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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 callncoset groups; they arise as orbit spaces of groupsG modulo a group of automorphisms withn elements.


We see that the theory ofnvalued groups is distinct from that of groups with a given automorphism group.


There are natural concepts of the action of annvalued group on a space and of a representation in an algebra of operators.


We introduce the (purely algebraic) notion of annHopf algebra and show that the ring of functions on annvalued group and, in the topological case, the cohomology has annHopf algebra structure.


The cohomology algebra of the classifying space of a compact Lie group admits the structure of annHopf algebra, wheren is the order of the Weyl group; the homology with dual structure is also annHopf algebra.


In general the group ring of annvalued group is not annHopf algebra but it is for anncoset group constructed from an abelian group.


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.


LetG be a connected, simplyconnected, 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 openG0orbit onZ is a semisimple symmetric space.


We investigate holomorphic selfmaps 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.

