A Steinberg Cross Section for Non–Connected Affine Kac—Moody Groups
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Publication:5480073
DOI10.4153/CJM-2006-026-7zbMATH Open1155.22301arXivmath/0401203MaRDI QIDQ5480073
Publication date: 25 July 2006
Published in: Canadian Journal of Mathematics (Search for Journal in Brave)
Abstract: We generalise the concept of a Steinberg cross-section to non-connected Kac-Moody group. As in the connected case, which was treated by G. Br"uchert, a quotient map w.r.t the conjugacy action exists only on a certain submonoid of the Kac-Moody group. Non-connected Kac-Moody groups appear naturally as semidirect product of C^* with a central extension of loop groups LG, where the underlying simple group G is no longer simply connected and might even be non-connected. In contrast to the connected case, the understanding of central extensions of non-connected loop groups is a rather complicated issue. Following the approach of V. Toledano Laredo, who dealt with the case of automorphisms coming from the fundamental group pi_1(G), we classify all of these central extensions for cyclic component group of LG. Then, we define the quotient map w.r.t conjugacy action. Furthermore, we construct the cross-section in every connected component of LG and show that, due the one-dimensional centre, it carries a natural C^*-action which does not exist in the finite dimensional case.
Full work available at URL: https://arxiv.org/abs/math/0401203
Related Items (3)
A note on the Tits systems of Kac-Moody Steinberg groups ⋮ On the Steinberg map and Steinberg cross-section for a symmetrizable indefinite Kac-Moody group ⋮ Conjugacy classes in affine Kac-Moody groups and principal \(G\)-bundles over elliptic curves
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