# Degenerate series of representations

The set of representations of a semi-simple Lie group $G$ induced by the characters of a non-minimal parabolic subgroup $P$ of it. Let $\Pi$ be a fundamental root system with respect to which the Lie algebra of a Borel subgroup $B \subset G$ is spanned by the root vectors $e _ \alpha$, $\alpha < 0$. The set of all parabolic subgroups containing $B$ is in one-to-one correspondence with the set of all subsystems $\Pi _ {0} \subset \Pi$; $P \neq B$ if $\Pi _ {0}$ is non-empty, and the Lie algebra of the group $P$ is generated by the $e _ \alpha$, $\alpha < 0$, and $e _ \alpha$, $\alpha \in \Pi _ {0}$. Let $\pi ( \chi )$ be the representation of the group $G$ induced by a character $\chi$ of $P$ in the class $C ^ \infty ( G)$. There exist characters $\chi$ for which $\pi ( \chi )$ can be extended to a unitary representation of the group $G$ in $L _ {2} ( Z)$, where $Z$ is a subgroup in $G$ whose Lie algebra is spanned by the vectors $e _ \alpha$, $\alpha > 0$, $\alpha \notin \Delta _ {0}$; here $\Delta _ {0}$ is the additive hull of $\Pi _ {0}$. Such representations are known as the representations of the basic degenerate series of representations. A supplementary degenerate series of representations is obtained by extending $\pi ( \chi )$( for certain values of $\chi$) with respect to other scalar products in $\pi ( \chi )$. The representations of a degenerate series of representations are irreducible for the group $G = \mathop{\rm SL} ( n, \mathbf C )$.
 [1] I.M. Gel'fand, M.A. Naimark, "Unitäre Darstellungen der klassischen Gruppen" , Akademie Verlag (1957) (Translated from Russian) [2] K.I. Gross, "The dual of a parabolic subgroup and a degenerate principal series of $Sp(n,\CC)$" Amer. J. Math. , 93 : 2 (1971) pp. 398–428 [a1] B. Speh, D.A., jr. Vogan, "Reducibility of general principal series representations" Acta Math. , 145 (1980) pp. 227–299 [a2] D.A., jr. Vogan, "The unitary dual of $GL(n)$ over an archimedean field" Invent. Math. , 83 (1986) pp. 449–505