Journal of Lie Theory Volume 6 (1996) 69–146 C ◯1996 Heldermann Verlag

On Hyperbolic Cones and Mixed Symmetric Spaces

Bernhard Krötz and Karl-Hermann Neeb

Communicated by J. Hilgert

Abstract. A real semisimple Lie algebra 𝔤 admits a Cartan involution, θ, for which the corresponding eigenspace decomposition 𝔤=𝔨+𝔭 has the property that all operators ad X, X∈𝔭 are diagonalizable over ℝ. We call such elements hyperbolic, and the elements X∈𝔨 are elliptic in the sense that ad X is semisimple with purely imaginary eigenvalues. The pairs (𝔤,θ) are examples of symmetric Lie algebras, i.e., Lie algebras endowed with an involutive automorphism, such that the −1-eigenspace of θ contains only hyperbolic elements. Let (𝔤,τ) be a symmetric Lie algebra and 𝔤= 𝔥+𝔮 the corresponding eigenspace decomposition for τ. The existence of “enough” hyperbolic elements in 𝔮 is important for the structural analysis of symmetric Lie algebras in terms of root decompositions with respect to abelian subspaces of 𝔮 consisting of hyperbolic elements. We study the convexity properties of the action of Inn_{𝔤}(𝔥) on the space 𝔮. The key role will be played by those invariant convex subsets of 𝔮 whose interior points are hyperbolic.

B. Krötz, K. - H. Neeb Mathematisches Institut Universität Erlangen-Nürnberg Bismarckstr. 1\frac{1}{2} D-91054 Erlangen Germany

Received May 5, 1996 and in final form June 11, 1996

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag