϕ is surjective.

14 PART II: THE WONDERFUL COMPACTIFICATION 14 Similarly, it is enough to show that ϕ is injective on orbits. Suppose that ϕ(g, h, z K ) = ϕ(e, e, z K ). Then (g, h) Stab G G (z K ) P K P K P I P I. It follows that in the fiber product G G PI P I X I. (g, h, z K ) (e, e, z K ) 7. Independence of regular dominant weight Suppose that λ and µ are two regular dominant weights of G, with corresponding irreducible representations V and W. They produce two compactifications: X 1 P(End V ) and X 2 P(End W ). Let v 0,... v n be the usual basis of V chosen in (3), and let w 0,... w n be the analogous basis of W. Choose identity basepoints [ ] [ ] x 1 = vi vi X 1 and x 2 = wi wi X 2 and define There are natural projections X = (G G)(x 1, x 2 ) X 1 X 2. X p 1 p 2 X 1 X 2. Theorem 7.1. The projections p 1 and p 2 are both isomorphisms, and they induce an isomorphism p 2 p 1 1 : X 1 X 2. We will apply superscripts to the notation of the previous sections, so that X0 i cell of X i, X i the closure of the torus in the big cell, etc. Define will be the big Z = T (x 1, x 2 ) X 1 0 X 2 0. Lemma 7.2. There is an isomorphism Z = C l and the projections p i : Z Z i are isomorphisms. Proof. The proof is exactly as in Proposition 3.4: ([ l t(x 1, x 2 ) = v 0 v0 1 + α i=1 i (t) v i vi + ] v i vi, i>l [ l w 0 w0 1 + α i=1 i (t) w i wi + ]) w i wi, i>l

15 where and are polynomials in PART II: THE WONDERFUL COMPACTIFICATION 15 1 α 1 (t),..., 1 α l (t). As before, there is an isomorphism C l Z given by ([ l (z 1,..., z l ) v 0 v0 + z i v i vi + ] v i vi, i>l Let X 0 i=1 = p 1 i (X0 i ) and define the action map [ w 0 w 0 + l z i w i wi + ]) w i wi. i>l i=1 χ : U U Z X. Lemma 7.3. The morphism χ is an isomorphism onto X 0. Proof. Consider the commutative diagram U U Z χ X Id Id p i p i χ i U U Z i X i. It is clear that χ is injective because the composition χ i (Id Id p i ) is injective. Let Y be the image of χ, and consider the composition σ = χ (Id Id p i ) 1 χ i 1 : X0 i Y. The diagram is commutative, so p i σ = Id X 0 i and σ is a section of p i on X0 i. The composition σ p i is defined only on X0 on X0 i. Because σ is a section, it follows that the restriction of, because σ is defined σ p i : X 0 X to the image Y of χ is also the identity. But X0 and Y are irreducible of the same dimension as X, so X0 Y is a dense subset of X0. Then the map σ p i is the identity on a dense subset of X0, so it is the identity on all of X 0. This shows that p i gives an isomorphism X0 Xi 0, so χ : U U Z X0 is surjective. Lemma 7.4. The restriction of p i to the set U = a G G ax 0

16 PART II: THE WONDERFUL COMPACTIFICATION 16 is injective. Proof. For any subset I {1,..., l}, let zi = (zi 1, zi 2 ) X 1 X 2. By Lemma 7.3, as in Section 5, X0 is U T U-stable and the U T U-orbits on X0 are exactly indexed by the basepoints zi. It is enough to check the statement of the lemma on the intersections of G G-orbits with X0. Using the G G-equivariance of p i, it is enough to suppose that p i ((g, h)zi ) = p i (zi ). Then so that (g, h) Stab G G (zi i ). But (g, h)z i I = z i I, Stab G G (z I ) = Stab G G (z 1 I ) Stab G G (z 2 I ) = Stab G G (z i I), so this means that (g, h) Stab G G (z I ) and (g, h)z I = z I. Proof of Theorem 7.1. The projection p i restricts to an isomorphism p i : X 0 X i 0, and by G G-equivariance it gives a surjection p i : U ax0 i = X i a G G which is injective by Lemma 7.4. Because X i is smooth, p i is an isomorphism. Then U X is a projective subvariety of the same dimension as X, so they are equal. It follows that p i is an isomorphism between X and X i. 8. Compactifications in more general spaces The results in this section are outlined in [EJ] Section 3.1. induces an action G G P(E). A point [x] P(E) whose stabilizer is the diagonal subgroup Any representation E of G G G = {(g, g) g G}, gives an embedding ψ E : G P(E) g (g, e) [x]

