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We study reductive group actions on complex affine quadrics.
      
Presentations for crystallographic complex reflection groups
      
We find presentations for the irreducible crystallographic complex reflection groupsW whose linear part is not the complexification of a real reflection group.
      
The presentations are given in the form of graphs resembling Dynkin diagrams and very similar to the presentations for finite complex reflection groups given in [2].
      
Invariant analytic domains in complex semisimple groups
      
LetH? be a real form of a complex semisimple group.
      
Gindikin that complex analytic objects related to these domains will provide explicit realizations of unitary representations ofH?.
      
Motivated by the physical concept of special geometry, two mathematical constructions are studied which relate real hypersurfaces to tube domains and complex Lagrangian cones, respectively.
      
If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (Theorem 1).
      
If the variety is a complex affine space and the ring of invariants is isomorphic to a polynomial ring, then the action is conjugate to a translation (Theorem 3).
      
We prove as the main result thatM is weakly symmetric with respect toG1 and complex conjugation.
      
Cayley transforms and orbit structure in complex flag manifolds
      
LetZ=G/Q be a complex flag manifold andG0 a real form ofG.
      
These results are applied to the theory ofcompactly causal symmetric spaces: we describe explicitly the complex domain Ξ associated to such a space.
      
We describe a procedure for constructing monomial bases for finite dimensional irreducible representations of complex semisimple Lie algebras.
      
We investigate holomorphic self-maps of complex manifolds of the formG/Γ whereG is a complex Lie group and Γ a lattice.
      
We construct essentially all the irreducible modules for the multiparameter quantum function algebraF?φ[G], whereG is a simple simply connected complex algebraic group, and ? is a root of unity.
      
Using the path model and the theory of crystals, we generalize the concept of patterns to arbitrary complex semi-simple algebraic groups.
      
LetG be a complex reductive Lie group with maximal compact subgroupK andG×X →X a holomorphic action on a Stein manifoldX.
      
We consider actions of compact real Lie GroupsK on complex spacesX such that the associated reducedK-space admits a semistable quotient, e.g.X is a Stein space.
      
 

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