# 2022-01-24

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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.