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Invariant orders in Lie groups

Neeb, Karl-Hermann — 1991

Proceedings of the Winter School "Geometry and Physics"

[For the entire collection see Zbl 0742.00067.]The author formulates several theorems about invariant orders in Lie groups (without proofs). The main theorem: a simply connected Lie group G admits a continuous invariant order if and only if its Lie algebra L ( G ) contains a pointed invariant cone. V. M. Gichev has proved this theorem for solvable simply connected Lie groups (1989). If G is solvable and simply connected then all pointed invariant cones W in L ( G ) are global in G (a Lie wedge W ⊂ L ( G ) is said to...

On the complex and convex geometry of Ol'shanskii semigroups

Karl-Hermann Neeb — 1998

Annales de l'institut Fourier

To a pair of a Lie group G and an open elliptic convex cone W in its Lie algebra one associates a complex semigroup S = G Exp ( i W ) which permits an action of G × G by biholomorphic mappings. In the case where W is a vector space S is a complex reductive group. In this paper we show that such semigroups are always Stein manifolds, that a biinvariant domain D ⊆ S is Stein is and only if it is of the form G Exp ( D h ) , with D h ⊆ i W convex, that each holomorphic function on D extends to the smallest biinvariant Stein domain containing D ,...

Globality in semisimple Lie groups

Karl-Hermann Neeb — 1990

Annales de l'institut Fourier

In the first section of this paper we give a characterization of those closed convex cones (wedges) W in the Lie algebra s l ( 2 , R ) n which are invariant under the maximal compact subgroup of the adjoint group and which are controllable in the associated simply connected Lie group S l ( 2 , R ) n ∼ , i.e., for which the subsemigroup S = ( exp W ) generated by the exponential image of W agrees with the whole group G (Theorem 13). In Section 2 we develop some algebraic tools concerning real root decompositions with respect to compactly...

On the complex geometry of invariant domains in complexified symmetric spaces

Karl-Hermann Neeb — 1999

Annales de l'institut Fourier

Let M = G / H be a real symmetric space and 𝔤 = 𝔥 + 𝔮 the corresponding decomposition of the Lie algebra. To each open H -invariant domain D 𝔮 ⊆ i 𝔮 consisting of real ad-diagonalizable elements, we associate a complex manifold Ξ ( D 𝔮 ) which is a curved analog of a tube domain with base D 𝔮 , and we have a natural action of G by holomorphic mappings. We show that Ξ ( D 𝔮 ) is a Stein manifold if and only if D 𝔮 is convex, that the envelope of holomorphy is schlicht and that G -invariant plurisubharmonic functions correspond to convex H -invariant...

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