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LOCAL SYSTEMS ON NILPOTENT ORBITS AND WEIGHTED DYNKIN DIAGRAMS
 

Summary: LOCAL SYSTEMS ON NILPOTENT ORBITS AND WEIGHTED
DYNKIN DIAGRAMS
PRAMOD N. ACHAR AND ERIC N. SOMMERS
1. Introduction
Let G be a reductive algebraic group over the complex numbers, B a Borel subgroup of
G, and T a maximal torus of B. We denote by = (G) the weight lattice of G with
respect to T, and by + = +(G) the set of dominant weights with respect to the positive
roots defined by B. Let g be the Lie algebra of G, and let N denote the nilpotent cone in g.
Now, let e N be a nilpotent element, and let Oe be the orbit of e in g under the adjoint
action of G. We write Ge for the centralizer of e in G. Let No denote the set of nilpotent
orbits in g, and No,r the set of G-conjugacy classes of pairs
{(e, ) | e N and an irreducible rational representation of Ge
}.
Lusztig [9] conjectured the existence of a bijection No,r + using his work on cells
in affine Weyl groups. From the point of view of Harish-Chandra modules, Vogan also
conjectured a bijection between No,r and +. Such a bijection has been established by
Bezrukavnikov in two preprints (the bijections in each preprint are conjecturally the same)
[2], [3]. Bezrukavnikov's second bijection is closely related to Ostrik's conjectural descrip-
tion of the bijection [12] (see also [4]). In the case of G = GL(n, C), the first author
[1] described an explicit combinatorial bijection between No,r and + from the Harish-

  

Source: Achar, Pramod - Department of Mathematics, Louisiana State University

 

Collections: Mathematics