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We apply these results to intersection theory on varieties with group actions, especially to Schubert calculus and its generalizations.
      
An important class consists of those that we calln-coset groups; they arise as orbit spaces of groupsG modulo a group of automorphisms withn elements.
      
We see that the theory ofn-valued groups is distinct from that of groups with a given automorphism group.
      
There are natural concepts of the action of ann-valued group on a space and of a representation in an algebra of operators.
      
We introduce the (purely algebraic) notion of ann-Hopf algebra and show that the ring of functions on ann-valued group and, in the topological case, the cohomology has ann-Hopf algebra structure.
      
The cohomology algebra of the classifying space of a compact Lie group admits the structure of ann-Hopf algebra, wheren is the order of the Weyl group; the homology with dual structure is also ann-Hopf algebra.
      
In general the group ring of ann-valued group is not ann-Hopf algebra but it is for ann-coset group constructed from an abelian group.
      
Using the properties ofn-Hopf algebras we show that certain spaces do not admit the structure of ann-valued group and that certain commutativen-valued groups do not arise by applying then-coset construction to any commutative group.
      
LetG be a connected, simply-connected, real semisimple Lie group andK a maximal compactly embedded subgroup ofG such thatD=G/K is a hermitian symmetric space.
      
In this paper we explicitly determine the virtual representations of the finite Weyl subgroups of the affine Weyl group on the cohomology of the space of affine flags containing a family of elementsnt in an affine Lie algebra.
      
Suppose thatG0 is the analytic automorphism group of an irreducible bounded symmetric domain and that some openG0-orbit onZ is a semisimple symmetric space.
      
We investigate holomorphic self-maps of complex manifolds of the formG/Γ whereG is a complex Lie group and Γ a lattice.
      
We construct essentially all the irreducible modules for the multiparameter quantum function algebraF?φ[G], whereG is a simple simply connected complex algebraic group, and ? is a root of unity.
      
LetG be a complex reductive Lie group with maximal compact subgroupK andG×X →X a holomorphic action on a Stein manifoldX.
      
We study discrete (Kleinian) subgroups of the isometry group Iso+H4 of the real hyperbolic space of dimension 4.
      
We prove that the group of invertible elements of an irreducible algebraic monoid is an algebraic group, open in the monoid.
      
Moreover, if this group is reductive, then the monoid is affine.
      
We prove that, for a real reductive algebraic group, they can be characterized as the spaces of real points of affine spherical homogeneous varieties of the complexified group.
      
As an application, under the same assumption on the transitive group, we show that weakly symmetric spaces are precisely the homogeneous Riemannian manifolds with commutative algebra of invariant differential operators.
      
LetG be a classical algebraic group defined over an algebraically closed field.
      
 

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