17 PART II: THE WONDERFUL COMPACTIFICATION 17 and a compactification X(E, [x]) = ψ E (G) P(E). In the previous section we showed that if V and W are regular irreducible representations of G, then X(End V, [Id V ]) = X(End W, [Id W ]). Suppose now that V is an irreducible G-representation of regular highest weight λ and that W 1,..., W r are irreducible G-representations of highest weight µ 1,..., µ r. Write W = W 1... W r and let F be any G-representation. As in the previous sections, write X = X(End V, [Id V ]). Theorem 8.1. Suppose that µ k λ for every k = 1,..., r. Then X(End V End W F, [Id V Id W 0]) = X. Remark 8.2. The G-orbit G [Id V Id W 0] lies in the image of the closed embedding so there is an identification P(End V End W ) P(End V End W F ), X(End V End W F, [Id V Id W 0]) = X(End V End W, [Id V Id W ]). Denote by ψ the injection ψ : G P(End V End W ) g g [Id V Id W ] and by X the closure of its image inside P(End V End W ). To prove the theorem it will be sufficient to show that X = X. As before, let v 0,..., v n be a basis of T -weight vectors for V satisfying the conditions (3). Let w 0,..., w m be a basis of T -weight vectors for W such that w i has weight µ i. This gives a basis for the space End V End W. Let {v i v j, w i w j } P 0 = {[ aij v i vj + b ij w i w j and let Z be the closure of the torus inside this affine space: Z = ψ(t ) P 0. ] } a 00 0,

18 PART II: THE WONDERFUL COMPACTIFICATION 18 Proposition 8.3. There is an isomorphism Z = C l. Proof. Let t T and t T be some preimage of t in the simply-connected cover G. Then [ ψ(t) = t vi vi + ] w i wi [ = λi ( t)v i vi + ] µ i ( t)w i wi [ l = v 0 v0 1 + α i=1 i (t) v i vi + λ i ( t) λ( t) v i vi + ] µ i ( t) λ( t) w i wi i>l The coefficients are polynomial in the terms λ i ( t) λ( t) and µ i( t) λ( t) 1 α 1 (t),..., 1 α l (t) because each µ i is less than some µ k in the partial ordering of the weight lattice, and each µ k λ by the assumption of Theorem 8.1. We can define an isomorphism C l Z just as in the proof of Proposition 3.4: (z 1,..., z l ) [ v 0 v 0 + l z i v i vi + v i vi + ] w i wi, i>l i=1 where and are polynomials in z 1,..., z l. Define P = {[A B] P(End V End W ) A 0}. Then there is a natural projection π : P P(End V ), and Proposition 8.3, together with Proposition 3.4, imply that the restriction π Z : Z Z is an isomorphism. Fix I {1,..., l}, and under the identification of Proposition 8.3 define z I = (z 1,..., z l ) Z 1, if i I, z i = 0, if i I. (Cf. the definitions at the start of Section 5.) Then π(z I) = z I, and each T -orbit on Z contains exactly one basepoint of the form z I. As in Section 6, let I = {α i i I} and let l I be the corresponding Levi subalgebra of g = Lie G. Define V I = Ul I v 0

19 PART II: THE WONDERFUL COMPACTIFICATION 19 to be the subspace of V generated by applying l I to the highest weight vector v 0, and recall that the unipotent radical U I of the corresponding positive parabolic acts on V I trivially. For each index k such that λ µ k is in the span of the simple roots I, let w0 k,..., wk n k be a basis of T -weight vectors for W k, such that wi k has weight µ k i and which satisfies the conditions of (3). Define W k I = Ul I w k 0 to be the subspace of W k generated by applying l I to the highest weight vector w k 0. In Section 6, the set of indices J = j {0,..., n} λ λ j = n i α i, n i Z 0 α i Φ I indexed a basis of weight vectors {v j j J} for V I. Similarly, for each W k as above, define J k = j {0,..., n} λ µk j = n i α i, n i Z 0. α i Φ I Because λ µ k j = (λ µ k ) + (µ k µ k j ), and each term is a linear combinations of roots with non-negative coefficients, an index j is in J k if and only if both λ µ k and µ k µ k j are in the span of I. Then the set of weight vectors {w k j j J k } is a basis for W k I, and it is guaranteed to be nonempty because 0 J k. Lemma 8.4. Let pr W k I End W denote the projection onto WI k. Then [ ( )] z I = pr VI prw k, I where the sum is taken over all k such that λ µ k is in the span of I. Proof. From Proposition 8.3, [ z I = v 0 v 0 + n δ i v i vi + i=1 r k=1 ( nk i=0 δ k i w k i w k i Here δ i = 1 if λ λ i is in the span of I, and 0 otherwise. Likewise, δi k = 1 if λ µk i is in the span of I, and 0 otherwise. It follows immediately that z I = v 0 v0 + v i vi + j J k )] wi k wi k, j J k where the only indices k appearing in the second sum are those for which λ µ k is in the span of I. Lemma 8.5. The points z I X and z I X have the same stabilizer in G G..

20 Proof. The inclusion PART II: THE WONDERFUL COMPACTIFICATION 20 Stab G G (z I) Stab G G (z I ). is clear, since z I P(End V End W ), z I P(End V ), and the action of G G is block-diagonal. Conversely, recall from Proposition 6.3 that Stab G G (z I ) = {(ux, vy) U I L I U I L I xy 1 Z(L I )}. For any point (ux, vy), [ ( )] (ux, vy) pr VI prw k = I [ xpr VI y 1 ( xprw k I y 1)], because u U I and v U I both act trivially. (Cf. the proof of Proposition 6.3.) Since V I is an irreducible representation of L I, the central element xy 1 acts on it by the scalar λ(xy 1 ). Likewise, xy 1 acts on each W k I by µ k (xy 1 ). Because xy 1 is central in L I, for any α i in the set I of simple roots that generate l I, α i (xy 1 ) = 1. But λ µ k is in the span of I, so it follows that λ(xy 1 ) µ k (xy 1 ) = 1. Retracing our steps, [ ( )] [ ( (ux, vy) pr VI prw k = xpr VI y 1 xprw k y 1)] I I [ ( )] = λ(xy 1 )pr VI µ k (xy 1 )pr W k I [ ( )] = pr VI prw k, I so (ux, vy) stabilizes z I. Consider the open subset X 0 = X P 0 of X. It is U T U-stable, and we define the action map χ : U U Z X 0. Proposition 8.6. The morphism χ is an isomorphism, and therefore X 0 = C dim G. Proof. Applying the construction of Lemma 3.6, there is a U U-equivariant morphism β : X 0 U U such that β (χ (u, v, z )) = (u, v). Then by Lemma 3.7, there is an isomorphism X 0 = U U β 1 (e, e), and Z β 1 (e, e).

21 PART II: THE WONDERFUL COMPACTIFICATION 21 But X 0 is irreducible of dimension n = dim G, so the fiber β 1 (e, e) is irreducible of dimension dim T, and the inclusion Z β 1 (e, e) is an equality. Now define Y = a G G to be the union of all G G-translates of the open affine cell X 0. The open subvariety Y of X is contained in P. Proposition 8.7. The restriction of to Y is injective. ax 0 π : P P(End V ) Proof. It is sufficient to check that π is injective on G G-orbits. Every T -orbit on Z contains a basepoint of the form z I. Then every U T U-orbit on X 0 contains some point z I, and therefore every G G-orbit on Y contains such a point. Suppose without loss of generality that Because π is G G-equivariant, π(gz Ih 1 ) = π(z I). gz I h 1 = z I, so (g, h) Stab G G (z I ). By Lemma 8.5, this is the same as the stabilizer of z I, so gz Ih 1 = z I. Proof of Theorem 8.1. The restriction π Z : Z Z is an isomorphism by Proposition 8.3. By Proposition 8.6 and 3.5, it follows that π X 0 : X 0 X 0 is also an isomorphism. So the restriction π Y : Y ax 0 = X a G G is surjective, and by Proposition 8.7 it is also injective. Since the wonderful compactification X is smooth, π Y is an isomorphism of algebraic varieties. This means that Y X is a projective (and therefore complete, and therefore closed) algebraic subvariety of X of the same dimension, so they are equal. Then π gives an isomorphism X = X. Remark 8.8. The compactification X is contained in the closed subvariety ( ( P End V End W k) ) F P (End V End W F ).

