2:00 pm, Seminar Hall
Introduction to geometric representation theory 1 (First Lecture)
University of Aarhus, Denmark.
We start by recalling that both conjugacy classes of nilpotent matrices and irreducible representations of the symmetric group are naturally enumerated by partitions or, more graphically, by Young diagrams. We devote our talk to the explanation of this fact. The main geometric construction is called Springer correspondence.
We introduce the nilpotent cone N, the Flag variety Fl, the Springer variety Sp, and the Stainberg variety Z for SL(n). Then we develop the general machinery of a convolution algebra due to Ginzburg. Namely, for a suitable map X --> Y, Borel-Moore homology of the fiber product of X with itself over Y obtains a structure of a convolution algebra. In particular, Borel-Moore homology of Z provides a geometric realization for the group algebra of the symmetric group. Irreducible representations of the symmetric group obtain a nice geometric parametrization due to this construction.
Finally we hint how the construction is generalized to other simple Lie algebras.