Abstract
Ol'shanskiǐ's semigroup plays a prominent role in the discussion of symmetric spaces. There are two important applications: the construction of the discrete series in representation theory and analysis on symmetric spaces. In this article the notion of an Ol'shanskiǐ wedge in a symmetric Lie algebra is defined. The tangent wedge of Ol'shanskiǐ's semigroup is an example of such a wedge. In Section 1 of this paper, a geometric characterization of Ol'shanskiǐ wedges is given and their relation to special Lie wedges is established. Section 2 deals with invariant Ol'shanskiǐ wedges and Ol'shanskiǐ semialgebras. We give a complete classification of symmetric Lie algebras supporting invariant Ol'shanskiǐ wedges, resp., Ol'shanskiǐ semialgebras.
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