22 PART II: THE WONDERFUL COMPACTIFICATION 22 Therefore we could replace X in the discussion above by ( ( X End V End W k) F, for some scalars c k C. [ Id V ( ck Id W k ) ]) 0, 9. The Lie algebra realization of the compactification This section outlines another realization of the wonderful compactification, using the results of Section 8, and following the construction in [EJ] Section 3.2. Let n be the dimension of G, and consider the action of G G on the Grassmannian Gr(n, g g). The stabilizer in G G of the diagonal subalgebra g = {(x, x) x g} g g is the diagonal subgroup G = {(g, g) g G} G G. The orbit of this diagonal subalgebra in the Grassmannian is (G G) g = (G G)/G = G, and we consider its closure G = (G G) g Gr(n, g g). Theorem 9.1. The compactification G is isomorphic to the wonderful compactification X. Consider the Plücker embedding Gr(n, g g) P ( n (g g)), which takes a subspace spanned by a basis u 1,..., u n to the line [u 1... u n ]. Let [g ] be the image of g. Because this is a closed embedding, G = (G G) g = (G G) [g ] P ( n (g g)). For a nonzero vector v [g ], define the subspace E = U(g g) v n (g g). It is clear that E does not depend on the choice of v inside [g ], and the compactification G is contained in the projectivization of E: We will show that E is of the form End V G P(E). ( End W k) F

23 PART II: THE WONDERFUL COMPACTIFICATION 23 for some irreducible representation V of G of highest weight λ and some irreducible representations W k of highest weights µ k with µ k λ in the partial order on the weight lattice. We will show that under this identification [g ] = [ ( ) ] Id V ck Id W k 0, so that Theorem 9.1 will follow from Theorem 8.1 and Remark 8.8. Let h 1,..., h l be a basis for the Cartan h = Lie T, and for each α Φ let e α g be a root vector of weight α. There is a basis of T T -weight vectors for g g: {(h i, ±h i ) i = 1,..., l} of weight (0, 0), {(e α, 0) α Φ} of weight (α, 0), {(0, e α ) α Φ} of weight (0, α). This gives a basis of T T -weight vectors of n (g g) indexed by triples (A, B, S), where A, B Φ are such that A + B n, S {(h i, ±h i ) i = 1,..., l} is such that A + B + S = n. The weight vector corresponding to such a triple is ( ) ( ) v ABS = (e α, 0) s (0, e α ), and it has weight α A s S α, β. α A β B β B Remark 9.2. Let B + G be a positive choice of Borel subgroup containing the maximal torus T, and let B be the opposite Borel. Denote by Φ + Φ the positive roots. Then the T T -weight vector v 0 = ( ) (e α, 0) (h i, h i ) (0, e α ), α Φ + i=1 l β Φ + has weight (λ, λ), where λ = α Φ + α is the sum of the positive roots. Any other T T -weight vector v ABS has weight (µ, µ ) with µ λ and µ λ, so v 0 is a highest weight vector with respect to the Borel subgroup B B G G. Proposition 9.3. The vector v 0 is in the subspace E = U(g g) v of n (g g). Let h h be a regular element such that α i (h) = 1, i = 1,..., l. This element induces an injection γ : C T

24 PART II: THE WONDERFUL COMPACTIFICATION 24 such that Lie C = Ch. The proof of Proposition 9.3 will follow from the following Lemma. Lemma 9.4. lim γ(z) [g ] = [v 0 ]. z Proof. First we decompose [g ] into a projectivized sum of T T -weight vectors. Write [( ) ( l )] [g ] = (e δ, e δ ) (h i, h i ). δ Φ i=1 Then ( ) ( l ) (9.1) v = ((e δ, 0) + (0, e δ )) (h i, h i ) = v ABS, δ Φ i=1 where the sum is taken over all triples (A, B, S) such that A Φ, B = Φ\A, S = {(h i, h i ) i = 1,..., l}. For any α = l i=1 n iα i Φ, γ(z) e α = z α(h) e α = z ht(α) e α, where ht(α) = l i=1 n i is the height of the root α. Then ( ) ( ) (9.2) γ(z) v ABS = z ht(α) (e α, 0) s (0, e α ) = z n A v ABS, where α A s S n A = α A ht(α) is the sum of the heights of the roots appearing in A. Let Then n 0 = α Φ + ht(α). n 0 n A for all A Φ, β B n 0 = n A if and only if A = Φ +.

