Kac-Moody symmetric spaces

• Applicant: Professor Dr. Ralf Köhl
• Subject Area: Mathematics
• Term: from 2016 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 321661801

• Final Report Year: 2022

Final Report Abstract

In quantum gravity, finite-dimensional representations of the maximal compact subalgebra k of the split real Kac–Moody-Algebra g of type E10 have been observed. In this project we obtained a uniform coordinate-free description of these representations via the Weyl group. Moreover, we showed that these representations do not lift to the maximal compact subgroup K of the split real Kac–Moody group G of type E10 , but instead only to the universal double cover of K – we are thus dealing with so-called spin representations. Further questions that we managed to answer concered their irreducibility and the irreducibility of some of their tensor products. In a second part of the project we studied the action of artihmetic Kac–Moody groups on the corresponding Kac–Moody symmetric space (Kac–Moody symmetric spaces have been introduced and studied thoroughly during the first funding period of this project). Here we showed that the stabilizer of the canonical basis point of the symmetric space within the canonical arithmetic Kac–Moody group equals the extended Weyl group. Further results concern general structural observations for axiomatically defined symmetric spaces with certain transitivity properties.

Publications

• Extending generalized spin representations. J. Lie Theory 28 (2018), 915 - 940
  Robin Lautenbacher, Ralf Köhl
  
• Modular groups and hyperbolic Weyl groups, PhD thesis, Justus-Liebig-Universität Gießen 2018
  Chengen Jiang
  
• An extension theorem for embedded Riemannian symmetric spaces of non-compact split type and an application to their universal property, Adv. Geometry 20 (2020), 499–506
  Julius Grüning, Ralf Köhl
  
• Kac–Moody symmetric spaces, Münster J. Math 13 (2020), 1–114
  Walter Freyn, Tobias Hartnick, Max Horn, Ralf Köhl
  
• Higher spin representations in Kac–Moody theory, PhD thesis, Justus-Liebig-Universität Gießen 2021
  Robin Lautenbacher
  
• Representations of involutory subalgebras of affine Kac-Moody algebras
  Axel Kleinschmidt, Ralf Köhl, Robin Lautenbacher, Hermann Nicolai
  
• Fundamental groups of split real Kac-Moody groups and generalized real flag manifolds. With appendices by Tobias Hartnick and Ralf Köhl and by Julius Grüning and Ralf Köhl, in: Transf. Groups (2022)
  Paula Harring, Ralf Köhl