Titlebook: Representation Theories and Algebraic Geometry; Abraham Broer,A. Daigneault,Gert Sabidussi Book 1998 Springer Science+Business Media B.V.

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Geometric methods in the representation theory of Hecke algebras and quantum groups,entations of the algebras in question, using our geometric construction and perverse sheaves methods..Similar techniques can be applied to other algebras, e.g. the double-affine Hecke algebras, elliptic algebras, quantum toroidal algebras.

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Standard Monomial Theory and applications,tandard Monomial Theory are made to prove geometrical properties of certain ladder determinantal varieties and certain quiver varieties. We sketch at the end an extension of the theory to Bott-Samelson varieties and configuration varieties.

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Book 1998 and the representation theoriesof various modern mathematical structures, such as reductive groups,quantum groups, Hecke algebras, restricted Lie algebras, and theircompanions. This interplay has been extensively exploited duringrecent years, resulting in great progress in these representationtheor

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Equivariant cohomology and equivariant intersection theory,, a much more recent topic initiated by D. Edidin and W. Graham..Our main aim is to obtain explicit descriptions of cohomology or Chow rings of certain manifolds with group actions which arise in representation theory, e.g. homogeneous spaces and their compactifications..As another appplication of e

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Lectures on decomposition classes,es of Jordan blocks. This notion generalizes to arbitrary complex semisimple Lie algebras. There are finitely many of them, each one is a smooth variety, and the closure of one is a disjoint union of decomposition classes. Each decomposition class is the union of all adjoint orbits isomorphic to a f

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Geometric methods in the representation theory of Hecke algebras and quantum groups, naturally in Representation Theory such as the group algebra of a Weyl group, the universal enveloping algebra of a complex semisimple Lie algebra, a quantum group or the Iwahori-Hecke algebra of bi-invariant functions (under convolution) on a .-adic group, are considered. We give a uniform geometr