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Cohomology of Flag Varieties and the Brylinski-Kostant Filtration

Let G be a semisimple complex algebraic group with Borel subgroup B and let P be a parabolic subgroup of G. Let T*(G/P) denote the cotangent bundle of G/P. Ranee Brylinski discovered a connection between cohomology of G-equivariant line bundles on T*(G/B) and the so-called Brylinski-Kostant filtration, which describes the action of principal sl_2 triples on G-representations. In this paper we generalize these results to a larger class of sl_2 triples. Along the way we also obtain generalizations of results due to Broer on cohomology of G-equivariant bundles on T*(G/P) for various parabolics P.


Introduction
Let G be a semisimple algebraic group over C. Given an irreducible representation V of G, Brylinski [5] constructed a filtration on weight spaces of V . This filtration, often referred to as the Brylinski-Kostant filtration, studies the action on V of certain special nilpotent elements of the Lie algebra of G (the principal nilpotents). This filtration was motivated by fundamental work of Kostant [15], [16] on adjoint orbits and actions of sl 2 -triples. Using higher cohomology vanishing of pullback bundles on the cotangent bundle of the flag variety G/B, Brylinski showed that certain polynomials -Lusztig's q-analogs of weight multiplicity -compute the weight spaces occuring in the various degrees of this filtration. These polynomials, as proven in [11], are Kazhdan-Lusztig polynomials, which are deep objects in combinatorial representation theory (see [12], [13], and [18] for a starting point in the large amount of literature on this subject).
We now consider a larger class of nilpotent elements, the even nilpotents. Given an even nilpotent X we define a generalized Brylinski-Kostant filtration corresponding to X. If X is principal we obtain the original Brylinski-Kostant filtration. As in the original filtration, certain polynomials -in this case, parabolic versions of Lusztig's q-analogs -arise in computing dimensions of the generalized filtration. In particular, these polynomials arise in computing multiplicities of irreducible G-modules in the global sections of certain equivariant bundles on cotangent bundles of partial flag varieties; we then relate these multiplicities to the generalized Brylinski-Kostant filtration.
Our results rely on cohomology vanishing results for equivariant bundles on cotangent bundles of flag varieties. In section 4 we obtain extensions of some results of Broer [3]. These vanishing results provide a basic technical tool in analyzing the generalized Brylinski-Kostant filtration but are interesting in their own right; another application of these cohomology vanishing results is in the geometry of G-orbits of nilpotent elements in Lie(G). These orbits and their closures are subvarieties of Lie(G) with rich geometric structure, and these cohomology vanishing results are a vital tool in studying their structures, cf [3], [9], [20], and [21].
In section 5 we conclude by explicitly giving examples of generalized Brylinski-Kostant filtrations on various irreducible representations.

Notation
Let G denote a complex semisimple simply-connected algebraic group over C. Fix a Borel B and maximal torus T ⊆ B of G (we assume G simplyconnected only for notational convenience; all results will generalize easily to G of arbitrary isogeny type). Let U denote the unipotent radical of B. Set g := Lie G, h := Lie T , and n := Lie U . Let W be the Weyl group of G. Let Λ ⊆ h * denote the weight lattice of G. Let Λ R ⊆ Λ denote the root lattice. Let ∆ (resp. ∆ − , ∆ + ), denote the roots (resp. positive and negative roots) with respect to T and B and let π ⊆ ∆ + denote the simple roots. Let ρ be the half sum of all elements of ∆ + . There is a shifted action of W on h * defined by w * λ := w(λ + ρ) − ρ. This action keeps Λ and Λ R stable.
For any T -module M and any weight µ of M , let M µ denote the µ-weight subspace of M . Set V λ (µ) := V (µ) λ . Let P be a standard parabolic in G and let V be a variety with a P -action. Let G × P V denote the G-equivariant fiber bundle on G/P with fiber V . Let f : G × V ։ G × P V be the quotient map. For g ∈ G and x ∈ V we set g * x := f (g, x) .
Then gp * x = g * px, for all g ∈ G, p ∈ P , and x ∈ V . For a P -module M , G × P M is a G-equivariant vector bundle on G/P . Let L P (M ) denote the sheaf of sections of this bundle.
