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 ) $.

References

[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 " Amer. J. Math. , 93 : 2 (1971) pp. 398–428

References

[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 over an archimedean field" Invent. Math. , 83 (1986) pp. 449–505