---
date: 2022-01-24
tags: [web/quick_notes]
---

# 2022-01-24

Tags:
#untagged

Refs:
?

## 16:09

- What is the [affine Grassmannian](affine%20Grassmannian.md) $\Gr_G$?
- What is the [Demazure character formula](Demazure%20character%20formula)?
- What is [geometric Satake](geometric%20Satake.md)?
- What are [Macdonald polynomials](Macdonald%20polynomials)?
- The Weyl group: $W \da N_G(T)/T$.
- Cocharacter lattice: $X_*(T) = \Hom(\CC\units, T)$, and the character lattice $X^*(T) = \Hom(T, \CC\units)$.
- For $K = \CC\fls{t}$, $\OO_K = \CC\fps{t}$.
- Loop groups: its $R$ points are $LG(R) = G(R\fls{t})$.
  - Define $L^+G(R) = G(R\fps{t})$.
- $L\GG_m(R)$ for $R\in \Alg\slice \CC$ 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](central%20charge) $c: \Pic(\Gr_G)\to \ZZ$.
- 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](Schubert%20varieties) have singularities along their boundary.