25 PART II: THE WONDERFUL COMPACTIFICATION 25 We can now compute lim γ(z)[g ] = lim [γ(z)v ] z z = lim z [ = lim z Proof of Proposition 9.3. The closed subvariety [ ] z n A v ABS A Φ v 0 + A Φ P(E) P( n (g g)) z n A n 0 v ABS ] = [v 0 ]. is T T -stable and closed, so It follows that [v 0 ] P(E), and v 0 E. (T T )[g ] P(E). Proof of Theorem 9.1. By Proposition 9.3 and Remark 9.2, v 0 E is a highest weight vector of weight (λ, λ), with λ = α Φ α. Then U(g g) v 0 = V V = End V E, where V is the irreducible G-representation of regular highest weight λ. Because G is semisimple, we can decmopose E = End V ( End W k ) F, where the second summand consists of all irreducible representations of G G of the form W W, and the third summand consists of all irreducible representations G G of the form U W with U = W. Each representation End W k has highest weight (µ k, µ k ), and from Remark 9.2 it follows that µ k λ. It remains to show that [g ] = This will follow from the next lemma. [ ( ) ] Id V ck Id W k 0. Lemma 9.5. An irreducible representation of G G has a G -stable one-dimensional subspace if and only if it is of the form End W for some irreducible representations W of G. In this case, the unique such space is CId W. Proof. Any irreducible representation of G G is of the form for irreducible representations U and W of G. U W = Hom(W, U)

26 PART II: THE WONDERFUL COMPACTIFICATION 26 There is a G -stable line in U W if and only if there is a G-equivariant homomorphism in Hom(W, U). By Schur s lemma, such a homomorphism exists if and only if U = W, in which case it is unique up to scaling. Because [g ] P(E) is G -fixed, the line Cv E = End V ( End W k ) F is G -stable. Then, its projection onto each summand is G -stable. Lemma 9.5 then implies that the projection of v onto End V is a 0 Id V for some a 0 C, that its projection onto End W k is a k Id W k for some a k C, and that its projection onto F is 0 because F has no one-dimensional G -stable subspaces. So we have But recall from (9.1) that v = c 0 Id V ( ck Id W k v = v 0 + v ABS ) 0. as a sum of T T -weight vectors in n (g g), so the projection of v onto End V is nonzero and so c 0 0. It follows that [g ] = [ ( ) ] Id V ck Id W k 0. Theorem 9.1 gives an isomorphism ϕ : X G Gr(n, g g) such that for any interior point g G X of the wonderful compactification, (9.3) ϕ(g) = (g, e) g. We will describe which n-dimensional subspaces of g g appear in the boundary of G in the Grassmannian. Because the map ϕ is G G-equivariant, it is enough to find the image ϕ(z I ) Gr(n, g g) of each G G-orbit basepoint z I. Recall the notation defined at the beginning of Section 6. Theorem 9.6. The image of the orbit basepoint z I under the isomorphism ϕ is the n-dimensional space ϕ(z I ) = {(u + x, v + x) u u I, v u I, x l I} = p I li p I.

27 PART II: THE WONDERFUL COMPACTIFICATION 27 Remark 9.7. Suppose I = {1,..., l} and z {1,...,l} is the basepoint of the unique closed G G-orbit of minimal dimension. Theorem 9.6 says that ϕ(z {1,...,l} ) = b h b, and the image of this subspace under the Plücker embedding is exactly the point defined in Remark 9.2. [v 0 ] P ( n (g g)) The proof is similar to the discussion in Lemma 9.4. Proof of Theorem 9.6. Let h h be such that 0, α i I α i (h) = 1, α i I This produces a one-parameter subgroup γ : C T such that Lie γ(c ) = Ch. Then α i (γ(z)) = z αi(h) 1, α i I = z, α i I. The in X P(End V ), this one-parameter subgroup is [ l γ(z) = v 0 v0 1 + α i=1 i (γ(z)) v i vi + ] v i vi, i>l (where is polynomial in the first l coefficients, cf. Proposition 3.4) and as z tends to infinity we obtain lim γ(z) = z v 0 v0 + v i vi + v i v i = z I. α i I So γ is a one-parameter subgroup that tends to the orbit basepoint z I in the boundary of the wonderful compactification. Then ϕ(z I ) = lim ϕ(γ(z)) = lim (γ(z), e) g, z z as in (9.3). To compute (γ(z), e) g we work in the projective space P( n (g g)) under the Plücker embedding. Recall that [g ] = [ vabs ], where as in (9.1) the sum is taken over all triples (A, B, S) such that A Φ, B = Φ\A, i>l