Let L denote the Levi factor of P containing T . Given λ ∈ Λ, we say that λ is P -dominant (resp. P -regular dominant) if λ(β ∨ ) ≥ 0 (resp. λ(β ∨ ) > 0) for all roots β of L. Note that for P = B these conditions are vacuous. For P -dominant λ let V P (λ) denote the irreducible P -module of highest weight λ, and set Note that L B (λ) is a line bundle on G/B for all λ ∈ Λ. Let N denote the nullcone (the variety of all nilpotent elements) of g. We say that a nilpotent element X ∈ N is in good position if (1) X ∈ n and (2) there is an sl 2 -triple {X, Y, H} ⊆ g with H ∈ D ′ . There is a unique dense open G-orbit in N ; elements of this orbit are called principal nilpotents.
(2) The B-orbit of X is dense in n.
(3) We have where the c β are constants such that c α = 0 for all α ∈ π.
Definition 3.3. The BGG category O consists of g representations V that have finite-dimensional weight spaces and are n-locally finite; i.e., for any v ∈ V , v lies in a finite-dimensional n-submodule of V .
Let V be an object in the BGG category O and let U ⊆ V be any vector subspace. Let e be a principal nilpotent in good position. In [5], Ranee Brylinski defined a filtration on U inside of V , called the Brylinski-Kostant filtration (or BK-filtration), as follows: set F n V (U ) := {v ∈ U : e n+1 v = 0} . In particular, Brylinski considered this filtration in the case where V is an irreducible representation V (µ) of G and U is a weight subspace V λ (µ) of V , for µ ∈ D and λ ∈ Λ. In this case we shall suppress the subscript and write . This jump polynomial counts the dimensions of the degrees of the filtration, and clearly dim (V λ (µ)) = r λ µ (1).

Kazhdan-Lusztig polynomials and Brylinski
, which counts the number of ways of writing γ as a sum of positive roots. The degree-n coefficient of p q (γ) counts the number of ways of writing γ as a sum of precisely n (not necessarily distinct) positive roots.
Using this setup, we have the following theorem of Brylinski which relies on a cohomology vanishing condition. Let p B : T * (G/B) → G/B be the cotangent bundle of G/B. Theorem 3.4. (Brylinski,[5]) Let e be a principal nilpotent in good position. Let µ, λ ∈ D. If H i T * (G/B), p * B L B (λ) = 0 for all i > 0 then m λ µ (q) = r λ µ (q). We also have the following result of Broer which is a special case of Theorem 4.3 below.
3.2.1. Nilpotent Orbits. In this section we collect some basic facts about sl 2triples and nilpotent orbits in characteristic 0. Our main reference for this section is [7]. It should be noted that many aspects of the theory change in positive characteristic; for a good overview of the theory in arbitrary characteristic, see [10].
Since an sl 2 -triple gives g the structure of an sl 2 -representation, we see that if {X, Y, H} is an sl 2 -triple then X and Y are nilpotent and H is semisimple. X is called the nilpositive element of the triple and H is the semisimple element of the triple. We have the following important theorem.
Let {X, Y, H} be an sl 2 -triple. Then, viewing g as an sl 2 -representation, we can write the eigenspace decomposition of g under the action of H as This decomposition depends only on the nilpositive element X and not on the rest of the elements of the triple. Thus, given any nilpotent element X, we obtain such a decomposition of g into eigenspaces. We say that X is even if g n = 0 for all odd n.
One may check that n≥0 g n is a parabolic subalgebra of g; we call this the parabolic subalgebra associated to X (or H). It has Levi factor g 0 = g H and nilradical n>0 g n (where g H denotes the centralizer of H in g). This associated parabolic is a standard parabolic iff X is in good position, and X is regular iff the associated parabolic is a Borel subalgebra of g. Let P ⊆ G be the parabolic subgroup of G such that Lie(P ) = p; we will call P the parabolic subgroup associated to X. Let p be a parabolic subalgebra of g. For X ∈ N let O X denote the G-orbit of X. We say that X is a Richardson element for p if O X ∩ n p is dense in n p . The Richardson elements for p form a unique orbit in N . Furthermore, every parabolic p has a Richardson element.
