Representation Theory and Mathematical Physics Seminar: Mapping class groups and difference operators.

Seminar | October 11 | 4-5 p.m. | 939 Evans Hall

 Shamil Shakirov, Harvard University

 Department of Mathematics

We review the representation of $SL(2,Z)$ - the mapping class group of the torus - by automorphisms of a simple algebra of difference operators. The algebra, known as spherical double affine Hecke algebra (DAHA) plays an important role in many developments in modern representation theory and mathematical physics. We will define a new algebra which is a direct analogue of spherical DAHA for a genus two surface, and sketch the proof of the corresponding mapping class group action. Time permitting, we will explain the connection to the Reshetikhin-Turaev construction, and possible generalizations to higher genus.