28 S = {(h i, h i ) i = 1,..., l}. PART II: THE WONDERFUL COMPACTIFICATION 28 For any root α Φ, write α = n i α i + α i I α j I n j α j and define ht I (α) = n j. α j I Then γ(z) e α = z α(h) e α = z hti(α) e α. Applying the one-parameter subgroup γ to T T -weight vectors in n (g g), γ(z) v ABS = z m A v ABS, where (Cf. the computation in (9.2).) Let Then m A = α A ht I (α). m 0 = α Φ + ht I (α). m 0 m A for all A Φ, m 0 = m A if and only if Φ + \Φ I A Φ + Φ I, that is, m 0 = m A if and only if A differs from Φ + by roots in Φ I, which do not contribute to the sum m A. Then we can compute lim γ(z)[g ] = lim [γ(z) ] v ABS z z = lim [v 0 + ] z m A m 0 v ABS = z where the last sum is taken over triples (A, B, S) such that [ va B S], Φ + \Φ I A Φ + Φ I B = Φ\A S = {(h i, h i ) i = 1,..., l}. This sum can be written (9.4) ( ) ( l ) (e α, 0) (h i, h i ) (0, e β ), α A β B (A,B,S) i=1

29 PART II: THE WONDERFUL COMPACTIFICATION 29 and we notice that every root vector e δ with δ Φ I appears as both (e δ, 0) and as (0, e δ ) in this sum. Then we can rewrite (9.4) as (e α, 0) ( l ) (e δ, e δ ) (h i, h i ) (0, e β ) α Φ + \Φ I δ Φ I i=1 β Φ + \Φ I The vectors in the first wedge give a basis for u I 0 g g, the vectors in the last wedge give a basis for 0 u I g g, and the diagonal vectors in the two middle wedges give a basis for the diagonal subspace l I = {(x, x) x l I } g g. It follows that ϕ(z I ) = lim z γ(z)[g ] = [p I li p I ]. 10. Log-homogeneous varieties In this section we introduce some general notions about log-homogeneous varieties, following the exposition in Sections 1.1 and 2.1 of the lecture notes [Bri1]. For now, let G be a connected complex algebraic group with Lie algebra g, and let X be a smooth connected G-variety. Denote by T X = Der(O X ) the tangent sheaf of X, whose sections are derivations of the ring of regular functions O X. This is the locally-free sheaf associated to the tangent bundle T X of X. The action of G on the variety X gives a map op X : g Γ(X, T X) ξ v ξ, where v ξ is the vector field induced by the differential of the G-action: v ξ (x) = d (exp( tξ)x). dt t=0 (The negative sign is necessary to make op X a homomorphism of Lie algebras.) There is a corresponding morphism of sheaves op X : O X g T X. Definition The variety X is homogeneous if the action of G on X is transitive. Proposition The variety X is homogeneous if and only if the morphism op X is surjective.

30 PART II: THE WONDERFUL COMPACTIFICATION 30 Proof. Choose a basepoint x X. If X is homogeneous, the action map ϕ x : G X g g x is surjective, and so is its differential dϕ x : g T x X. But dϕ x = op X,x, so it follows that op X is also surjective. Conversely, suppose that op X is surjective, so that the induced map on stalks dϕ x is surjective at every x. Then ϕ x is a submersion, and its image G x is open in X. Since X is connected and x X was chosen arbitrarily, it follows that X is homogeneous. Definition An effective reduced divisor D X has normal crossings if at each x X there exist local coordinates x 1,..., x n such that D = {(x 1,..., x n ) x 1... x k = 0}. That is, in the completed local ring Ô X,x = C[[x 1,..., x n ]] the ideal of D is generated by x 1... x k. Definition Suppose that D X is a normal crossing divisor. The logarithmic tangent sheaf is the subsheaf T X ( log D) T X whose sections are the derivations of O X that preserve the ideal sheaf of D. In other words, these sections are vector fields on X that are tangent to the divisor D, called logarithmic vector fields. Example Let X = C n and let D = {x 1... x k = 0} be the union of the first k coordinate hyperplanes. At the origin, the logarithmic tangent sheaf is generated by x 1 1,..., x k k, k+1,..., n. Remark (1) Because D is a normal crossing divisor, the logarithmic tangent sheaf is locally-free of rank dim X, and the associated vector bundle is the logarithmic tangent bundle T X( log D). It is not a subbundle of the tangent bundle on the contrary, the two bundles have the same rank.

