that 
The purpose of this note is to prove, as Lusztig stated, that the Euler characteristic of the variety of Iwahori subalgebras containing a certain nilelliptic elementnt istcl wherel is the rank of the associated finite type Lie algebra.


In the present paper we study the remaing nontrivial case, that of a negative central chargeN.


In this note we present a very simple method of proving that some hyperbolic manifoldsM have finite sheeted covers with positive first Betti number.


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


In the case of 4dimensional anticommutative algebras a construction is given that links the associated cubic surface and the 27 lines on it with the structure of subalgebras of the algebra.


We prove that if a reductive group action on an affine quadric with a 1dimensional quotient has a linear model, then the action is linearizable.


As a consequence, we obtain that determinantal varietes degenerate to (normal) toric varieties.


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


Finally we show for more than half of the infinite series that a presentation for the fundamental group of the space of regular orbits ofW can be derived from our presentations.


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.


Using these monomial bases we prove that the image of the transfer for a general linear group over a finite field is a principal ideal in the ring of invariants.


As an application we produce complete hyperbolic 5manifolds that are nontrivial plane bundles over closed hyperbolic 3manifolds and conformally flat 4manifolds that are nontrivial circle bundles over closed hyperbolic 3manifolds.


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


This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues.


In the second example, we obtain a proof of the ChalyhVeselov conjecture that the CalogeroMoser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko.


We conjecture that this is also true for the exceptional reflection groups and then sketch a proof for the group of typeF4.


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.


As a corollary we obtain that the moduli space of rank 3 and degree 0 bundles on a smooth projective curve of genusg is CM.


In this paper we prove that the homogeneous spaceG/K has a structure of a globally symmetric space for every choice ofG andK, especially forG being compact.

