The affine-grassmannian tag has no wiki summary.

**3**

votes

**1**answer

115 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 ...

**7**

votes

**0**answers

173 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

**1**answer

161 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 $T \subset G$ ...

**4**

votes

**1**answer

179 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

**1**answer

141 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

**0**answers

106 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 ...

**5**

votes

**0**answers

191 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

**1**answer

344 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

**2**answers

327 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

**0**answers

52 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

**0**answers

306 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

**0**answers

228 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

**0**answers

283 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

**2**answers

742 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

**1**answer

348 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 ...