2022-01-24

Tags: #untagged

Refs: ?

16:09

• What is the affine Grassmannian ${\operatorname{Gr}}_G$?

• What is the Demazure character formula?

• What is geometric Satake?

• What are Macdonald polynomials?

• The Weyl group: $W \coloneqq N_G(T)/T$.

• Cocharacter lattice: $X_*(T) = \mathop{\mathrm{Hom}}({\mathbf{C}}^{\times}, T)$, and the character lattice $X^*(T) = \mathop{\mathrm{Hom}}(T, {\mathbf{C}}^{\times})$.

• For $K = {\mathbf{C}}{\left(\left( t \right)\right) }$, ${\mathcal{O}}_K = {\mathbf{C}}{\left[\left[ t \right]\right] }$.

• Loop groups: its $R$ points are $LG(R) = G(R{\left(\left( t \right)\right) })$.

  • Define $L^+G(R) = G(R{\left[\left[ t \right]\right] })$.

• $L{\mathbf{G}}_m(R)$ for $R\in \mathsf{Alg} _{/ {{\mathbf{C}}}}$ are formal Laurent series with coefficients in $R$?

• Idea: get $LG$ to act on cohomology of the affine Grassmannian to produce representations.

  • Only acts projectively, so pass to central extensions. Produces a central charge $c: \operatorname{Pic}({\operatorname{Gr}}_G)\to {\mathbf{Z}}$.

• Heisenberg algebras: central extensions of an abelian algebra, and some analog of the Stone-von-Neumann theorem classifying representations.

• ADE groups: simply laced.

• Affine Schubert varieties have singularities along their boundary.