Each semisimple orbit in g has a unique element in D ′ . If H is a semisimple element in D ′ such that H is in some sl 2 -triple {X, Y, H} then we say that H is distinguished semisimple. Further, if H ∈ D ′ is distinguished semisimple then α(H) ∈ {0, 1, 2} for all α ∈ π. If X ∈ N is in good position then there is an sl 2 -triple {X, Y, H} such that H is distinguished.
Lemma 3.8. (Kostant) Let X ∈ N and let p be the associated parabolic. Let P ⊆ G be the parabolic subgroup with Lie algebra p. Set Then Remark 3.9. Lemma 3.8 implies that X is even nilpotent iff X is a Richardson element for its associated parabolic, since X is even iff p 2 = n p .

Definitions and main result.
We now generalize the BK-filtration to the case of an arbitrary nilpotent element. Choose X ∈ N and V in the category O. Let U ⊆ V be any vector subspace. Define a filtration F X,V on U by F n X,V (U ) := {v ∈ U : X n+1 v = 0} . If e is a principal nilpotent in good position then clearly F n e,V (U ) = F n V (U ), the original BK-filtration.
Let X ∈ n be an even nilpotent element in good position and let P be its associated standard parabolic. Let L be the Levi factor of P containing T . Let ∆ P , ∆ + P , and π P denote the roots, the positive roots, and the simple roots, respectively, of L. For any µ ∈ D and weight λ of V (µ) let . We obtain a jump polynomial as before: set Remark 3.10. Note that p B q (γ) = p q (γ) and m B, λ µ (q) = m λ µ (q). Thus the polynomials m P,λ µ (q) are P -generalized versions of Kazhdan-Lusztig polynomials.
Let p P : T * (G/P ) → G/P be the cotangent bundle of G/P . The following theorem is the main theorem in this section; it generalizes Theorem 3.4 above.
Theorem 3.11. Let X ∈ n be a standard even nilpotent. Let µ, λ ∈ D. If for all i > 0, then r X,λ µ (q) = m P,λ µ (q), where P is the parabolic associated to X.
In section 4 below, we will prove this cohomology vanishing for various λ.
Remark 3.12. Note that this implies, in the presence of the cohomology vanishing condition, that the coefficients of m P,λ µ (q) are positive when µ, λ ∈ D. This is by no means clear from the definition and, in fact, can fail if λ / ∈ D.
Remark 3.13. One can note that in the proof of Theorem 3.11 below, we do not explicitly use the fact that X is a standard even nilpotent; we only use the following a priori weaker conditions: (i) There is H ∈ D ′ with [H, X] = X; and (ii) X is Richardson for the parabolic associated to H. However, these conditions are equivalent to X being even nilpotent. Indeed, by the proof of the Jacobson-Morozov theorem, if H ′ is a semisimple element of g such that [H ′ , X] = 2X, then H ′ is the semisimple element in some sl 2triple containing X. Furthermore, if X is Richardson for the corresponding associated parabolic, then X is even by Remark 3.9.

Outline of proof.
Remark 3.14. Our proof is an adaptation of Brylinski's proof of her Theorem 3.4 in [5] (which is stated as Theorem 3.4 above).
Let us first give a sketch of the proof. Choose an even nilpotent element X in good position and let P be the parabolic subgroup corresponding to X. Let λ be a P -dominant weight and let p : T * (G/P ) → G/P be the cotangent bundle of G/P . We first construct two G-equivariant locally-free sheaves q * A P (λ) and p * p * L P (λ) (defined below) of infinite rank on G/P such that (a) q * A P (λ) is filtered by G-equivariant locally-free sheaves q * A P (λ) ≤n of finite rank and (b) p * p * L P (λ) is graded by Gequivariant locally-free sheaves p * p * L P (λ) n of finite rank.
