Summary: Zentralblatt MATH Database 1931 2009
c 2009 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag
Vogan, David A.jun.
Isolated unitary representations. (English)
Sarnak, Peter (ed.) et al., Automorphic forms and applications. Papers of the IAS/Park
City Mathematics Institute, Park City, UT, USA, July 120, 2002. Providence, RI:
American Mathematical Society (AMS). IAS/Park City Mathematics Series 12, 379-
398 (2007). ISBN 978-0-8218-2873-1/hbk
Let G be a real reductive group with (complexified) Lie algebra g and maximal compact
subgroup K. Let G denote the unitary dual of G. One of the outstanding unsolved
problems in the harmonic analysis of G is to understand G. The main theorem of this
paper shows that most of the cohomological unitary representations of G are isolated,
where G is endowed with the Fell topology.
To elaborate, suppose q = l + u is a -stable parabolic subalgebra of g where is the
Cartan involution corresponding to K. Fix a Cartan subalgebra h l, and a weight
. Assume the positivity condition: Re < , > > 0, for all (u, h). Let (u)
be half the sum of roots of h in u. Let L be the normalizer of q in G. Let R denote
the Zuckerman cohomological induction functor from M(l, L K)-(u) to M(g, K).