Journal of Lie Theory 18 (2008), No. 1, 083--091
Copyright Heldermann Verlag 2008
A Simple Proof of the Algebraic Version of a Conjecture by Vogan
Tim Bratten
Facultad de Ciencias Exactas, UNICEN, B-7000 Tandil - Buenos Aires, Argentina
bratten@exa.unicen.edu.ar
Sergio Corti
Facultad de Ciencias Exactas, UNICEN, B-7000 Tandil - Buenos Aires, Argentina
csdaniel2003@yahoo.com.ar
D. Vogan ["Unitary representations and complex analysis", Notes from the Cime summer school, Venice, Italy 2004] conjectured that four canonical globalizations of Harish-Chandra modules commute with certain n-cohomology groups. In this article we prove that Vogan's conjecture holds for one of the globalizations if and only if it holds for the dual. Using a previously published result of one of the authors, which establishes the conjecture for the minimal globalization, we can therefore deduce Vogan's conjecture for the maximal globalization.
Keywords: Representations of reductive Lie groups, n-homology groups, globalizations of Harish-Chandra modules.
MSC: 22E46
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