Questions tagged [affine-grassmannian]
The affine-grassmannian tag has no usage guidance.
43
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Smooth unipotent algebraic groups over $\mathbb A^n$
Let $G\to \mathbb A^n_{\mathbb C}$ be a smooth morphism whose fibers at any point of $\mathbb A^n$ are unipotent groups. Can we conclude that $G\simeq \mathbb A^{n+N}_{\mathbb C}$ for some $N$, as a ...
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Factoring out an element of a root subgroup to make a conjugation integral
Fix a nonarchimedean local field $L/\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, uniformizer $\varpi$, and residue field $k$. If I take a matrix
$$\begin{pmatrix} a & \varpi b \\ c & d \...
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What is Pic of the torus global affine Grassmannian?
Let $T$ be a torus and $X$ a proper smooth curve over characteristic $0$ algebraically closed field $k$.
What is $\text{Pic}(\text{Gr}_{T,X^n})$?
Here $\text{Gr}_{T,X^n}$ is the BD Grassmannian over $...
- 5,565
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In which sense affine Grassmannian is "affine"
A pretty naïve question: Which meaning has the term "affine" in the notion of affine Grassmanian. Especially, I do not see any immediate connection to the concept of an "affine scheme&...
- 5,780
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Levi quotients of parahorics in loop group
I am looking for some references on Levi quotients of parahorics in $LG = G(\mathbb{C}((t)))$, $G$ being an algebraic group with Weyl group $W$.
I have read that parahoric subgroups of $LG$ are in ...
- 57
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A pairing between “Schubert” $H_*(\operatorname{Gr})$ and “Langlands” $H^*(\operatorname{Gr})$
Let $\operatorname{Gr}$ be an affine Grassmanian of some complex semisimple group $G$. Of course, there is a well-known description of $H^*(\operatorname{Gr})$ in terms of Langlands dual Lie algebra.
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Does the space of hyperplanes in the Grassmannian have a name?
A way of defining the Grassmannian $Gr(k,n)$ is to consider the space of $k\times n$ matrices mod $GL(k)$ transformations on the rows. I'm interested in the space of $k\times 2n$ matrices mod $GL(k)$ ...
- 49
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Transition maps between coordinate charts on the Grassmann manifold
Let $\mathbf{Gr}_{n,k}$ be the manifold of $k$-dimensional subspaces of $\mathbb{R}^n$, and let $\mathbf{col}$ be the map that takes a matrix in $\mathbb{R}^{n\times k}$ to its columnspace. The map
\...
- 21
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What is the factorization algebra/space of an affine W algebra?
The affine vertex algebra $V_k(\mathfrak{g})$ factorizes, i.e. comes from a factorisation space, the Beilinson Drinfeld Grassmannian. Similarly, lattice vertex algebras have a factorization analogue.
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Error in Proposition 8.7.1 of Pressley–Segal
Let $G$ be a connected, compact Lie group, $T$ a maximal torus. Let $LG$ be the group of smooth (or polynomial) loops and $X=LG/T$ the affine flag variety ($T$ acts say by right multiplication). In ...
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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....
- 435
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1
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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 ...
- 401
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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 ...
- 5,565
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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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