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 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 are the 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 continue the study of character sheaves on disconnected reductive groups and bring the theory to the same level of completeness as that in the connected case. This study is necessary to put the classification of unipotent representations of adjoint p-adic groups (not necessarily inner forms of split groups) on a firm foundation. The more general theory of character sheaves will be also needed in the study of irreducible characters of the group of rational points of a reductive group with a cyclic group of components defined over a finite field.It is also proposed to continue the study of unipotent elements in small characteristic, in particular to try to give a uniform description of the group of components of centralizers of such elements. Progress on the topics above 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.
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. During the period covered by this grant I have discovered a new way to understand conjugacy classes of a group of Lie type (that is various types of symmetry) in terms of conjugacy classes in Weyl groups (which are finite groups whose structure is easily amenable to computation). This is a new application of representation theory of groups defined over a finite field. During the same period I have found (with Vogan) some new algebraic formalism which helps to understand the classification of unitary representations of groups over real numbers. A part of the formalism is the discovery of a new family of polynomials attached to a pair of involutions in the symmetric group (a pair of permutations with square one of n objects). I have also found a new type of Fourier transform attached to a Lie group which conjecturally should explain the characters of groupsover p-adic numbers.
