show that 
In all these cases we actually show that Γ=π1(M) has a finite index subgroup which is mapped onto a nonabelian free group.


We show that wonderful varieties are necessarily spherical (i.e., they are almost homogeneous under any Borel subgroup ofG).


In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues.


As an example of how this can be used, we show that the ring of invariants (under the adjoint action of SL (3)) ofg copies ofM3 is CM.


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.


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.


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.


For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalqdifference equation.


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.


We show that there is a complex spaceXcendowed with a holomorphic action of the universal complexificationG ofK that containsX as an openKstable subset.


We show that on each Schubert cell, the corresponding Kostant harmonic form can be described using only data coming from the Bruhat Poisson structure.


In this paper we compute the cohomology with trivial coefficients for the Lie superalgebraspsl(n, n), p (n) andq(2n); we show that the cohomology ring ofq(2n+1) is of Krull dimension 1 and we calculate the ring forq(3) andq(5).


We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.


The main goal of this paper is to show that this construction produces many new Gelfand pairs associated with nilpotent Lie groups.


First we show that the representation ofG×G on eachGbiinvariant irreducible reproducing kernel Hilbert space in Hol(D) is a highest weight representation whose kernel is the character of a highest weight representation ofG.


We show that the algebras of semiinvariants of a finite connected quiverQ are complete intersections if and only ifQ is of Dynkin or Euclidean type.


In the course of the proof we show that one can reduce the study of generating semiinvariants to the case when the quiver has no oriented paths of length greater than one.


We describe a basis forY, show that it is a polynomial algebra and describe its rank, which we compute explicitly in a number of cases.


We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in ?(G) are diagonalizable.


For this class of Lie Groups we prove the Auslander Conjecture whenever the dimensionn of the group is odd and we show that it is false forn even andn>amp;gt;2.

