Questions tagged [affine-grassmannian]

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8
votes
0answers
224 views

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 ...
8
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0answers
116 views

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'...
1
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0answers
71 views

Coefficient ring of Satake isomorphism

Let $G$ be a split reductive group over local field $F$, $G^L$ be the (complex) Langlands dual group of $G$. Denote $H$ to be the $\mathbb{Z}$-Hecke algebra of $G$, that is the ring of $G(\mathcal{O}...
7
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0answers
132 views

Explicit computation for perverse cohomology

To construct the convolution product for two ($G(O)$-equivariant) perverse sheaves $\mathcal{F}, \mathcal{G}$ on affine grassmanian, the first thing we need to compute is $^PH^0(\mathcal{F} \boxtimes^...
10
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1answer
432 views

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 ...
10
votes
2answers
516 views

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 ...
8
votes
1answer
446 views

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 [1]. Now we ...
13
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0answers
644 views

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 ...
7
votes
1answer
207 views

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 $\mathbb{C}$-...
3
votes
1answer
284 views

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 $R$...
4
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0answers
172 views

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 ...
2
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0answers
173 views

obstructions to embeddings of manifolds into Grassmannians

Let $G_k(\mathbb{R}^n)$ be the Grassmannian consisting of $k$-dimensional subspaces in $\mathbb{R}^n$ and $AG_k(\mathbb{R}^n)$ the "affine Grassmannian" consisting of $k$-dimensional planes in $\...
1
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0answers
127 views

Symplectic structures on the grassmannian model of the based loop group

$\newcommand{\Ad}{\operatorname{Ad}}$ In the study of (smooth/algebraic) based loop spaces of compact groups, one often uses a Grassmannian model to study the space. In particular, the Grassmannian ...
5
votes
1answer
601 views

Singular/Smooth locus of Schubert variety of the affine grassmannian

Let $G$ be a connected, simply connected, semisimple, complex linear algebraic group with maximal torus $T$ and affine Grassmannian $\mathcal Gr$. It is well known that $\mathcal Gr$ admits a Bruhat ...
3
votes
1answer
282 views

Do the following two filtrations of the affine Grassmannian agree?

Let $H = L^{2}(S^{1},\mathbb{C}^{n})$, $H_{0}\subseteq H$ the subset of maps that extend holomorphically to the unit disc, and $H_{m} = z^{m}H_{0}$. Consider the affine Grassmannian for $GL_{n}$ in ...
12
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0answers
842 views

Reconciling the affine grassmannian and the based loop group

I'm trying to reconcile the differences between the (algebraic) based loop group and the affine grassmannian. I once believed that I understood the relationship, but I just read a paper which has ...
2
votes
1answer
560 views

Various definitions of the Bruhat decomposition of the affine Grassmannian

Let $G$ be a connected, simply connected, complex, semi-simple group with affine Grassmannian $\mathcal Gr \cong G(\mathbb C[t,t^{-1}])/G(\mathbb C[t])$. Fix a choice of maximal torus $T \subset G$ ...
6
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1answer
1k views

Vanishing patterns of minors of matrix

Let $M$ be a $m\times n$ matrix with entries in, say $\mathbb{C}$; assume $n\leq m$. Denote by $I\subseteq\{1,2,\ldots, n\}$ a subset of the columns of M. I am interested in positive results to the ...
5
votes
1answer
462 views

Are Strata of the affine Grassmannian total spaces of equivariant vector bundles over flag varieties

This question is closely related to Peter Crooks question. Strata of the Affine Grassmannian Let $G$ be a complex reductive group, $\mathcal{K}:= \mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and ...
7
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0answers
157 views

Are Schubert varieties for Kac-Moody groups cut out by linear equations?

Let $G$ be a reductive group, and let $X$ be a partial flag variety for $G$. Then it is known that for any projective embedding of $X$, that the equations scheme-theoretically cutting out a Schubert ...
10
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1answer
762 views

What are global sections of the determinant bundle on the Beilinson-Drinfeld Grassmannian?

Let $X$ be a smooth proper algebraic curve over $\mathbb{C}$, and let $G$ be a reductive group over $\mathbb{C}$. Let $Gr_{X,n}$ be the Beilinson-Drinfeld Grassmannian (for n points in $X$), which ...
7
votes
1answer
905 views

The Bialynicki-Birula Stratification of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple group with affine Grassmannian $\mathcal{G}r$. Fix a maximal torus and Borel $T\subseteq B\subseteq G$. I am reading "Loop Grassmannian ...
8
votes
2answers
894 views

Relations between affine Grassmannian and Grassmannian

Let $\mathcal K = k((t))$ be the field of formal Laurent series over $k$, and by $\mathcal O = k[[t]]$ the ring of formal power series over $k$. Let $G$ be an algebraic group over $k$. The affine ...
1
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0answers
79 views

open immersion, affine grassmanian and negative loop group

Let $G$ a semisimple group over $k=\bar{k}$. Let the $k$-indgroup, $LG^{-}\subset G(k[t^{-1}])$ be the kernel of the reduction. We know by Faltings that the multiplication map: $LG^{-}\times G(k[[t]]...
2
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0answers
448 views

Transverse slices to orbits in the nilpotent cone and affine Grassmanian in type A

Background My question is about the paper http://arxiv.org/abs/0712.4160; specifically about the isomorphism in Theorem $1.2$ (in the Introduction), $T_{\lambda} \cap \overline{\mathcal{O}_{\mu}} \...
5
votes
0answers
264 views

Is it possible to describe the ideals of the Iwahori decomposition in a loop group using generalized minors?

Let $G$ be your favorite complex reductive algebraic group, and consider its (algebraic) loop group $G((t))$. A role very similar to the Borel $B\subset G$ is played in the loop group by the Iwahori ...
7
votes
0answers
355 views

Reference for the Thick Affine Grassmanian

Let $G$ be a reductive group and $LG$ be the algebraic loop group of $G$; i.e. $LG(k) = G( k((t)) )$. There is a fair amount of literature on the affine Grassmanian $LG(k)/G(k[[t]])$ and its Picard ...
11
votes
2answers
1k views

The affine Grassmannian and the Bogomolny equations

In "Electric-Magnetic Duality and The Geometric Langlands Program", Sections 9 and 10, Kapustin and Witten describe certain convolution varieties in the affine Grassmannian (and more generally, in the ...
5
votes
1answer
587 views

What is the Picard group of a Schubert variety in the affine Grassmannian?

I'm not sure I have a lot more to say than the title. Let $G$ be your favorite simple algebraic group over $\mathbb{C}$, and let $$\overline {\mathrm{Gr}}_\lambda= \overline{G(\mathbb{C}[[t]])\cdot t^...