14 research outputs found
Prehomogeneous vector spaces and ergodic theory II
We apply M. Ratner's theorem on closures of unipotent orbits to the study of
three families of prehomogeneous vector spaces. As a result, we prove analogues
of the Oppenheim Conjecture for simultaneous approximation by values of certain
alternating bilinear forms in an even number of variables and certain
alternating trilinear forms in six and seven variables
Computing the associatied cycles of certain Harish-Chandra modules
Let be a simple real linear Lie group with maximal compact
subgroup and assume that . In \cite{MPVZ} we proved that for any representation
of Gelfand-Kirillov dimension ,
the polynomial on the dual of a compact Cartan subalgebra given by the
dimension of the Dirac index of members of the coherent family containing
is a linear combination, with integer coefficients, of the multiplicities of
the irreducible components occurring in the associated cycle. In this paper we
compute these coefficients explicitly
The Dirac cohomology of a finite dimensional representation
International audienc
Smooth components of Springer fibers
This article studies components of Springer fibers for gl(n) that are associated to closed orbits of GL(p) X GL(q) on the flag variety of GL(n), n = p + q. These components occur in any Springer fiber. In contrast to the case of arbitrary components, these components are smooth varieties. Using results of Barchini and Zierau we show these components are iterated bundles and are stable under the action of a maximal torus of GL(n). We prove that if L is a line bundle on the flag variety associated to a dominant weight, then the higher cohomology groups of the restriction of L to these components vanish. We derive some consequences of localization theorems in equivariant cohomology and K-theory, applied to these components. In the appendix we identify the tableaux corresponding to these components, under the bijective correspondence between components of Springer fibers for GL(n) and standard tableaux.Peer reviewedMathematic