The goal of these lectures will be to start from scratch and learn what the
affine Grassmannian of a reductive group is, and how its geometry is
related to the representation theory of the Langlands dual group. I will
emphasize the examples of GL_n and SL_2 throughout. Here is an outline of
the three lectures:
Lecture 1: definitions; lattice model for GL_n; orbits
Lecture 2: convolution product; Lusztig's q-analogue of the weight multiplicity
Lecture 3: geometric Satake equivalence; MV cycles
Abstract: Arc spaces of algebraic varieties turn out to be very useful for defining some topological invariants of algebraic varieties via a motivic integration on their smooth birational models. The arc spaces of spherical homogeneous spaces have been used implicitly in the Luna-Vust theory (1983) of equivariant valuations and spherical embeddings. The purpose of my talks is to explain my recent joint results with Anne Moreau on arc spaces of spherical embeddings. These results allow in particular to derive a combinatorial formula for computing Betti numbers of an arbitrary smooth projective spherical variety.
1) Local theory of chiral algebras.
2) Factorization approach and chiral homology.
3) Applications to the local Langlands theory.
R-matrices are solutions of the quantum Yang-Baxter equation. At the origin of the theory
of quantum groups, they can be interpreted as intertwining operators in representation theory.
After reviewing standard constructions from quantum affine algebras, we will present recent
development in the theory.
Maulik-Okounkov gave a general geometric construction of stable basis and R-matrices.
In another direction, monoidal categorifications of cluster algebras have been established
using R-matrices to categorify Fomin-Zelevinsky mutations relations.
If time allows, we will also discuss recent advances on transfer matrices derived from R-matrices,
which give new informations on corresponding quantum integrable systems as well as on the ODE/IM
correspondence seen in the context of affine opers.
Here is a tentative plan for the lectures :
1) R-matrices, algebraic and geometric constructions.
2) Categorification of mutation relations.
3) Transfer matrices and spectra of quantum integrable models.
4) Langlands duality and affine opers.
In these lectures we will introduce spherical varieties and discuss some of their basic properties; we will also introduce related combinatorial objects and see how they govern the geometry of such varieties. The topics will be:
1. Definitions, examples, first properties.
2. Local structure theorem.
3. The Luna-Vust theory of embeddings.
4. Spherical roots and the multiplication of regular functions.