Varieties Related to Algebraic Group Actions

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ABSTRACT, WEYMAN This proposal consists of several parts. They are all connected to the study of varieties related to algebraic groups and representation theory. The first part is to study equivariant modules supported in varieties related to algebraic groups, like determinantal varieties and the closures of nilpotent conjugacy classes in simple Lie algebras. The investigator proposes to investigate the Grothendieck groups of categories of such modules and to apply these results to study the coordinate rings of orbit closures by means of degeneration sequences. The second part is the study of singularities of generalized discriminants and resultants. The investigator proposes to extend his classification ( with Zelevinsky ) of irreducible components of hyperdeterminantal varieties to arbitrary homogeneous spaces for the reductive group. The third part is concerned with representations of Z_2 graded Lie algebras gl(m,n). The investigator proposes to study the connection of such representations with syzygies of equivariant ideals in the polynomial ring of generic $m\times n$ matrices. The subject of this proposal is the study of the varieties related to classical groups. In general terms they can be described as the varieties with a lot of symmetries. These varieties play an important role in Commutative Algebra, serving as basic examples and also providing links to other fields, like Combinatorics, Representation Theory and Singularities. Historically many important general theorems were proved first for these kinds of varieties. The investigator proposes to study the algebraic invariants associated to these varieties, in particular the structure of equivariant modules supported in them.