Universität zu Köln

Rappel, Valentin Maximilian (2020). The path model and Bott–Samelson manifolds in the context of loop groups. PhD thesis, Universität zu Köln.

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Abstract

We realize the Littelmann path model and the associated root operators on the loop groups of the torus in a compact, simple Lie group. For integral loops - those loops with a good combinatorial description of the root operators - we define a geometric interpolation of the root operators through Bott–Samelson manifolds, whose definition we generalize for this purpose. We embed these manifolds in the loop group of the simple group and give a criterion under which the symplectic structure of the loop group restricts to a symplectic structure of the Bott–Samelson manifold. For loops in dominant direction we compute the image under the moment map. To establish a complex structure we give a diffeomorphism between the Bott–Samelson manifolds and the Bott–Samelson–Demazure–Hansen variety associated to a gallery in the affine building. This map is compatible with the root operators, and we interpret the results of Gaussent and Littelmann in the context of the gallery model anew. By means of this interpretation we define isotopic embeddings of Mirković-Vilonen cycles into the differential-geometric loop group. For this purpose we investigate the behavior of the Bott–Samelson manifold under homotopies of the underlying loop. A consequence of this is another criterion to determine the image of the moment map.