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In our paper [KR] we began a systematic study of representations of the universal central extension[InlineEquation not available: see fulltext.] of the Lie algebra of differential operators on the circle.
      
Finally we show for more than half of the infinite series that a presentation for the fundamental group of the space of regular orbits ofW can be derived from our presentations.
      
We apply our methods to the Dickson, upper triangular and symmetric coinvariants.
      
Andersen's theorem about the ideal of negligible modules which in our notations is nothing else then.
      
The main idea for our approach relies on a study of the boundary theory we established for the general CAT(-1) spaces.
      
As our main result, we prove that every coherentK-sheaf onX extends uniquely to a holomorphicG-sheaf onXc.
      
Our proof is based on a different approach and is much more rigorous.
      
Our main achievement is that any Schubert variety admits a flat deformation to a union of normal toric varieties.
      
For example, if our groupG isSn, these objects are field extensions; ifG=On, they are quadratic forms; ifG=PGLn, they are division algebras (all of degreen); ifG=G2, they are octonion algebras; ifG=F4, they are exceptional Jordan algebras.
      
Of course, our proof would work also in the finite type case.
      
Our proof uses the connection between this variety and the punctual Hilbert scheme of a smooth algebraic surface.
      
Our method relies on the canonical Frobenius splittings of Mathieu.
      
Our basic tool is Lusztig's canonical basis and the string parametrization of this basis.
      
Our results show that in general Cartan components are small.
      
Our lower bound comes from applying a decomposition theorem for "hyperbolic localization" [Br] to this torus action.
      
Our basic tool is the representation theory of the Burkhardt group G = G25 920, which acts on our varieties.
      
In the examples which have been studied so far, our semistability concept reproduces the known ones.
      
Therefore, our results give in particular a unified construction for many moduli spaces considered in the literature.
      
The case of some quasi-projective toric surfaces such as the affine plane are described by our method too.
      
For G simple, our main result is the classification of the G-modules V and integers k ≥ 2 such that
      
 

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