Free Resolution Invariants for Finite Groups

Given a finite group G and a G-free resolution F _* of Z, then d _G (Im(F _m+1F _m ))–(–1) ^m–i d _G (F _i ) is almost always an invariant of G.     

The Bott-Borel-Weil Theorem for direct limit groups

We show how highest weight representations of certain infinite dimensional Lie groups can be realized on cohomology spaces of holomorphic vector bundles. This extends the classical Bott-Borel-Weil Theorem for finite-dimensional compact and complex Lie groups. Our approach is geometric in nature, in the spirit of Bott's original generalization of the Borel-Weil Theorem. The groups for which we prove this theorem are strict direct limits of compact Lie groups, or their complexifications. We previously proved that such groups have an analytic structure. Our result applies to most of the familiar examples of direct limits of classical groups. We also introduce new examples involving iterated embeddings of the classical groups and see exactly how our results hold in those cases. One of the technical problems here is that, in general, the limit Lie algebras will have root systems but need not have root spaces, so we need to develop machinery to handle this somewhat delicate situation.

     

Hilbert schemes, polygraphs and the Macdonald positivity conjecture

We study the isospectral Hilbert scheme , defined as the reduced fiber product of with the Hilbert scheme of points in the plane , over the symmetric power . By a theorem of Fogarty, is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over . We derive two important consequences.

(1) We prove the strong form of the conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients . This establishes the Macdonald positivity conjecture, namely that .

(2) We show that the Hilbert scheme is isomorphic to the -Hilbert scheme of Nakamura, in such a way that is identified with the universal family over . From this point of view, describes the fiber of a character sheaf at a torus-fixed point of corresponding to .

The proofs rely on a study of certain subspace arrangements , called polygraphs, whose coordinate rings carry geometric information about . The key result is that is a free module over the polynomial ring in one set of coordinates on . This is proven by an intricate inductive argument based on elementary commutative algebra.

     

Extensions of Modules over Schur Algebras, Symmetric Groups and Hecke Algebras

We study the relation between the cohomology of general linear and symmetric groups and their respective quantizations, using Schur algebras and standard homological techniques to build appropriate spectral sequences. As our methods fit inside a much more general context within the theory of finite-dimensional algebras, we develop our results first in that general setting, and then specialize to the above situations. From this we obtain new proofs of several known results in modular representation theory of symmetric groups. Moreover, we reduce certain questions about computing extensions for symmetric groups and Hecke algebras to questions about extensions for general linear groups and their quantizations.     

Cohomology of line bundles on Schubert varieties: The rank two case

In this paper we describe vanishing and non-vanishing of cohomology of “most” line bundles over Schubert subvarieties of flag varieties for rank 2 semisimple algebraic groups.     

Support Varieties for Selfinjective Algebras

Support varieties for any finite dimensional algebra over a field were introduced in (Proc. London Math. Soc. 88 (3) (2004) 705–732) using graded subalgebras of the Hochschild cohomology ring. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, the variety of periodic modules are lines and for symmetric algebras a generalization of Webbs theorem is true. As a corollary of a more general result we show that Webbs theorem generalizes to finite dimensional cocommutative Hopf algebras.Received November 2003Mathematics Subject Classifications (2000) Primary: 16E40, 16G10, 16P10, 16P20; Secondary: 16G70.     

Exponential sums on , II

We prove a vanishing theorem for the -adic cohomology of exponential sums on . In particular, we obtain new classes of exponential sums on that have a single nonvanishing -adic cohomology group. The dimension of this cohomology group equals a sum of Milnor numbers.

     

Hopf cyclic cohomology and Hodge theory for proper actions on complex manifolds

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.     

On approximations of the de Rham complex and their cohomology

For a commutative algebra R, its de Rham cohomology is an important invariant of R. In the paper, an infinite chain of de Rham-like complexes is introduced where the first member of the chain is the de Rham complex. The complexes are called approximations of the de Rham complex. Their cohomologies are found for polynomial rings and algebras of power series over a field of characteristic zero.