# Questions tagged [affine-grassmannian]

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### Stratified fibration property of the "Ran" affine Grassmannian

Let us consider the so-called Ran Grassmannian $Gr_{Ran}$, i.e. the geometric object defined e.g. in [Zhu, An Introduction to the affine Grassmannian and the Geometric Satake equivalence, Definition 3....
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### Cartan decomposition of loop group

Let $G$ be a complex reductive group. Let $LG$ and $L^+ G$ denote the formal loop spaces given by maps from the punctured formal disk and the formal disk, respectively, to $G$. The quotient $LG/L^+ G$ ...
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### Is the affine Grassmannian manifold a symmetric homogeneous space?

I am interested in the manifold of affine subspaces of dimension $k$ of $\mathbb{R}^n$, which can be viewed as the homogeneous space $$E(n)/(E(k)\times O(n-k)),$$ where $E$ refers to rigid motions ...
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### Drinfeld Sokolov and the semiinfinite flag variety

For a long time I've been confused about Drinfeld Sokolov/BRST reduction/semiinfinite cohomology for affine Lie algebras. Most treatments define it in what to me feels like a fairly ad-hoc way, by ...
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### What is wrong with $A^{(2)}_{2n}$?

When dealing with affine Kac-Moody groups, especially geometrically (e.g. by examining their affine flag varieties or affine Grassmannians) I've been taught that time and time again, issues arise in ...
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### Comparison of two well-known bases of the integral homology group of based loop group

Let $G$ be a compact simply-connected Lie group. Then one can look at the homology $H_*(\Omega G;\mathbb{Z})$ of the based-loop space $\Omega G$ in at least two different ways: (1) Via Bott-Samelson'...
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### how to view homology of affine Grassmannian as a subring of symmetric function

Let $G=SL_n$, it is proven that $R:=H_*(Gr_G)\cong \mathbb{C}[\sigma_1,...,\sigma_{n-1}]$ where $\sigma_i$ are of degree $2i$ as a polynomial ring generated by $n-1$ variables and the ring structure ...
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### Relation between affine flag and Grassmannian Steinberg variety

Let $\mathcal{K}=\mathbb{C}((t))$ be the field of formal Laurent series over $\mathbb{C}$, and by $\mathcal{O}=\mathbb{C}[[t]]$ the ring of formal power series over $\mathbb{C}$. Given a semi-simple ...
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### Affine vs Yokonuma

Let $G=GL_n$. Let us start with the Hecke algebra $H_n$. It acts on K(constructible sheaves on $G/B$) by Hecke correpondences and on K(coherent sheaves on $G/B$) by Lusztig's construction . Now we ...
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### Borel-Weil-Bott, Langlands and Hitchin

Let $G$ be a compact semi-simple Lie group and $G_\mathbb{C}$ be its complexification. We denote by $B$ a Borel subgroup of $G_\mathbb{C}$. Given a dominant weight $\lambda$, one can construct a line ...
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### Homological contractibility of a prestack

This question is in reference to Gaitsgory's preprint Contractibility of the space of rational maps. On p. 5 of the preprint, Gaitsgory defines a prestack $\mathscr{Y}$ (say over affine $\mathbb{C}$-...
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### A technical question about affine grassmanian

For a commutative ring $R$, consider $R[[t]]$-modules $$t^k R[[t]]^n \subset M \subset t^{-k} R[[t]]^n \subset R((t))^n.$$ It is known that if $t^{-k} R[[t]]^n / M$ is finitely generated projective $R$...
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### Fibers of torus equivariant moment maps

Given a closed (possibly singular) projective variety $V$ with a symplectic structure and a torus action, there is a moment map $\mu: V \rightarrow Lie(T)^*$. Note that the dimension of $T$ could be ...