The loop-groups tag has no usage guidance.

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### Geometric Satake and Restriction

The Geometric Satake correspondence (due to Lusztig, Ginzburg, Mirkovic-Vilonen) relates perverse sheaves on the Loop Group $\hat{G}$ (with their convolution product) to the Representations of the ...

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### For a new operation on a finite group of odd order giving a loop structure, when does this also gives a group

For finite groups $G$ of odd order, as $x \mapsto x^2$ is bijection (but no automorphism in general) then, we can define for each $g \in G$ the element $x^{1/2}$ by requiring $(x^{1/2})^2 = x$. Then ...

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### Does Kähler structure on X imply Kähler structure on the loop space of X?

Does Kähler structure on $X$ imply Kähler structure on the loop space ($LX$) of $X$? Since the loop space of $X$ is the space of maps from the circle $S^1$ to $X$, I suspect one may use the pullback ...

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### 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 ...

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### 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 ...

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### Based loop groups as stacks?

I have been stuck for some time, thinking about the following question.
Let $G$ be a Lie group. Its classifying space $BG$ can be seen as the differentiable stack $[pt/G]$, which is of dimension $-...

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### Concise mathematical definition of the fusion product on the Verlinde ring?

The Verlinde ring of a (let us say) simply connected simple compact Lie group has as underlying additive group the Grothendieck group of representations of the central extension $\widehat{LG}$ of the ...

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### Equivalence of Kahler structures of based loop group and its Grassmannian model

In Pressley-Segal's Loop Groups, we have the following spaces equipped with Kahler structures. Let $G$ be a compact, connected, (simply connected) group with Lie algebra $\mathfrak g$.
Let $\...

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### 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 ...

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### Estimate for the Iwasawa decomposition in loop groups

Let $GL(n,\mathbb{C})$ be the general linear group and let $U(n)$ be the unitary group in it, which is a maximal compact subgroup.
I consider the loop group $\Lambda GL(n,\mathbb{C})$ of maps from $S^...

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### 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 ...

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### Lattice model for Affine Grassmannians of non type A

There is a Lattice model for affine Grassmannians of type A, due to Lusztig. It describes affine Grassmannians of type A as the moduli space of certain subspaces in an infinite-dimensional $\mathbb{C}-...

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### What is the level of a positive energy loop group representation?

I am trying to learn a bit about loop group representation theory to understand its role in string geometry.
Let $G$ be a Lie group. I am thinking of $\text{Spin}(n)$, so you may assume $G$ to be ...

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### Why should affine lie algebras and quantum groups have equivalent representation theories?

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}}$ be the Kac-Moody algebra obtained as the canonical central extension of the algebraic loop algebra $\mathfrak{...

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### adjoint orbit of (twisted) loop group

Let $G$ be the twisted loop group of $SL_2(\mathbb C)$ and let $g$ be its Lie algebra, where diagonal entries are even functions and off diagonal entries are odd functions (of loop parameter lambda).
...

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### Real varieties with enough algebraic loops

Let $(X,\sigma)$ be a complex variety with complex conjugation (equivalently, an algebraic variety over $\mathbb R$).
We use the notations $X(\mathbb R):=X^\sigma$ for the set of fixed points of $X$ ...

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### Holomorphic convergence conditions on $\mathbb C((z))$-valued points of a group $G$

Let $G$ be a complex, connected, simply connected, semisimple group. I'm trying to compare the following two spaces: The free loop space $LG$ of $G$, and the $\mathbb C((z))$-valued points of $G$, $G(\...

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### Equivariant Poincare Series of Based Loop Group of SU(2)

Let $\Omega SU(2)$ denote the based loop group of $SU(2)$, and consider the action of $S^1$ on $\Omega SU(2)$ as a maximal torus of $SU(2)$. (This is not the "loop rotation" action.) Is there an ...

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### $\text{mod} \, p^2$ trace identity

Let $p$ be a prime, and let $\text{GL}_n \big( \Bbb{Z} / p^2 \Bbb{Z} \big)$ be the group of $n \times n$ invertible matrices over the ring $\Bbb{Z} / p^2 \Bbb{Z}$. Does there exist a positive integer $...

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### Central extension of the algebraic loop group

I'm doing some constructions with the universal central extension $\widehat{\Omega G}$ of the loop group $\Omega G$ (here $G$ is a matrix group), where a priori the loops involved are just smooth, but ...

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### differentiating positive energy LG reps

Background:Let $G$ be a cscsc¹ Lie group, and let $\widetilde{LG}$ be the universal central extension (center = $S^1$) of $LG:=Map_{C^\infty}(S^1,G)$, with the topology inherited from the $C^\infty$ ...

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### Is Pic( G((z)) ) = $\mathbb{Z}$?

There are a fair number of papers by Beauville, Laszlo, Sorger, Kumar and others on the geometry of $LG/L^+G = G((z))/G[[z]]$ where $G$ is a simply connected and simple group over $\mathbb{C}$. In ...

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### Is there a fusion rule in positive characteristic?

Verlinde's fusion gives a certain "tensor product" of representations of loop groups. The category of representations of loop groups has (essentially equivalent) two incarnations. One is analytic, ...