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The set of representations of a semi-simple Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d0308701.png" /> induced by the characters of a non-minimal parabolic subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d0308702.png" /> of it. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d0308703.png" /> be a fundamental root system with respect to which the Lie algebra of a Borel subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d0308704.png" /> is spanned by the root vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d0308705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d0308706.png" />. The set of all parabolic subgroups containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d0308707.png" /> is in one-to-one correspondence with the set of all subsystems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d0308708.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d0308709.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087010.png" /> is non-empty, and the Lie algebra of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087011.png" /> is generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087013.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087015.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087016.png" /> be the representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087017.png" /> induced by a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087019.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087020.png" />. There exist characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087021.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087022.png" /> can be extended to a unitary representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087025.png" /> is a subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087026.png" /> whose Lie algebra is spanned by the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087029.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087030.png" /> is the additive hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087031.png" />. Such representations are known as the representations of the basic degenerate series of representations. A supplementary degenerate series of representations is obtained by extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087032.png" /> (for certain values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087033.png" />) with respect to other scalar products in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087034.png" />. The representations of a degenerate series of representations are irreducible for the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030870/d03087035.png" />.
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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====
 
====References====
 
<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>
 
<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>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Revision as of 17:32, 5 June 2020


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

Comments

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
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