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Refs: ?


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

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

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

    • Define \(L^+G(R) = G(R{\left[\left[ t \right]\right] })\).
  • \(L{\mathbb{G}}_m(R)\) for $R\in {\mathsf{Alg}}_{/ {{\mathbb{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 \(c: {\operatorname{Pic}}({\operatorname{Gr}}_G)\to {\mathbb{Z}}\).
  • 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.

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