The proposal consists of 4 parts. In the first part the investigator and his colleagues study the asymptotic affine Hecke algebra introduced by G. Lusztig. The asymptotic Hecke algebra is a suitable limit of the usual Hecke algebra as parameter tends to zero. Its representation theory is closely related with representation theory of the Hecke algebra itself. One of the aims here is a proof of Lusztig's Conjecture describing the asymptotic Hecke algebra in the elementary K-theoretic terms. In the second part the investigator studies module categories over monoidal categories. This subject is closely related with modern physics where module categories appear in the context of the Boundary Conformal Field Theory. In the third part the investigator and collaborators study distinguished involutions in the affine Weyl group. In particular they make extensive explicit calculations of canonical distinguished involutions in number of cases. In the fourth part the investigator and his colleagues study the Double Affine Hecke Algebra. The aim here is to describe Intersection Cohomology of certain infinite dimensional algebraic varieties in terms of Kazhdan-Lusztig type combinatorics of this algebra.
In this proposal the investigator studies various questions of Representation Theory. Representation Theory is a part of mathematics that studies all possible ways in which symmetry can be used for solving concrete physical or technical problems. Many physical and technical systems do not change under some transformations (e.g. the gravitational field of the Sun depends only on the distance from the Sun and so it does not change under rotating of the space around the Sun). Such transformations are called symmetries of the system. In many cases symmetries can be used to simplify the study of such systems. So it is not surprising that Representation Theory has many applications in physics (where continuous symmetry is one of the most fundamental concepts), chemistry (especially in quantum chemistry where it is used in computations of chemical forces inside molecules), computer science (for example, Fourier analysis, which can be considered as a simplest case of Representation Theory, is one of the most widely used of all calculation techniques), and inside of mathematics itself, in number theory (where Representation Theory is an essential part of the Langlands program). One of the central objects of study in Representation Theory is the affine Hecke algebra, because answers to many seemingly unrelated questions are encoded in the structure of this algebra. This proposal is mainly devoted to the study of the affine Hecke algebra.
