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In the second example, we obtain a proof of the Chalyh-Veselov conjecture that the Calogero-Moser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko.
      
As an application of the results we prove a generalization of Chevalley's restriction theorem for the classical Lie algebras.
      
We apply these results to intersection theory on varieties with group actions, especially to Schubert calculus and its generalizations.
      
In this paper, we prove three types of rigidity results related to CAT(-1) spaces, namely the rigidity of the isometric actions on CAT(-1) spaces under the commensurability subgroups, the higher rank lattices and certain ergodic cocycles.
      
This extends the results and simplifies the proof for the classical orbit structure description of [10] and [11], which applies whenF=Z.
      
These results are applied to the theory ofcompactly causal symmetric spaces: we describe explicitly the complex domain Ξ associated to such a space.
      
We also use known results about canonical bases forUq2 to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmannians), due to Lascoux-Schützenberger and Zelevinsky.
      
This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of.
      
We also prove cohomology vanishing results for line bundles on the compactification.
      
In the final section the theorem is applied to gradient actions on other homogeneous spaces and we show, that Hilgert's Convexity Theorem for moment maps can be derived from the results.
      
Results of this kind were first obtained by Moore and Seiberg, but their paper contains serious gaps.
      
We study their general properties and apply these results to Schubert varieties.
      
In the last section we give an exposition of results, communicated to us by J.-P.
      
In this paper we establish two results concerning algebraic (?,+)-actions on ?n.
      
This provides a generalization of results of [9], [7], [6].
      
The proof uses basic results about algebraic surfaces.
      
Our results show that in general Cartan components are small.
      
Therefore, our results give in particular a unified construction for many moduli spaces considered in the literature.
      
Finally, we study their reducibility of the action of the Casimirs on the zero-weight spaces of self-dual g-modules and obtain complete classification results for g = sln and g2.
      
Whether the corresponding results hold in positive characteristic is not known.
      
 

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