31 PART II: THE WONDERFUL COMPACTIFICATION 31 (2) The restriction of T X ( log D) to the open piece X = X\D is the usual tangent sheaf T X. (3) The dual of T X ( log D) is the sheaf Ω 1 X (log D) of logarithmic 1-forms with poles along D. (A logarithmic form is an algebraic form with simple poles whose differential also has simple poles.) In Example 10.5, this sheaf is locally generated by dx 1 x 1,..., dx k x k, dx k+1,..., dx n. Its associated bundle is the logarithmic cotangent bundle T X( log D). Now let G act on X and let D X be a G-stable normal crossing divisor. Then the differential of the action map induces the morphism of Lie algebras and the associated morphism of sheaves op X,D : g Γ(X, T X( log D)) op X,D : O X g T X ( log D). Definition The pair (X, D) is log-homogeneous if the morphism op X,D is surjective. Example (1) Suppose that X = C n is affine space, D = {x 1... x n = 0 is the union of the coordinate hyperplanes, and G = (C ) n acts on X by coordinate-wise multiplication. Then (X, D) is log-homogeneous. (2) Suppose that X is a smooth projective toric variety for a torus G = T, so that T sits inside X as an open T -orbit. The boundary D = X\T is a normal crossing divisor, and X can be covered by open T -stable affine spaces C n on which T acts by coordinate multiplication. It follows that (X, D) is log-homogeneous. (See [Ful].) Remark Suppose that (X, D) is log-homogeneous. Then the restriction op X,D X = op X : O X X g T X ( log D) X = T X is surjective, so X is a homogeneous space. Construct a stratification of the divisor D as follows: let X 1 = D, X 2 = Sing(D),..., X m = Sing(X m 1 ),..., and let the strata be the connected components of X m \X m+1. They are smooth, locally-closed, and G-stable because G is connected. Fix a stratum S and a point x S, and let x 1,..., x n be coordinates at x such that the divisor D is given by D = {x 1... x k = 0}. Then X m \X m+1 is the locus where exactly m coordinates are zero, and S = {x 1 =... = x k = 0}

32 PART II: THE WONDERFUL COMPACTIFICATION 32 has codimension k. The stratum S is the intersection of the stratum closures S i = {x j = 0 j k, j i}. The normal space of S in X at x N S/X,x = T x X/T x S decomposes as a sum of lines (10.1) N S/X,x = L 1... L k, where each L i is the normal space to S in S i at x. The stabilizer Stab G (x) = G x of x in G acts on all these spaces, and its identity component preserves each line L i. The action map ρ x : (G x ) (C ) k has differential dρ x : g x C k. The following gives a criterion for log-homogeneity. (See [Bri1], Proposition ) Proposition The following are equivalent: (1) The pair (X, D) is log-homogeneous. (2) Each stratum S is a single G-orbit and the differential dρ x is surjective at every x S. If these conditions hold, there is a short exact sequence of Lie algebras 0 ker(dρ x ) g op X,D T x X( log D) 0. Proof. Because T X ( log D) preserves the ideal sheaf of S, there is a morphism of sheaves T X ( log D) S T S that descends to a linear map on fibers p x : T x X( log D) T x S. In coordinates x 1,..., x n at x, p x is the projection {x 1 1,..., x k k, k+1,..., n } { k+1,..., n }. Since p x op X,D = op S : g T x S, the composition p x op X,D factors through the injection ι x : g/g x T x S.