We then show in Theorem 3.23 below that for any µ ∈ D the multiplicity is precisely the degree-n coefficient of the jump polynomial r X,λ µ (q). In Proposition 3.26 we show that the cohomology vanishing condition implies that for all n ≥ 0 the multiplicity of V (µ) * in H 0 G/P, p * p * L P (λ) n is the degree-n coefficient of m P,λ µ (q). To complete the proof of Theorem 3.11, we show that the cohomology vanishing condition gives an isomorphism 3.3.2. The proof. For the remainder of this section fix X, P and λ as above. Let L be the Levi factor of P containing T . Recall that D ′ ⊆ h is the dominant chamber. Let H ′ ∈ D ′ be the unique distinguished semisimple element in D ′ occuring in an sl 2 -triple with nilpositive element X and set Let G/L ։ G/P be the projection map. Via this map we have a Gequivariant isomorphism G/L ∼ = G × P (P/L) and hence we will consider G/L as a G-equivariant fiber bundle on G/P . Lemma 3.15. Let P act on H + n p through the adjoint action. Then there is an isomorphism Proof. It suffices to show that P/L ∼ = H + n p as P -varieties. As a variety, Note that n p .H ⊆ n p , so that U P .H ⊆ H + n p (cf [6], Lemma 1.4.12(i)). As U P and H + n p are both isomorphic to A n for n = dim(n p ), the variety injection U P .H ֒→ H + n p must be an isomorphism. Thus we have Remark 3.16. From here on, we will use the G-equivariant isomorphism G/L ∼ = G × P (H + n p ) to write elements of G/L in the form g * (H + Z), for g ∈ G and Z ∈ n p . Notation 3.17. Let p : T * (G/P ) → G/P and q : G/L → G/P be the bundle maps and set A P (λ) := q * L P (λ) .
Remark 3.18. By the projection formula, since Definition 3. 19. Define a gradation on p * p * L P (λ) by there is a well-defined notion of the top degree of f , which is a homogeneous polynomial on n p . Define the degree of f to be the degree of this polynomial. This gives rise to a natural P -invariant of graded sheaves on G/P .
Proof. This follows immediately from an argument similar to that of Theorem 5.5 of [5].
Recall that if λ is a weight of V (µ) then W P λ (µ) denotes the subspace of Denote by ϕ the isomorphism Definition 3. 22. For any µ ∈ D we define a filtration H on W P λ (µ) as follows. Set Theorem 3.23. Choose µ ∈ D. We have Proof. We give a sketch of the proof; the details follow from arguments similar to those in the proofs of Lemmas 5.6 and 5.7 and Proposition 5.9 of [5]. Choosef this is a line in the fiber of G/L over eP . For g ∈ G we may interpret f v (gP ) as a regular function on the fiber of G/L over gP . By an argument similar to that of Lemma 5.7 in [5], one checks that there exists g ∈ G such that the degree of the polynomial t → f v (gP ) g. N (t) is n. By G-equivariance this implies that the degree of the polynomial t → f gv (eP )(N (t)) is n.
By an analog of Lemma 5.6 in [5], we have that the degree of the polynomial t → f gv (N (t)) is the same as the degree of the polynomial In particular, the degree of this polynomial is n. As the fiber of G/L over eP is isomorphic to V P (λ) * we obtain that n is the maximum of the degrees of the V P (λ) * -valued polynomials exp(−tX).f v (eP )(H) as v runs over the elements of V (µ) * . By an argument similar to that of Proposition 5.9 of [5], this implies that the degree of the polynomial exp(tX). ϕ(f ) is also n. Since we see that X n .f = 0 and X m .f = 0 for all m > n, as desired.
of G-modules. Furthermore, the first isomorphism is an isomorphism of graded G-modules, where the grading on the first module is obtained from the grading on S(n * p ) and the grading on the second is obtained from the grading on p * p * L P (λ).
Proof. By Remark 3.18 we have that Let pr : G/B → G/P be the projection map. Then , since both are G-equivariant bundles on G/B with fiber S(n * p ). By Remark 3.24, we have that Since P/B ∼ = L/B L , we have and The Leray spectral sequence and the projection formula now imply the first isomorphism (note that all isomorphisms here preserve the grading). The second isomorphism follows from the Leray spectral sequence again, as p is an affine morphism.
Let X denote the group ring of Λ over Z written multiplicatively; ie, X is the Z-algebra with generators {e λ : λ ∈ Λ} and relations e λ 1 e λ 2 = e λ 1 +λ 2 . for all µ ∈ D.
