Geometric methods in representation theory

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Principal Investigator: George Lusztig Proposal Number: 0243345Institution: Massachusetts Institute of TechnologyABSTRACTGeometric methods in representation theoryRepresentation theory of groups of Lie type is a central part of mathematics. It is concerned with understanding systems with symmetry by representing them in matrix form. One of the most difficult areas of representation theory is that of groups over p-adic fields, which has strong connections with number theory. One of the main tools in the study of these groups is the use of affine Hecke algebras. G. Lusztig proposes to continue the study of affine Hecke algebras with unequal parameters and in particular to establish a geometric interpretation for their canonical basis. Also it is proposed to establish the existence of the corresponding asymptotic Hecke algebras. This should give new information on the representation theory of groups over p-adic fields. It is also proposed to further investigate the analogue of the Deligne-Lusztig theory in the case where finite fields are replaced by certain finite rings. This again should have applications to the representation theory of groups over p-adic fields. It is proposed to further investigate the canonical bases of quantized enveloping algebras from the point of view of perverse sheaves. It is also proposed to continue the study of character sheaves on reductive groups. Progress in these topics is expected to have applications to various parts of mathematics and theoretical physics.The theory of group representations attempts to study the idea of symmetry by means of matrices, which are more amenable to computation. One of the oldest applications of representation theory is the theory of Fourier series, widely used in engineering and applied science. More recently, ideas from representation theory have been used in chemistry (study of crystals) and physics (theory of elementary particles). G. Lusztig's research is concerned with applications of methods of algebraic topology (study of shapes by means of algebra) and algebraic geometry (geometric study of equations) to obtain new results on group representations which could not be obtained by other methods.