Degenerate series of representations

The set of representations of a semi-simple Lie group induced by the characters of a non-minimal parabolic subgroup of it. Let be a fundamental root system with respect to which the Lie algebra of a Borel subgroup is spanned by the root vectors , . The set of all parabolic subgroups containing is in one-to-one correspondence with the set of all subsystems ; if is non-empty, and the Lie algebra of the group is generated by the , , and , . Let be the representation of the group induced by a character of in the class . There exist characters for which can be extended to a unitary representation of the group in , where is a subgroup in whose Lie algebra is spanned by the vectors , , ; here is the additive hull of . Such representations are known as the representations of the basic degenerate series of representations. A supplementary degenerate series of representations is obtained by extending (for certain values of ) with respect to other scalar products in . The representations of a degenerate series of representations are irreducible for the group .

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