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7
votes
0answers
115 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
210 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 $R$...
3
votes
0answers
90 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
votes
0answers
98 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
vote
0answers
64 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
255 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
177 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 ...
9
votes
0answers
382 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
309 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 $T \subset G$ ...
5
votes
1answer
294 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 ...
4
votes
1answer
233 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 ...
6
votes
0answers
129 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 ...
6
votes
0answers
339 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 ...
2
votes
1answer
461 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 ...
5
votes
2answers
477 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
vote
0answers
61 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
votes
0answers
354 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
242 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 ...
6
votes
0answers
318 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 ...
10
votes
2answers
820 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
416 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^...