204 research outputs found
The global nilpotent variety is Lagrangian
The purpose of this note is to present a short elementary proof of a theorem
due to Faltings and Laumon, saying that the global nilpotent cone is a
Lagrangian substack in the cotangent bundle of the moduli space of G-bundles on
a complex compact curve. This result plays a crucial role in the Geometric
Langlands program, due to Beilinson-Drinfeld, since it insures that the
D-modules on the moduli space of G-bundles whose characteristic variety is
contained in the global nilpotent cone are automatically holonomic, hence, e.g.
have finite length.Comment: LaTeX, 9pp. Final version, to appear in Duke Math.
Nil Hecke algebras and Whittaker D-modules
Given a reductive group G, Kostant and Kumar defined a nil Hecke algebra that
may be viewed as a degenerate version of the double affine nil Hecke algebra
introduced by Cherednik. In this paper, we construct an isomorphism of the
spherical subalgebra of the nil Hecke algebra with a Whittaker type quantum
Hamiltonian reduction of the algebra of differential operators on G. This
result has an interpretation in terms of the geometric Satake and the Langlands
dual group. Specifically, the isomorphism provides a bridge between very
differently looking descriptions of equivariant Borel-Moore homology of the
affine flag variety (due to Kostant and Kumar) and of the affine Grassmannian
(due to Bezrukavnikov and Finkelberg), respectively.
It follows from our result that the category of Whittaker D-modules on G
considered by Drinfeld is equivalent to the category of holonomic modules over
the nil Hecke algebra, and it is also equivalent to a certain subcategory of
the category of Weyl group equivariant holonomic D-modules on the maximal
torus.Comment: Final version, 34p
Principal Nilpotent pairs in a semisimple Lie algebra, I
This is the first of a series of papers devoted to certain pairs of commuting
nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable
properties and which are expected to play a major role in Representation
theory. The properties of these pairs and their role is similar to those of the
principal nilpotents. To any principal nilpotent pair we associate a
two-parameter analogue of the Kostant partition function, and propose the
corresponding two-parameter analogue of the weight multiplicity formula.
In a different direction, each principal nilpotent pair gives rise to a
harmonic polynomial on the Cartesian square of the Cartan subalgebra, that
transforms under an irreducible representation of the Weyl group. In the
special case of sl_n, the conjugacy classes of principal nilpotent pairs and
the irreducible representations of the Symmetric group, S_n, are both
parametrised (in a compatible way) by Young diagrams. In general, our theory
provides a natural generalization to arbitrary Weyl groups of the classical
construction of simple S_n-modules in terms of Young's symmetrisers.Comment: new section on Partial Slices added; minor corrections made; 46 pp.,
AMS-TeX, 1 eps figur
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