by 
The method applies to the standard arithmetic subgroups ofSO(n,1) (a case which was proved previously by Millson [Mi]), to the nonarithmetic lattices inSO(n,1) constructed by Gromov and PiatetskiShapiro [GPS] and to groups generated by reflections.


Open subsets of projective spaces with a good quotient by an action of a reductive group


As in the case of affine Weyl groups, they can be obtained by adding a further node to the diagram for the linear part.


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.


Then the ring of invariants ofG is a polynomial ring if and only ifG is generated by pseudoreflections and the pointwise stabilizer inG of any nontrivial subspace has a polynomial ring of invariants.


For the case of positive characteristic we use the classification of finite irreducible groups generated by pseudoreflections due to Kantor, Wagner, Zalesski? and Sere?kin.


A localglobal principle is proved by the second named author in the adjacent paper of this volume.


In the interesting case when the group is of Coxeter typeDn (n≥4) we use higher polarization operators introduced by Wallach.


The setX+ of dominant weights is a union of closed alcoves numbered by the elementsw∈Wf of a certain subset of affine Weyl groupW.


For a cellA?Wf we consider a full subcategory formed by direct sums of tilting modulesQ(λ) with highest weights.


The proof is an application of a recent result by W.


Using the properties ofnHopf algebras we show that certain spaces do not admit the structure of annvalued group and that certain commutativenvalued groups do not arise by applying thencoset construction to any commutative group.


As a result, this provides new examples of weakly symmetric spaces that are nonsymmetric, including those already discovered by Selberg (cf.


Then theG0orbit structure ofZ is described explicitly by the partial Cayley transforms of a certain hermitian symmetric subflagF?Z.


The application arises because of a very strong homological property enjoyed by certain cell filtrations forqpermutation modules.


In the present article we propose a more detailed proof of this fact than the one given by Varagnolo and Vasserot.


Similarly, by a specific choice of the parameter, the level (1,0) vertex representation of the quantum totoidal algebra gives rise to a structure on irreducible level1 highest weightmodules.


We show that they are induced by automorphisms ofG and that a surjective holomorphic selfmap can be nonbijective only in the directions of the nilradical ofG.


Finally the cubic surfaces of each component ofQ are studied in details by determining their stabilizers, their rational representations and whether they can be expressed as the determinant of a 3×3 matrix of linear forms.


LetRo andR1 be two KempfNess sets arising from moment maps induced by strictly plurisubharmonic,Kinvariant, proper functions.

