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Representation Theory

ISSN 1088-4165



The Dynkin diagram $R$-group

Author: Dana Pascovici
Journal: Represent. Theory 5 (2001), 1-16
MSC (2000): Primary 22E46; Secondary 22E45
Published electronically: January 18, 2001
MathSciNet review: 1826426
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We define an abelian group from the Dynkin diagram of a split real linear Lie group with abelian Cartan subgroups, $G$, and show that the $R_{\delta, 0}$-groups defined by Knapp and Stein are subgroups of it. The proof relies on Vogan's approach to the $R$-groups. The $R$-group of a Dynkin diagram is easily computed just by looking at the diagram, and so it gives, for instance, quick proofs of the fact that the principal series with zero infinitesimal character of the split groups $E_6$, $E_8$, $G_2$ or $F_4$ are irreducible. The Dynkin diagram subgroup also implicitly describes a small Levi subgroup, which we hope might play a role in computing regular functions on principal nilpotent orbits. We present in the end a conjecture and some evidence in this direction.

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Additional Information

Dana Pascovici
Affiliation: Department of Mathematics, Purdue University, MATH 602, West Lafayette, Indiana 47906

Keywords: $R$-groups, reducible principal series
Received by editor(s): May 31, 2000
Received by editor(s) in revised form: October 10, 2000
Published electronically: January 18, 2001
Article copyright: © Copyright 2001 American Mathematical Society