Proof. By Lemma 3.25 we have isomorphisms By the vanishing assumption, Hence For any n ≥ 0 let p n,P q be the degree-n coefficient of the polynomial p P q and let m n,P,λ µ be the degree-n coefficient of m P,λ µ (q) (recall the definition of p P q from section 3.2.2). For all n ≥ 0 a standard computation now gives That is, for all γ ∈ D the multiplicity of V (γ) * in is m n,P,λ γ , as desired.
We can now prove Theorem 3.11.
Proof. (of Theorem 3.11) Recall that we want to show the following: If for all i > 0, then r X,λ µ (q) = m P,λ µ (q) . First note that, by Theorem 3.23, we have By Proposition 3.26 it now suffices to show that n for all n ≥ 0 and µ ∈ D.
Fix µ ∈ D. Recall that Using that ϕ is an isomorphism and that the functor Finally, note that by Proposition 3.20 and the cohomology vanishing assumption, an easy induction shows that for all n ≥ 0 there is an isomorphism 4.1.1. Definitions. Let P be a standard parabolic subgroup of G and let L be the Levi subgroup of P containing T . Recall that ∆ P , ∆ + P , and π P denote the roots, the positive roots, and the simple roots, respectively, of L. Let ρ P denote the half-sum of all positive roots of L. Let W P ⊆ W denote the Weyl group of L.
In this section we assume without loss of generality that all parabolics are standard. 4.1.2. We begin by providing an outline of the results in this section. For any parabolic P ⊆ G and λ ∈ Λ let r P : G × B n p → G/B be the bundle map and consider the cohomology groups

By Remark 3.18 and Lemma 3.25, when λ is P -dominant we have
, where p P : T * (G/P ) → G/P is the bundle map.
Broer [3] showed that these cohomology groups vanish for i > 0 in the following cases (see Theorem 4.3 and Propositions 4.10 and 4.11 below): • P = B and cht (λ) = 0 (see Lemma 4.2(i) below for a characterization of the weights for which cht = 0). In particular, we have higher cohomology vanishing when P = B and λ ∈ D. • P is a minimal parabolic corresponding to a short simple root and λ ∈ D. • P is any parabolic and λ is any weight such that λ(α ∨ ) = −1 for some α ∈ π P (in fact, we have cohomology vanishing for all i in this case). • P is any parabolic and C λ is a 1-dimensional P -module, i.e. λ(α ∨ ) = 0 for all α ∈ π P .
In this chapter we show that these cohomology groups vanish for i > 0 in the following additional cases: • P is any minimal parabolic and λ ∈ D (section 4.2.2).
• P is any parabolic and λ = µ − 2ρ P for some P -regular dominant µ ∈ D (section 4.3). • P is any parabolic in type A and λ ∈ D is regular (section 4.4).
In section 4.3 we also collect some corollaries that give vanishing for some higher cohomologies in certain cases.
The following is a more natural way of classifying the weights of combinatorial height 0. Part (i) is due to Broer ([4], Proposition 2), and (ii) follows readily from (i).

Vanishing Theorem.
Remark 4.9. Let P be a minimal parabolic corresponding to a simple root α. We have a short exact sequence of B-modules 0 → n p → n → C α → 0 which gives rise (upon taking the dual) to a short exact Koszul sequence 0 → S(n * ) ⊗ C −α → S(n * ) → S(n * p ) → 0 . Let µ be any weight. Tensoring with C −µ we obtain the short exact sequence The following proposition is implicit in [3]. We provide a proof here for completeness.
Proposition 4.10. Let G be simple. Let P be a minimal parabolic corresponding to a short simple root α. Let µ ∈ D. Then H i,np (µ) = 0 for all i > 0.
Proof. Using the long exact sequence of cohomology associated to the short exact sequence of sheaves in Remark 4.9 above, the result now follows from Lemma 4.2 (ii), Theorem 4.3 (ii), and Lemma 4.5.
The following result is a reformulation of Broer's Lemma 3.1 in [3].
Proposition 4.11. Let P be a standard parabolic and choose λ ∈ Λ. If there is w ∈ W P such that w * λ is P -dominant then Further, if there does not exist w ∈ W P such that w * λ is P -dominant, then Remark 4.12. Choose λ ∈ Λ. Then there does not exist w ∈ W P such that w * λ is P -dominant if and only if λ(β ∨ ) = −ρ P (β ∨ ) for some β ∈ ∆ + P . In particular, if λ(α ∨ ) = −1 for some α ∈ π P then by Proposition 4.11, H i,np (λ) = 0 for all i.
