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As in the case of Mumford's geometric invariant theory (which concerns projective good quotients) the problem can be reduced to the case of an action of a torus.
      
We also show how to distinguish examples of open subsets with a good quotient coming from Mumford's theory and give examples of open subsets with non-quasi-projective quotients.
      
Cones of highest weight vectors, weight polytopes, and Lusztig's q-analog
      
As an application of the results we prove a generalization of Chevalley's restriction theorem for the classical Lie algebras.
      
Let g be a Lie algebra,S(g) the symmetric algebra,U(g) the universal enveloping algebra, andZ(g) the center ofU(g).
      
The aim of this paper is to discuss a construction of a class of linear isomorphisms σ:S(g)→U(g) which commute with the adjoint representation.
      
We study Edidin and Graham's equivariant Chow groups in the case of torus actions.
      
Andersen's theorem about the ideal of negligible modules which in our notations is nothing else then.
      
We consider 3-parametric polynomialsPμ*(x; q, t, s) which replace theAn-series interpolation Macdonald polynomialsPμ*(x; q, t) for theBCn-type root system.
      
Ass → ∞ the polynomialsPμ*(x; q, t, s) becomePμ*(x; q, t).
      
The proof is based on a variant of Moser's method using time-dependent vector fields.
      
Lichtenstein in the caseu =(n, ?) or(n?), we prove that ?(q).ζ1/2 is non zero for all harmonic polynomialsq ∈S() \ {0}.
      
We also prove the shifted cocycle condition for the twistors, thereby completing Fr?nsdal's findings.
      
Coordinates on Schubert cells, Kostant's harmonic forms, and the Bruhat Poisson structure onG/B
      
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.
      
Another proof of Joseph and Letzter's separation of variables theorem for quantum groups
      
Joseph and Letzter extended Kostant's theorem to the case of the quantized enveloping algebra of g.
      
Using the theory of crystal bases as the main tool, we prove a quantum analogue of Richardson's theorem.
      
From it, we recover Joseph and Letzter's result by a kind of "quantum duality principle".
      
A counterexample to Hilbert's Fourteenth Problem in dimension six
      
 

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