### A Cartan decomposition for p-adic loop groups.

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We study algebraic loop groups and affine Grassmannians in positive characteristic. The main results are normality of Schubert-varieties, the construction of line-bundles on the affine Grassmannian, and the proof that they induce line-bundles on the moduli-stack of torsors.

By applying the Hamiltonian reduction technique we derive a matrix first order differential equation that yields the classical r-matrices of the elliptic (Euler-) Calogero-Moser systems as well as their degenerations.

A unitary representation $\pi $ of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\mathtt{d}\pi \left(x\right)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $\U0001d524$ of $G$. We classify all irreducible semibounded representations of the groups ${\widehat{\mathcal{L}}}_{\phi}\left(K\right)$ which are double extensions of the twisted loop group ${\mathcal{L}}_{\phi}\left(K\right)$, where $K$ is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant) and $\phi $ is...

Equivalence is established between a special class of Painlevé VI equations parametrized by a conformal dimension $\mu $, time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The isomonodromic tau function and solutions of the Euler top equations are explicitly constructed in terms of Wronskian solutions of the 2-vector 1-constrained symplectic Kadomtsev-Petviashvili (CKP) hierarchy by means of Grassmannian formulation. These Wronskian solutions give...