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)
- Introduction (basic definitions, properties, etc.)
Lecture 1: 1.1, 1.2, 1.6, Lecture 2: 2.1 and 2.6 in X. Zhu: An introduction to affine Grassmannians and the geometric Satake equivalence
- Geometric Satake equivalence
I. Mirkovic, K. Vilonen: Geometric Langlands duality and representations of algebraic groups over commutative rings
- Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangians
J. Kamnitzer, K. Pham, A. Weekes: Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangians
W. L. Gan, V. Ginzburg: Quantization of Slodowy slices
- Minimal degeneration singularities
- Minimal degeneration singularities in the nilpotent cones of classical Lie algebras: H. Kraft and C. Procesi, On the geometry of conjugacy classes in classical groups
- Minimal degeneration singularities in the affine Grassmannian A. Malkin, V. Ostrik, M. Vybornov The minimal degeneration singularities in the affine Grassmannians
- Beilinson-Drinfeld Grassmannian
Lecture 3 in X. Zhu An introduction to affine Grassmannians and the geometric Satake equivalence
A. Beilinson and V. Drinfeld, Quantization of Hitchinâ€™s integrable system and Hecke eigensheaves
- Mirkovic-Vilonen polytopes
J. Anderson: A polytope combinatorics for semisimple groups
- Affine Springer fibers
Lectures 1 and 2 in Z. Yun: Lectures on Springer theories and orbital integrals
- Classification of toric bundles on affine toric schemes over DVRs
Other helpful resources
Talks
- Sep. 16, Boris Tsvelikhovskiy: Introduction to affine Grassmannians.
- Sep. 23, Boris continues.
- Sep. 30, Boris finishes, Complete notes
- 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.
- 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.
- Oct. 28, Boris Tsvelikhovskiy: Nakajima quiver varieties & affine Grassmannians, slides
References:
V. Ginzburg: Lectures on Nakajima's quiver varieties.
I. Mirkovic and M. Vybornov: Quiver varieties and Beilinson Drinfeld Grassmannians of type A.
- Nov. 5, Carl Wang-Erickson: An appearance of the affine Grassmannian in p-adic Hodge theory.
Abstract.
We give an overview of a few fundamental ideas in p-adic Hodge theory and conclude with an example of how an
affine Grassmannian contains moduli spaces of certain p-adic Hodge theoretic objects.
- Nov. 12, Kiumars Kaveh: Bruhat-Tits building of SL(2).
Abstract.
Following Bill Casselman's notes, I will discuss how points in the affine Grassmannian for SL(2) and over a p-adic field
(up to equivalence) can be thought of as nodes on an infinite tree. This is a special case of the Bruhat-Tits building of
a reductive group over a local field.
We will use theory of modules over a DVR. The material are rather elementary and accessible to graduate students.
Reference:
B. Casselman: The Bruhat-Tits tree of SL(2), notes
- Dec. 02, Boris Tsvelikhovskiy: On categories O for conical symplectic resolutions, slides
- Dec. 09, Joel Kamnitzer: Mirkovic-Vilonen polytopes from geometry and algebra
Abstract.
A famous foundational problem concerns finding combinatorial expressions for tensor products
of irreducible representations. One way to solve this problem is to use the combinatorics of
Mirkovic-Vilonen polytopes. These polytopes were originally defined as moment map images
of MV cycles, but they also arise from the study of submodules of preprojective algebra modules.
With Pierre Baumann and Allen Knutson, we defined measures supported on these polytopes.
These measures allow us to perform computations which distinguish among different bases
for representations.