The Dynkin diagram \(R\)-group (Q2761173)

It was shown by \textit{A. W. Knapp} and \textit{E. M. Stein} [Irreducibility theorems for the principal series, Lect. Notes Math. 266, 197-214 (Berlin etc. 1972; Zbl 0248.22017)] that understanding the irreducibility of principal series representations of real reductive Lie groups boils down to computing certain small abelian subgroups of the Weyl group, called \(R\)-groups. The author concentrates on a particular class of principal series for linear and split groups, those with infinitesimal character equal to zero. They are obtained by parabolic induction as \(\text{Ind}^G_{\text{MAN}}\delta\otimes 0\), for all representations \(\delta\) of \(M\), which is a finite abelian group for all cases which the author considers. It is shown that one can reduce the understanding of the decomposition into irreducible components for all principal series with infinitesimal character zero at once, to the same problem for a small Levi subgroup which at the Lie algebra level consists of several copies of \(sl(2, \mathbb{R})\). As a by-product, the author describes a finite group which can be constructed combinatorially by looking at the Dynkin diagram of a simple split group, and shows that each \(R_{\delta, 0}\)-group has to be a subgroup of it.