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4
votes
0answers
122 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
215 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
250 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
48 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 ...
2
votes
0answers
266 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
219 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 ...
5
votes
0answers
263 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 ...
7
votes
2answers
688 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
322 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 ...