University of Sydney Algebra Seminar

Pramod Achar (Louisiana State University)

Friday 21st June, 12:05-12:55pm, Carslaw 373

The topology of the affine Grassmannian and the Mirkovic-Vilonen conjecture

Given a complex reductive group \(G\), one can form its "affine Grassmannian" (or "loop Grassmannian"), a certain infinite-dimensional space that in many ways resembles ordinary (finite-dimensional) flag manifolds or Grassmannians. A deep result of Mirkovic-Vilonen (building on earlier work of Ginzburg and Lusztig) asserts that the algebraic topology of the affine Grassmannian encodes the representation theory of the dual group to \(G\) over any field. More precisely, certain intersection cohomology groups with coefficients in a field \(k\) realize the irreducible representations of the dual group over \(k\). This result raises the possibility of using the "universal coefficient theorem" of topology to compare representations over different fields. With that in mind, Mirkovic and Vilonen conjectured in the late 1990's that the local intersection cohomology of the affine Grassmannian with integer coefficients is torsion-free. I will try to explain these ideas with examples, and I will discuss the recent proof of (a slight modification of) the Mirkovic-Vilonen conjecture. This is joint work with Laura Rider.