Borel Weil Bott Theorem Research Articles

Cohomology of line bundles on the incidence correspondence

For a finite dimensional vector space V V of dimension n n , we consider the incidence correspondence (or partial flag variety) X ⊂ P V × P V ∨ X\subset \mathbb {P}V \times \mathbb {P}V^{\vee } , parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on X X in characteristic p > 0 p>0 . If n = 3 n=3 then X X is the full flag variety of V V , and the characterization is contained in the thesis of Griffith from the 70s. In characteristic 0 0 , the cohomology groups are described for all V V by the Borel–Weil–Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo–Mumford regularity. When n = 3 n=3 , we recover the recursive description of characters from recent work of Linyuan Liu, while for general n n we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters.

Invariants of algebraic group actions from differential point of view

In this paper we suggest an approach to study actions of semisimple (or reductive) algebraic groups in their algebraic complex representations. We use differential-geometric methods instead of classical algebraic constructions. Namely, according to Borel–Weil–Bott theorem, every algebraic representation of semisimple algebraic group is isomorphic to the action of this group on the module of holomorphic sections of some reductive bundle over homogeneous space. Using this, we give a complete description of the field of differential invariants for this action and obtain a criterion, which separates regular orbits.

Orthosymplectic Lie superalgebras, Koszul duality, and a complete intersection analogue of the Eagon–Northcott complex

We study the ideal of maximal minors in Littlewood varieties, a class of quadratic complete intersections in spaces of matrices. We give a geometric construction for a large class of modules, including all powers of this ideal, and show that they have a linear free resolution over the complete intersection and that their Koszul dual is an infinite-dimensional irreducible representation of the orthosymplectic Lie superalgebra. We calculate the algebra of cohomology operators acting on this free resolution. We prove analogous results for powers of the ideals of maximal minors in the variety of length 2 complexes when it is a complete intersection, and show that their Koszul dual is an infinite-dimensional irreducible representation of the general linear Lie superalgebra. This generalizes work of Akin, Józefiak, Pragacz, Weyman, and the author on resolutions of determinantal ideals in polynomial rings to the setting of complete intersections and provides a new connection between representations of classical Lie superalgebras and commutative algebra. As a curious application, we prove that the cohomology of a class of reducible homogeneous bundles on symplectic and orthogonal Grassmannians and 2-step flag varieties can be calculated by an analogue of the Borel–Weil–Bott theorem.

Differential contra algebraic invariants: Applications to classical algebraic problems

In this paper we discuss an approach to the study of orbits of actions of semisimple Lie groups in their irreducible complex representations,which is based on differential invariants on the one hand, and on geometry of reductive homogeneous spaces on the other hand. According to the Borel–Weil–Bott theorem, every irreducible representation of semisimple Lie group is isomorphic to the action of this group on the module of holomorphic sections of some one–dimensional bundle over homogeneous space. Using this, we give a complete description of the structure of the field of differential invariants for this action and obtain a criterion which separates regular orbits.

The representation theory of the exceptional Lie superalgebras [formula omitted] and [formula omitted

This paper is a resolution of three related problems proposed by Yu.I. Manin and V. Kac for the so-called exceptional Lie superalgebras F(4) and G(3). The first problem posed by Kac (1978) is the problem of finding character and superdimension formulae for the simple modules. The second problem posed by Kac (1978) is the problem of classifying all indecomposable representations. The third problem posed by Manin (1981) is the problem of constructing the superanalogue of Borel–Weil–Bott theorem.

On differential invariants of actions of semisimple Lie groups

In this paper we suggest an approach to the study of orbits of actions of semisimple Lie groups in their irreducible complex representations. This approach is based on differential invariants on the one hand, and on geometry of reductive homogeneous spaces on the other hand. According to Borel–Weil–Bott theorem, every irreducible representation of semisimple Lie group is isomorphic to the action of this group on the module of holomorphic sections of some one-dimensional bundle over homogeneous space. Using this, we give a complete description of the structure of the field of differential invariants for this action and obtain a criterion, which separates regular orbits.

Harder–Narasimhan Filtrations which are not split by the Frobenius maps

We will produce a smooth projective scheme X over ℤ, a rank 2 vector bundle V on X with a line subbundle L having the following property. For a prime p, let Fp be the absolute Fobenius of Xp, and let Lp ⊂ Vp be the restriction of L ⊂ V. Then for almost all primes p, and for all t ≥ 0, \((F_p^*)^t L_P \subset (F_p^*)^t V_p\) is a non-split Harder-Narasimhan filtration. In particular, \((F_p^*)^t V_p\) is not a direct sum of strongly semistable bundles for any t. This construction works for any full flag veriety G/B, with semisimple rank of G ≥ 2. For the construction, we will use Borel–Weil–Bott theorem in characteristic 0, and Frobenius splitting in characteristic p.

Embeddings of semisimple complex Lie groups and cohomological components of modules

Let G↪G˜ be an embedding of semisimple complex Lie groups, B⊂B˜ a pair of nested Borel subgroups and G/B↪G˜/B˜ the associated embedding of flag manifolds. Let O˜(λ˜) be an equivariant invertible sheaf on G˜/B˜ and O(λ) be its restriction to G/B. Consider the G-equivariant pullbackπλ˜:H(G˜/B˜,O˜(λ˜))→H(G/B,O(λ)). The Borel–Weil–Bott theorem and Schurʼs lemma imply that πλ˜ is either surjective or zero. If πλ˜ is nonzero, the image of the dual map (πλ˜)⁎ is a G-irreducible component in a G˜-irreducible module, called a cohomological component.We establish a necessary and sufficient condition for nonvanishing of πλ˜. Also, we prove a theorem on the structure of the set of pairs of dominant weights (μ,μ˜) with V(μ)⊂V˜(μ˜) cohomological. Here V(μ) and V˜(μ˜) denote the respective highest weight modules. Simplified specializations are formulated for regular and diagonal embeddings. In particular, we give an alternative proof of a recent theorem of Dimitrov and Roth. Beyond the regular and diagonal cases, we study equivariantly embedded rational curves and we also show that the generators of the algebra of ad-invariant polynomials on a semisimple Lie algebra can be obtained as cohomological components. Our methods rely on Kostantʼs theory of Lie algebra cohomology.

