Symmetric spaces of exceptional groups

Symmetric Spaces of Exceptional Groups

We adress the problem of the reasons for the existence of 12 symmetric spaces with the exceptional Lie groups. The 1+2 cases for G2 and F4 respectively are easily explained from the octonionic nature of these groups. The 4+3+2 cases on the E6,7,8 series require the magic square of Freudenthal and, for the split case, an appeal to the supergravity chain in 5, 4 and 3 spacetime dimensions.

Groups to Symmetric Spaces

This paper is based on a talk given at the conference ”Representation theory of real reductive groups”, Salt Lake City, July 2009. We fix an algebraically closed field k of characteristic exponent p. (We assume, except in §17, that either p = 1 or p ≫ 0.) We also fix a symmetric space that is a triple (G, θ,K) where G is a connected reductive algebraic group over k, θ : G −→ G is an involution ...

From Groups to Symmetric Spaces

This paper is based on a talk given at the conference ”Representation theory of real reductive groups”, Salt Lake City, July 2009. We fix an algebraically closed field k of characteristic exponent p. (We assume, except in §18, that either p = 1 or p ≫ 0.) We also fix a symmetric space that is a triple (G, θ,K) where G is a connected reductive algebraic group over k, θ : G −→ G is an involution ...

Rational Homotopy Groups of Generalised Symmetric Spaces

We obtain explicit formulas for the rational homotopy groups of generalised symmetric spaces, i.e., the homogeneous spaces for which the isotropy subgroup appears as the fixed point group of some finite order automorphism of the group. In particular, this gives explicit formulas for the rational homotopy groups of all classical compact symmetric spaces.

Discrete Groups, Symmetric Spaces, and Global Holonomy