Representation 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 tools which has proved to be highly effective for the study of representations is the theory of character sheaves which the PI initiated in the early 1980's. This theory provides a geometrization of the theory of irreducible characters of finite simple groups of Lie type. It is proposed to continue the project of studying character sheaves on reductive algebraic groups, including disconnected ones. In particular it is proposed to continue the study of affine Hecke algebras with unequal parameters and in particular to establish a geometric interpretation for their canonical basis. This should give new information on the representation theory of groups over p-adic fields. It is also proposed to continue the study of characters of semisimple p-adic groups continuing the author's earlier work with J.L. Kim. 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 application 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.
