GEOMETRY OF FLAG VARIETIES AND REPRESENTATION THEORY

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

ABSTRACT OF THE NSF PROJECT OF SHRAWAN KUMAR Shrawan Kumar intends to continue work in the general area of 'Lie Theory and Geometry'. He has proposed to work on five projects requiring diverse and extensive mathematical tools from Algebraic Geometry, Classical and Quantum Representation Theory, and Topology. Briefly described, these projects are as follows: In an attempt to understand the Kostka-Macdonald coefficients, Garsia and Haiman proposed a remarkable conjecture, known as the n! conjecture. This asserts that the dimension of a certain space of polynomials in 2n variables is exactly n!. The validity of the conjecture will imply that the coefficients of the Kostka-Macdonald polynomials are non-negative integers which remains an open challenging problem. The first project aims at proving this conjecture (jointly with Prof. J. Thomsen from Aarhus) by using the geometry of Hilbert schemes. The second project involves proving the non-existence of non-constant holomorphic maps between certain homogeneous compact complex manifolds. Kumar already has a precise conjecture in this direction. Lusztig has defined a certain Frobenius morphism for quantized enveloping algebras at roots of unity and also a certain 'splitting' of this morphism defined on the positive parts of the algebras involved. On the other hand, any algebraic variety defined over a field of positive characteristic admits a Frobenius morphism. Mehta-Ramanathan introduced the concept of Frobenius splitting of such a variety and showed that the flag varieties associated to any semisimple algebraic group do admit Frobenius splitting. This leads to some important results on the geometry of flag varieties. Now Kumar-Littelmann have obtained a very precise connection between these two (quantum and geometric) notions of Frobenius morphisms and Frobenius splittings and the third project involves completing this ongoing work. The fourth project involves proving a conjecture given by Kumar himself on the cohomology of 'thick' flag varieties associated to affine Kac-Moody groups with coefficients in homogeneous vector bundles. A 'direct' proof of this conjecture will lead to uniform proofs of several important results related to the Moduli of Vector Bundles and Verlinde Formula. The fifth is a long continuing project which is due for completion by the end of this summer. It involves writing the book "Kac-Moody Groups, their Flag Varieties and Representation Theory," to be submitted to the Graduate Texts in Mathematics series of Springer-Verlag. To explain these to a general scientific community, the main underlying theme behind Kumar's projects is to exploit 'symmetry' or in more technical terms 'invariance' in problems arising in mathematics and mathematical physics. Several natural phenomena exhibit symmetry, from crystals to the surface of the earth we live on (where the latter of course is invariant under any rotation). Now consider a mathematical or a physical problem, say studying the geometric properties of an object (from a cell to the universe), or solving some complex equations arising in diverse situations in mathematics and other sciences. An effective method has been to study the collection of all the symmetries of the problem under consideration (called a Group) and use the properties of the Group to shed light on the original problem. This method has been very successfully employed in a wide variety of problems leading to some pioneering works since the nineteenth century.