Informal Seminar on Affine Grassmanians
Fall 2021

Meetings

The seminar meets at 1 pm on Thursdays via Zoom, Meeting ID: 928 0506 1304

Overview

The seminar will be devoted to affine Grassmannians and their appearances in various branches of mathematics, which will be chosen according to the interests of the participants.

Preliminary list of topics (with references)

Other helpful resources

Talks

  1. Sep. 16, Boris Tsvelikhovskiy: Introduction to affine Grassmannians.
  2. Sep. 23, Boris continues.
  3. Sep. 30, Boris finishes, Complete notes
  4. Oct. 7, Roman Fedorov: Geometric Satake correspondence, slides
    Abstract: Let G be a reductive group and Gr_G be its affine Grassmannian.
    Following a paper of Mirkovic and Vilonen, I will explain that the
    category of G_O-equivariant perverse sheaves on Gr_G is equivalent to
    the category of representations of the Langlands dual group of G.
    Somewhat surprisingly, this can be taken as the definition of the
    Langlands dual group. I will not assume that the listeners are
    familiar with perverse sheaves but I will assume some knowledge of
    affine Grassmannians.
  5. Oct. 21, Boris Tsvelikhovskiy: Mirkovic-Vilonen cycles and polytopes, slides
    Abstract: We will start by quickly reviewing the (classical) Satake isomorphism and then
    the geometric Satake correspondence. Under this correspondence the IC sheaves
    (intersection homology) of affine Schubert cells are associated with irreducible representations
    of the Langlands dual group of G. The Mirkovic-Vilonen cycles are subvarietiesin the affine
    Schubert cells, which give bases in the intersection homologies of those cells, thus playing a
    fundamental role in the correspondence. Similar to the Weyl (weight) polytopes, the polytopes arising
    as the images of Mirkovic-Vilonen cycles under the moment map w.r.t. the action of maximal compact torus T
    allow to (combinatorially) read off some information on representations of G (dimension of weight spaces,
    multiplicities of irreducibles in tensor products, etc.) I will assume only basic knowledge on affine
    Grassmannians (these notes are more than sufficient). Examples will be given.