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Affine modifications and affine hypersurfaces with a very transitive automorphism group
      
We obtain a criterion for rational smoothness of an algebraic variety with a torus action, with applications to orbit closures in flag varieties, and to closures of double classes in regular group completions.
      
The symmetric varieties considered in this paper are the quotientsG/H, whereG is an adjoint semi-simple group over a fieldk of characteristic ≠ 2, andH is the fixed point group of an involutorial automorphism ofG which is defined overk.
      
Let g be a simple finite-dimensional complex Lie algebra and letG be the corresponding simply-connected algebraic group.
      
It is well-known that the ring of invariants associated to a non-modular representation of a finite group is Cohen-Macaulay and hence has depth equal to the dimension of the representation.
      
Weyl group extension of quantized current algebras
      
We prove the following converse: ifG is a reductive group andK[V]G is Cohen-Macaulay for every moduleV, thenG is linearly reductive.
      
More generally, we prove that if Γ is an irreducible arithmetic non-cocompact lattice in a higher rank group, then Γ containsf·g·p·d·f groups.
      
We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.
      
We then go on to identify the group up to commensurability class.
      
The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type.
      
First we prove that in characteristicp>amp;gt;0 a module with good filtration for a group of type E6 restricts to a module with good filtration for a group of type F4.
      
LetG be a reductive algebraic group and letH be a reductive subgroup ofG.
      
For this class of Lie Groups we prove the Auslander Conjecture whenever the dimensionn of the group is odd and we show that it is false forn even andn>amp;gt;2.
      
We study the dynamics of the automorphisms group of K3 surfaces.
      
LetD be a Hermitian symmetric space of tube type,S its Shilov boundary andG the neutral component of the group of bi-holomorphic diffeomorphisms ofD.
      
In the model situationD is the Siegel disc,S is the manifold of Lagrangian subspaces andG is the symplectic group.
      
It is also proved that the group of holomorphic automorphisms ofG?υ which commute withG? acts transitively on the set of polynomially convexG-orbits.
      
We generalize to the case of a symmetric variety the construction of the enveloping semigroup of a semisimple algebraic group due to E.
      
LetG be a simply connected semisimple complex algebraic group.
      
 

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