We now have the following theorem.
Proof. By Proposition 4.10, we need to check this result for parabolics corresponding to long simple roots. Let α be a long simple root and let P be the standard parabolic defined by α. By Theorem 4.3 and the long exact sequence in cohomology associated to the short exact sequence → 0 of sheaves on G/B (cf Remark 4.9 above), to prove the theorem for p it suffices to show that cht (α + µ) ≤ 1 for all µ ∈ D.
We now check the cht condition case by case. We shall label the simple roots as π = {α 1 , . . . , α n }.
For any k ∈ N with m ≤ k < n, set An easy induction on n utilizing Proposition 4.4 (iv) shows that Further, β k is a short root, so cht (β k + µ) = 0 by Lemma 4.5. So, by Lemma 4.8 and the fact that (µ , β k − α k ) = 0, we have (2) Type C n .
By Propositions 4.11 and 4.13 we now have for all i > 0. Thus, by the short exact sequence above, for all i > 1. Now, from the short exact Koszul complex → L B S(n * p ) ⊗ L B (µ) → 0 and the fact that µ ∈ D we obtain that H i,np (µ) = 0 for all i > 0, as desired.
Theorem 4.14. (Grauert-Riemenschneider) Let X be a smooth complex variety and let f : X → Y be a proper morphism of complex varieties. Let ω X denote the canonical bundle of X. If n is the dimension of a generic fiber of f then R i f * (ω X ) = 0 for all i > n.
The proof of the following theorem uses ideas from Broer's proof of his theorem 2.2 in [3] and appears here in a strengthened form due to a new proof from the referee.
Theorem 4.15. Let P be any standard parabolic and let µ ∈ D be P -regular dominant. Then H i,np (µ − 2ρ P ) = 0 for all i > 0.
Proof. Set Y := G × B (n p × C µ ). The natural map is birational, since the stabilizer in P of V µ (µ) is B by the P -regular dominance of µ. From this and the isomorphism we now obtain a birational map Now, the natural map G × P n p → g is generically finite; choose X ∈ g such that the fiber over X is finite. Then the fiber over (X, 0) in the natural map is also finite, sof generically has finite fibers.
By (4.3.1) and (4.3.2) we now obtain a morphism This map is proper, since it factors as followed by the projection onto the last two coordinates; and it is also generically finite, since the maps in (4.3.1) and (4.3.2) generically have finite fibers. We now obtain from the Grauert-Riemenschneider theorem that for all i > 0. By an argument similar to the proof of Theorem 2.2 in [3], this implies that This completes the proof.
We now collect a few corollaries of Theorem 4.15.
Corollary 4.16. Let P be any standard parabolic. Recall that w P 0 is the longest element of the Weyl group W P ⊆ W corresponding to π P . Choose µ ∈ Λ such that w P 0 (µ) ∈ D and w P 0 is of minimal length with this property. Then H i,np (µ) = 0 for all i = l(w P 0 ). Proof. If µ(α ∨ ) = −1 for any α ∈ π P then the result follows immediately from Proposition 4.11. Thus we may assume that µ(α ∨ ) ≤ −2 for all α ∈ π P and hence w P 0 µ (α ∨ ) ≥ 2 for all α ∈ π P . This implies that w P 0 * µ α ∨ = w P 0 µ − 2ρ P α ∨ ≥ 0 for all α ∈ π P . Since w P 0 * µ ∈ D this gives w P 0 * µ + 2ρ P = w P 0 µ ∈ D . Thus H i,np (w P 0 * µ) = 0 for all i > 0 by Theorem 4.15. The result now follows from Proposition 4.11.
Note that Thus, by the induction hypothesis, H i,n (µ + α) = 0 for all i > 1. Furthermore, H i,np (µ) = 0 for all i > 1 by Corollary 4.16. The result now follows easily from the Koszul complex above.

Vanishing in Type A.
