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Difference between revisions of "Degenerate series of representations"

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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Gel'fand,  M.A. Naimark,  "Unitäre Darstellungen der klassischen Gruppen" , Akademie Verlag  (1957)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K.I. Gross,  "The dual of a parabolic subgroup and a degenerate principal series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087036.png" />"  ''Amer. J. Math.'' , '''93''' :  2  (1971)  pp. 398–428</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Gel'fand,  M.A. Naimark,  "Unitäre Darstellungen der klassischen Gruppen" , Akademie Verlag  (1957)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Speh,  D.A., jr. Vogan,  "Reducibility of general principal series representations"  ''Acta Math.'' , '''145'''  (1980)  pp. 227–299</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.A., jr. Vogan,  "The unitary dual of $GL(n)$ over an archimedean field"  ''Invent. Math.'' , '''83'''  (1986)  pp. 449–505</TD></TR></table>
 
 
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====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Speh,  D.A., jr. Vogan,  "Reducibility of general principal series representations"  ''Acta Math.'' , '''145'''  (1980)  pp. 227–299</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.A., jr. Vogan,  "The unitary dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087037.png" /> over an archimedean field"  ''Invent. Math.'' , '''83'''  (1986)  pp. 449–505</TD></TR></table>
 

Latest revision as of 11:26, 26 March 2023


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
How to Cite This Entry:
Degenerate series of representations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degenerate_series_of_representations&oldid=46616
This article was adapted from an original article by D.P. Zhelobenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article