Tags: #untagged
Refs: ?
16:09
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What is the affine Grassmannian \({\operatorname{Gr}}_G\)?
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What is the Demazure character formula?
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What is geometric Satake?
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What are Macdonald polynomials?
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The Weyl group: \(W \coloneqq N_G(T)/T\).
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Cocharacter lattice: \(X_*(T) = \mathop{\mathrm{Hom}}({\mathbf{C}}^{\times}, T)\), and the character lattice \(X^*(T) = \mathop{\mathrm{Hom}}(T, {\mathbf{C}}^{\times})\).
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For \(K = {\mathbf{C}}{\left(\left( t \right)\right) }\), \({\mathcal{O}}_K = {\mathbf{C}}{\left[\left[ t \right]\right] }\).
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Loop groups: its \(R\) points are \(LG(R) = G(R{\left(\left( t \right)\right) })\).
- Define \(L^+G(R) = G(R{\left[\left[ t \right]\right] })\).
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\(L{\mathbf{G}}_m(R)\) for $R\in \mathsf{Alg} _{/ {{\mathbf{C}}}} $ are formal Laurent series with coefficients in \(R\)?
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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 \(c: \operatorname{Pic}({\operatorname{Gr}}_G)\to {\mathbf{Z}}\).
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Heisenberg algebras: central extensions of an abelian algebra, and some analog of the Stone-von-Neumann theorem classifying representations.
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ADE groups: simply laced.
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Affine Schubert varieties have singularities along their boundary.