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ORBIT CLOSURES IN THE ENHANCED NILPOTENT CONE PRAMOD N. ACHAR AND ANTHONY HENDERSON
 

Summary: ORBIT CLOSURES IN THE ENHANCED NILPOTENT CONE
PRAMOD N. ACHAR AND ANTHONY HENDERSON
Abstract. We study the orbits of G = GL(V ) in the enhanced nilpotent cone
V × N, where N is the variety of nilpotent endomorphisms of V . These orbits
are parametrized by bipartitions of n = dim V , and we prove that the closure
ordering corresponds to a natural partial order on bipartitions. Moreover, we
prove that the local intersection cohomology of the orbit closures is given by
certain bipartition analogues of Kostka polynomials, defined by Shoji. Finally,
we make a connection with Kato's exotic nilpotent cone in type C, proving
that the closure ordering is the same, and conjecturing that the intersection
cohomology is the same but with degrees doubled.
1. Introduction
Many features of the representation theory of an algebraic group are known to
be controlled by the geometry of its nilpotent cone. In particular, the Springer
correspondence, as developed by Borho­MacPherson and Lusztig, relates the local
intersection cohomology of the nilpotent orbit closures to composition multiplicities
in representations of the associated Weyl group (see the survey article [Sh1]). The
correspondence in types B/C is more complicated than that in type A, in a number
of respects: for instance, Weyl group representations are no longer in bijection with
nilpotent orbits, and the concise algebraic description of the Weyl group action on

  

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

 

Collections: Mathematics