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We use the theory of tilting modules for algebraic groups to propose a characteristic free approach to "Howe duality" in the exterior algebra.
      
The purpose of this note is to prove, as Lusztig stated, that the Euler characteristic of the variety of Iwahori subalgebras containing a certain nil-elliptic elementnt istcl wherel is the rank of the associated finite type Lie algebra.
      
The method applies to the standard arithmetic subgroups ofSO(n,1) (a case which was proved previously by Millson [Mi]), to the non-arithmetic lattices inSO(n,1) constructed by Gromov and Piatetski-Shapiro [GPS] and to groups generated by reflections.
      
We define a map from an affine Weyl group to the set of conjugacy classes of an ordinary Weyl group.
      
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.
      
The aim of the paper is to describe all open subsets of a projective space with an action of a reductive group which admits a good quotient.
      
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.
      
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.
      
Degenerations of flag and Schubert varieties to toric varieties
      
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.
      
As a consequence, we obtain that determinantal varietes degenerate 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.
      
Gindikin that complex analytic objects related to these domains will provide explicit realizations of unitary representations ofH?.
      
We develop methods that give rise to natural monomial bases for such rings over their ground fields and explicitly determine precisely which monomials are zero in the ring of coinvariants.
      
We apply our methods to the Dickson, upper triangular and symmetric coinvariants.
      
Motivated by the physical concept of special geometry, two mathematical constructions are studied which relate real hypersurfaces to tube domains and complex Lagrangian cones, respectively.
      
The theory is applied to the case of cubic hypersurfaces, which is the one most relevant to special geometry, obtaining the solution of the two classification problems and the description of the corresponding homogeneous special K?hler manifolds.
      
 

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