Remote Access Representation Theory
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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.

References [Enhancements On Off] (What's this?)

  • 1. A.W. Knapp and E.M. Stein, Irreducibility theorems for the principal series, in Conference on Harmonic Forms, Lecture Notes in Mathematics 266, Springer-Verlag, 1972, 197-214. MR 54:10499
  • 2. A.W. Knapp and G. Zuckerman, Classification of irreducible tempered representations of semi-simple Lie groups, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2178-2180. MR 57:538
  • 3. G. Lusztig and N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. (2) 19 (1979), 41-52. MR 82g:20070
  • 4. D. Pascovici, Regular functions on Principal Nilpotent Orbits and $R$-groups, Ph.D. Disertation, Cambridge, MIT, 2000.
  • 5. I. Satake, On representations and compactifications of symmetric Riemannian spaces, Annals of Math. 71 (1960), 77-110. MR 22:9546
  • 6. W. Schmid, On the characters of the discrete series. The Hermitian symmetric case, Inventiones Math. 30 1975, 47-144. MR 53:714
  • 7. J. Sekiguchi, Remarks on real nilpotent orbits of a symmetric pair, J. Math. Soc. Japan 39, (1987), 127-138. MR 88g:53053
  • 8. D. Vogan, Representations of Real Reductive Lie Groups, Birkhäuser, Boston, 1981. MR 83c:22022
  • 9. D. Vogan, The algebraic structure of the representations of semisimple Lie groups I, Annals of Math. 109, (1979), 1-60. MR 81j:22020

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 22E46, 22E45

Retrieve articles in all journals with MSC (2000): 22E46, 22E45

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

American Mathematical Society