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Tilting modules for classical groups and howe duality in positive characteristics
      
We use the theory of tilting modules for algebraic groups to propose a characteristic free approach to "Howe duality" in the exterior algebra.
      
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.
      
This study was continued in the paper [FKRW] in the framework of vertex algebra theory.
      
In the present paper we study the remaing nontrivial case, that of a negative central charge-N.
      
In this note we present a very simple method of proving that some hyperbolic manifoldsM have finite sheeted covers with positive first Betti number.
      
In all these cases we actually show that Γ=π1(M) has a finite index subgroup which is mapped onto a nonabelian free group.
      
Affine weyl groups and conjugacy classes in Weyl groups
      
We express them in terms of generatorsEij ofU(gl(n)) and as differential operators on the space of matrices These expressions are a direct generalization of the classical Capelli identities.
      
They result in many nontrivial properties of quantum immanants.
      
In the case of 4-dimensional anticommutative algebras a construction is given that links the associated cubic surface and the 27 lines on it with the structure of subalgebras of the algebra.
      
In the case of 3-dimensional commutative algebras a new proof of a recent theorem of Katsylo and Mikhailov about the 28 bitangents to the associated plane quartic is given.
      
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.
      
Such an action is called linearizable if it is equivalent to the restriction of a linear orthogonal action in the ambient affine space of the quadric.
      
In this paper, we prove the degenerations of Schubert varieties in a minusculeG/P, as well as the class of Kempf varieties in the flag varietySL(n)/B, to (normal) toric varieties.
      
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].
      
As in the case of affine Weyl groups, they can be obtained by adding a further node to the diagram for the linear part.
      
We then classify the reflections in the groupsW and the minimal number of them needed to generateW, using the diagrams.
      
Invariant analytic domains in complex semisimple groups
      
We study certain naturally-defined analytic domains in the complexified groupHC which are invariant under left and right translation byH?.
      
 

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