Fix a prime p; for the rest of this section we assume that all schemes areF p -schemes unless otherwise specified, where F p is the finite field with p elements andF p is its algebraic closure. The main reference for this section is [1]. 4.4.1. Frobenius splitting. Let X be a scheme overF p . We define a morphism F X of schemes over F p as follows. Set F X (x) = x for all x ∈ X and define F # X : O X → F X * O X to be the p th power map f → f p ; this is clearly an F p -linear map. Note that F X is not a morphism of schemes overF p . This morphism is called the absolute Frobenius morphism. Generally when the context is clear we'll drop the subscript and just write F . Definition 4. 19. We say that X is Frobenius split if there is an O X -linear map ϕ : The following is the essential result we require from the theory of Frobenius splitting.

4.4.2.
Application to cohomology of cotangent bundles of flag varieties. We now apply Frobenius splitting methods to flag varieties in type A.
The following theorem is a result due to van der Kallen and is a strengthening of Theorem 3.8 in [19]. We now come to the main result, which is a modification of Theorem 5.2.11 in [1] (see also [17]).  for all regular λ ∈ D and i > 0. Hence H i T * (G/P ), p * P L P (λ) = 0 for all i > 0 and regular λ ∈ D (recall that p P : T * (G/P ) → G/P is the bundle map).
Proof. For the moment, assume that G is a semisimple algebraic group over F p with all components of type A. Consider the inclusion Let q : G/B × g → G/B be projection onto the first coordinate. Since g is affine, the bundle q * L B (λ) is ample on G/B × g, since L B (λ) is ample on G/B. Note that the restriction of q * L B (λ) to G × B n p is p * P L B (λ); hence p * p L B (λ) is ample on G × B n p . Note that there is a proper morphism G × B n p → g given by By Corollary 4.21 and Theorem 4.22, we have that The result in characteristic 0 now follows from base change, cf [1], section 1.6.

Examples
In this section we explicitly compute some examples to illustrate Theorem 3.11. All three examples will be generalized BK-filtrations on L-highest weight subspaces of V 0 (µ) for some µ ∈ D; we will always consider the 0 weight space since it doesn't make the examples any more interesting to consider other weight spaces.
If the partition is such that p i = 1 for all 1 ≤ i ≤ k then the associated orbit is the 0 orbit, so we assume that p = [1, . . . , 1]. For any β ∈ ∆ + let X β ∈ g β be a Chevalley basis element. Let {α 1 , . . . , α n−1 } denote the simple roots from left to right in the Dynkin diagram of type A n−1 .
Given a partition p as above, set m = max{j : p j > 1}. For each p l−1 (where we set p −1 = 0) and set We now obtain an orbit representative corresponding to p by setting We may explicitly describe the matrix corresponding to X p as follows. For any q > 0 let J(q) be the q × q matrix with 1 ′ s on the superdiagonal and 0 ′ s everywhere else. Then X p is the matrix with Jordan blocks of size p 1 , . . . , p k , i.e. the block diagonal matrix The bijection between partitions and orbits is given by p → O Xp . Even nilpotent orbits correspond to partitions consisting of only even or only odd parts.
Let G be of type A 3 . There are 5 partitions of n = 4 and hence 5 nilpotent orbits in G. All but one are even; the non-even orbit corresponds to the partition [2, 1, 1]. Let π = {α 1 , α 2 , α 3 } be the standard ordering of the simple roots.
Consider the nilpotent orbit corresponding to the partition [3, 1]. By the construction above we see that X := X α 1 + X α 2 is an orbit representative. However, one may check (cf [7]) that a semisimple element H in an sl 2 triple containing X is given by H = 2χ ∨ 1 + 2χ ∨ 2 − 2χ ∨ 3 . As H is not dominant we now conjugate the triple to obtain a representative of our nilpotent orbit in good position.

Example 2.
Let G be of type G 2 . Let α denote the short simple root and β the long simple root.
For any semisimple Lie algebra g there is a unique dense orbit in N \O reg , where O reg is the regular nilpotent orbit (cf [7]). This orbit is called the subregular nilpotent orbit. By Example 8.2.13 in [7], the subregular orbit in G is even and the corresponding weighted Dynkin diagram has 2 on the node corresponding to α and 0 on the node corresponding to β. Set H := 2χ ∨ α ∈ h; then there is an sl 2 -triple {X, Y, H} with X ∈ O subreg ∩ n p , and X is clearly in good position.