A chiral Borel–Weil–Bott theorem

We compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold. This is applied to (1) finding the character of G-integrable irreducible highest weight modules over the affine Lie algebra at the critical level, and (2) computing a certain elliptic genus of the flag manifold. The main tool is a result that interprets the Drinfeld–Sokolov reduction as a derived functor.

On the cohomology of certain homogeneous vector bundles of [formula omitted] in characteristic zero

In his famous paper Demazure (1976) [2], Demazure gave a very short proof of the Borel–Weil–Bott theorem, giving the cohomology of line bundles on a generalized flag variety over a field of characteristic zero. To do this Demazure worked with certain natural modules denoted by V λ , α and their associated vector bundles. He then used dimension shifting and the modules V λ , α to explicitly derive the cohomology of line bundles on a generalized flag variety. He did this without describing the cohomology of the vector bundles corresponding to these auxiliary modules. However we believe the homological properties of V λ , α to be of independent interest. Here we calculate the cohomology of vector bundles corresponding to these modules and some generalizations.

Radon transforms for quasi-equivariant [formula omitted]-modules on generalized flag manifolds

In this paper we deal with Radon transforms for generalized flag manifolds in the framework of quasi-equivariant D -modules. We shall follow the method employed by Baston–Eastwood and analyze the Radon transform using the Bernstein–Gelfand–Gelfand resolution and the Borel–Weil–Bott theorem. We shall determine the transform completely on the level of the Grothendieck groups. Moreover, we point out a vanishing criterion and give a sufficient condition in order that a D -module associated to an equivariant locally free O -module is transformed into an object of the same type. The case of maximal parabolic subgroups is studied in detail.

A Holomorphic Version of Borel–Weil–Bott Theorem

A new geometric realization is obtained for finite dimensional representations of a complex reductive group. The representations are realized as holomorphic objects on a homogeneous Stein manifold. Fundamental to the construction is a holomorphic version of a Hodge decomposition. In certain important cases, the key underlying geometric structure in this decomposition is a certain kind of generalized conformal structure, which generalizes the twistor structures on Gr(2, 4).

Geometric phase, bundle classification, and group representation

The line bundles that arise in the holonomy interpretations of the geometric phase display curious similarities to those encountered in the statement of the Borel–Weil–Bott theorem of the representation theory. The remarkable relationship between the mathematical structure of the geometric phase and the classification theorem for complex line bundles provides the necessary tools for establishing the relevance of the Borel–Weil–Bott theorem to Berry’s adiabatic phase. This enables one to define a set of topological charges for arbitrary compact connected semisimple dynamical Lie groups. These charges signify the topological content of the phase. They can be explicitly computed. In this paper, the problem of the determination of the parameter space of the Hamiltonian is also addressed. It is shown that, in general, the parameter space is either a flag manifold or one of its submanifolds. A simple topological argument is presented to indicate the relation between the Riemannian structure on the parameter space and Berry’s connection. The results about the fiber bundles and group theory are used to introduce a procedure to reduce the problem of the nonadiabatic (geometric) phase to Berry’s adiabatic phase for cranked Hamiltonians. Finally, the possible relevance of the topological charges of the geometric phase to those of the non-Abelian monopoles is pointed out.

Multiplicity formulae for discrete series

In a series of papers [20, 211 and [22], Schmid obtained several important results on the discrete series for semisimple Lie groups. The purpose of this paper is to prove Schmid's results by somewhat different methods and to relax as much as possible the restriction imposed on the regularity of the parameters of discrete classes. Though the basic line is similar to Schmid's, the main difference is that our methods do not rely upon complex analysis on the non-compact flag manifold G/T as developed in [20] and [21]. Rather, we just rely on some elementary differential calculus on the symmetric space G/K. This difference gives rise to less restrictive assumptions since the results on the vanishing of L2-cohomologies in the symmetric space situation seem to be sharper, so far. Our work also leads to some interesting results concerning the multiplicity formula of discrete classes in L 2 (/' \ G). In our development, we shall give an alternative proof of the key fact (Theorem 1, w 4) which is obtained in [20] using the complex analysis on G/T. Our proof, given in w 5, will be carried out directly in the symmetric space situation, and some standard theory of sheaf cohomology centering the Borel-WeilBott theorem on the compact flag manifold KIT will be used as in [20]. In this proof, our methods seem to be quite elementary. We now give a more precise description of the contents of the paper. Let G be a non-compact real semisimple Lie group with discrete series g2 4= qS. Assume, for simplicity throughout the paper, that G is a connected real form of a simply connected complex semisimple Lie group G e. Harish-Chandra 1-7] showed that there exists a compact Cartan subgroup T of G and that, if one denotes by T' the set of regular characters of T, there exists a distinguished surjection co: T'---, g2. We fix T and a maximal compact subgroup K containing T once and for all. Let ge, t e denote the complexifications of the Lie algebras g, t of G, T respectively. In considering the discrete class co (A) for a given regular character A e T', we always choose a positive root system P for the pair (go, t~) such that A is regular dominant with respect to 19, i.e., P = {~; (A, c0>0 }. Here as