Crystals Learning Seminar
In this seminar we will learn about crystals, which are nice combinatorial models for representations of Lie algebras. The first half of the seminar will focus on quantum groups and construction of crystal bases from this perspective (Kashiwara's "grand loop argument"). The second half will focus on combinatorial (Young tableaux, Littelmann path model) and geometric (quiver varieties, MV polytopes) constructions.
The affine Grassmannian and geometric Satake correspondence
University of Toronto
15:00 (Monday, Nov. 21, 2011
BA6180, Bahen Center, 40 St George St
Given a reductive group G, the affine Grassmannian Gr(G) is a certain infinite-dimensional "variety" which can be thought of as a partial flag variety for the loop group of G. It has a natural stratification by finite-dimensional subvarieties labelled by the dominant coweights of G. At the same time, these coweights label the irreducible representations of the Langlands dual group G'. This coincidence is a shadow of the geometric Satake correspondence, which gives an equivalence of pivotal categories between Rep(G') and the category of perverse sheaves on Gr(G) with respect to the aforementioned stratification. We will sketch a proof of this correspondence, along the way giving definitions (or at least some intuition) of the relevant objects.
Dates in this series
- · Monday, Sep. 26, 2011:
Introduction to quantum groups II (Iva Halacheva)
- · Monday, Nov. 21, 2011:
The affine Grassmannian and geometric Satake correspondence (Brad Hannigan-Daley)
- · Monday, Nov. 28, 2011:
Crystals via Mirković-Vilonen cycles (Brad Hannigan-Daley)
- · Monday, Dec. 05, 2011:
MV polytope model for crystals (Joel Kamnitzer)