33 PART II: THE WONDERFUL COMPACTIFICATION 33 We obtain a commutative diagram in which the rows are short exact sequences: 0 g x g g/g x 0 dρ x op X,D ι x 0 x i i T x X( log D) T x S 0. Because ι x is injective, it follows by the Snake Lemma that (1) op X,D is surjective if and only if (2) both dρ x and ι x are surjective. The latter is equivalent to the condition that S is a single G-orbit. Moreover, we have ker(dρ x ) = ker(op X,D ). If op X,D is surjective, this gives the short exact sequence p x 0 ker(dρ x ) g op X,D T x X( log D) The logarithmic cotangent bundle of G Now let G be a semisimple connected algebraic group with trivial center, and let X once again be the wonderful compactification of G. Write D X for the boundary divisor, which is a normal crossing divisor by Theorem 5.3. Proposition The pair (X, D) is log-homogeneous. Proof. The stratification of the divisor D given above is exactly the stratification of the boundary of X into G G-orbits from Section 5. It is enough to check the criterion in Proposition at the orbits basepoints z I, I {1,..., l}. Recall from Theorem 4.4 that X is covered by G G-translates of the big cell X 0 = U U Z, where U and U are the unipotent radicals of a fixed pair of opposite Borels, and Z is the closure of the resulting maximal torus in X 0, isomorphic to C l by Proposition 3.4. Moreover, each orbit basepoint z I is contained in Z. Keeping the notation of Section 5, assume without loss of generality that I = {1,..., k}. Then the basepoints z I is of the form z I = (0,..., 0, 1,..., 1) and we have the following tangent spaces: T zi X = T zi X 0 = u u C l T zi (G G)z I = u u C l k. By Proposition 3.4, the torus T acts on Z = C l on the left via ( α 1,..., α l ),

34 PART II: THE WONDERFUL COMPACTIFICATION 34 and so it acts on the normal space T zi X 0 /T zi (G G)z I = C k by ( α 1,..., α k ). Recall from Proposition 6.3 that the stabilizer of z I in G G is Stab G G (z I ) = {(ux, vy) U I L I U I L I xy 1 Z(L I )}. It acts on the normal space by fixing each line in the decomposition (10.1), and it acts on the line L i by the central character α i. It follows that the map dρ zi : Lie(Stab G G (z I )) C k (u + x, v + y) (α 1 (y x),..., α k (y x)) is surjective, and so it follows by Proposition the wonderful compactification X is loghomogeneous. Corollary The isotropy Lie algebra of the orbit basepoint z I is ker(dρ zi ) = p I li p I. Now consider the vector bundle R X on X, with fiber at x X given by R X,x = ker(dρ zi ). It is called the bundle of isotropy subalgebras. By Proposition 9.6, this vector bundle is isomorphic to the restriction to X of the tautological bundle on the Grassmannian Gr(n, g g). Moreover, by Proposition 10.10, there is a short exact sequence of vector bundles on X: (11.1) 0 R X X g g T X( log D) 0. Proposition There is an isomorphism of vector bundles on X between the bundle of isotropy subalgebras and the logarithmic cotangent bundle of X: R X = T X( log D). Proof. (See [Bri2], Example 2.5.) Let β be a nondegenerate G-invariant symmetric bilinear form on g. The form (β, β) is a nondegenerate G-invariant symmetric bilinear form on g g, and the fiber R X,e = g is Lagrangian. Then R X is a Lagrangian subbundle of X g g, and from the short exact sequence (11.1) we get an isomorphism R X = R X = (X g g/rx ) = T X( log D).

35 PART II: THE WONDERFUL COMPACTIFICATION Cohomology of the wonderful compactification We compute the cohomology of X by decomposing it into a union of affine cells using the Bialynicki-Birula decomposition (see Part 1 of these notes, Theorem 2.2). This section follows [EJ], Sections 4.1 and 4.2. See also [DS]. As before, let T be a maximal torus of G and let W be the associated Weyl group. For every element w W, choose a coset representative ẇ N G (T ). Let z 0 = z {1,...,l} X be the basepoint of the unique closed G G-orbit in X. Proposition The T T -fixed points in X are exactly the points {z y,w = (ẏ, ẇ) z 0 y, w W }. Proof. Decompose and suppose X = (G G) z I I {1,...,l} x (G G) z I is fixed by T T. Then the stabilizer of x in G G contains a torus of dimension 2l, and so does the stabilizer of z I. But the maximal torus of Stab G G (z I ) = {(ux, vy) U I L I U I L I xy 1 Z(L I )} is the subgroup {(x, y) T T xy 1 Z(L I )}, which has dimension l + I. It follows that I = l and that I = {1,..., l}, so x is contained in the G G-orbit of minimal dimension. By Remark 6.4, this orbit if G G-isomorphic to the product of two copies of the flag variety. By Theorem 2.1 in Part 1, the T T -fixed points in G/B G/B are exactly the point (ẏb, ẇb ). The Proposition follows. Proposition The T T -weights on T z0 X are (1) ( α, 0), α Φ +. (2) (0, α), α Φ +. (3) ( α i, α i ), α i. Proof. Recall once again that the point z 0 is contained in the big cell X 0 = U U Z, and that this isomorphism is U T U-equivariant. Then T z0 X 0 = u u C l