from 
We define a map from an affine Weyl group to the set of conjugacy classes of an ordinary Weyl group.


We also show how to distinguish examples of open subsets with a good quotient coming from Mumford's theory and give examples of open subsets with nonquasiprojective quotients.


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.


This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry.


However, there are many examples that do not arise from this construction.


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


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


We prove a more general version of a result announced without proof in [DP], claiming roughly that in a partially integrable highest weight module over a KacMoody algebra the integrable directions from a parabolic subalgebra.


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


This fact is deduced from results about equivariantDmodules supported on the nilpotent cone of.


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 the final section the theorem is applied to gradient actions on other homogeneous spaces and we show, that Hilgert's Convexity Theorem for moment maps can be derived from the results.


From it, we recover Joseph and Letzter's result by a kind of "quantum duality principle".


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.


These are analogous to "fusion rules" in tensor product decomposition and their derivation obtains from an analysis of theRmatrix.


From Lie algebras of vector fields to algebraic group actions


Lower bounds for KazhdanLusztig polynomials from patterns


Our lower bound comes from applying a decomposition theorem for "hyperbolic localization" [Br] to this torus action.


For this, we show that these multiplicities are bounded from above by the dimensions of certain Demazure modules.


The operation adj on matrices arises from the (n  1)st exterior power functor on modules; the analogous factorization question for matrix constructions arising from other functors is raised, as are several other questions.

