# 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 $Sp(n,\CC)$" 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 |

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Degenerate series of representations.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Degenerate_series_of_representations&oldid=53414