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The family of irreducible continuous unitary representations of a locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023680/c0236801.png" />, the non-zero matrix elements of which cannot be approximated by finite linear combinations of matrix elements of the regular representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023680/c0236802.png" /> in the topology of uniform convergence on compact sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023680/c0236803.png" />. The complementary series of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023680/c0236804.png" /> is non-empty if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023680/c0236805.png" /> is not amenable, i.e. if the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023680/c0236806.png" /> contains no non-trivial left-invariant mean [[#References|[2]]]. A connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023680/c0236807.png" /> has a non-empty complementary series if and only if the semi-simple quotient group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023680/c0236808.png" /> by its maximal connected solvable normal subgroup is non-compact (cf. [[Levi–Mal'tsev decomposition|Levi–Mal'tsev decomposition]]). A complementary series was first discovered for the complex classical groups [[#References|[1]]]. At the time of writing (1987) complementary series have been fully described only for certain locally compact groups. Certain problems in number theory (see, for example, [[#References|[5]]]) are equivalent to problems in the theory of representations connected with the complementary series of adèle groups of linear algebraic groups.
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The family of irreducible continuous unitary representations of a locally compact group $G$, the non-zero matrix elements of which cannot be approximated by finite linear combinations of matrix elements of the regular representation of $G$ in the topology of uniform convergence on compact sets in $G$. The complementary series of the group $G$ is non-empty if and only if $G$ is not amenable, i.e. if the space $L_\infty(G)$ contains no non-trivial left-invariant mean [[#References|[2]]]. A connected Lie group $G$ has a non-empty complementary series if and only if the semi-simple quotient group of $G$ by its maximal connected solvable normal subgroup is non-compact (cf. [[Levi–Mal'tsev decomposition|Levi–Mal'tsev decomposition]]). A complementary series was first discovered for the complex classical groups [[#References|[1]]]. At the time of writing (1987) complementary series have been fully described only for certain locally compact groups. Certain problems in number theory (see, for example, [[#References|[5]]]) are equivalent to problems in the theory of representations connected with the complementary series of adèle groups of linear algebraic groups.
  
 
====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"> F.P. Greenleaf,   "Invariant means on topological groups and their applications" , v. Nostrand (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.A. Naimark,   "Linear representations of the Lorentz group" , Macmillan (1964) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> B. Kostant,   "On the existence and irreducibility of certain series of representations" ''Bull. Amer. Math. Soc.'' , '''75''' (1969) pp. 627–642</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> H. Petersson,   "Zur analytische Theorie der Grenzkreisgruppen I" ''Math. Ann.'' , '''115''' (1937–1938) pp. 23–67</TD></TR></table>
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<table>
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<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) {{MR|}} {{ZBL|}} </TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top"> F.P. Greenleaf, "Invariant means on topological groups and their applications" , v. Nostrand (1969) {{MR|0251549}} {{ZBL|0174.19001}} </TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top"> M.A. Naimark, "Linear representations of the Lorentz group" , Macmillan (1964) (Translated from Russian) {{MR|0170977}} {{ZBL|0100.12001}} {{ZBL|0084.33904}} {{ZBL|0077.03602}} {{ZBL|0057.02104}} {{ZBL|0056.33802}} </TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top"> B. Kostant, "On the existence and irreducibility of certain series of representations" ''Bull. Amer. Math. Soc.'' , '''75''' (1969) pp. 627–642 {{MR|0245725}} {{ZBL|0229.22026}} </TD></TR>
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<TR><TD valign="top">[5]</TD> <TD valign="top"> H. Petersson, "Zur analytische Theorie der Grenzkreisgruppen I" ''Math. Ann.'' , '''115''' (1937–1938) pp. 23–67 {{MR|}} {{ZBL|}} </TD></TR>
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.W. Knapp,   "Representation theory of semisimple groups" , Princeton Univ. Press (1986)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Press (1986) {{MR|0855239}} {{ZBL|0604.22001}} </TD></TR>
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[[Category:Topological groups, Lie groups]]

Latest revision as of 18:23, 26 October 2014

The family of irreducible continuous unitary representations of a locally compact group $G$, the non-zero matrix elements of which cannot be approximated by finite linear combinations of matrix elements of the regular representation of $G$ in the topology of uniform convergence on compact sets in $G$. The complementary series of the group $G$ is non-empty if and only if $G$ is not amenable, i.e. if the space $L_\infty(G)$ contains no non-trivial left-invariant mean [2]. A connected Lie group $G$ has a non-empty complementary series if and only if the semi-simple quotient group of $G$ by its maximal connected solvable normal subgroup is non-compact (cf. Levi–Mal'tsev decomposition). A complementary series was first discovered for the complex classical groups [1]. At the time of writing (1987) complementary series have been fully described only for certain locally compact groups. Certain problems in number theory (see, for example, [5]) are equivalent to problems in the theory of representations connected with the complementary series of adèle groups of linear algebraic groups.

References

[1] I.M. Gel'fand, M.A. Naimark, "Unitäre Darstellungen der klassischen Gruppen" , Akademie Verlag (1957) (Translated from Russian)
[2] F.P. Greenleaf, "Invariant means on topological groups and their applications" , v. Nostrand (1969) MR0251549 Zbl 0174.19001
[3] M.A. Naimark, "Linear representations of the Lorentz group" , Macmillan (1964) (Translated from Russian) MR0170977 Zbl 0100.12001 Zbl 0084.33904 Zbl 0077.03602 Zbl 0057.02104 Zbl 0056.33802
[4] B. Kostant, "On the existence and irreducibility of certain series of representations" Bull. Amer. Math. Soc. , 75 (1969) pp. 627–642 MR0245725 Zbl 0229.22026
[5] H. Petersson, "Zur analytische Theorie der Grenzkreisgruppen I" Math. Ann. , 115 (1937–1938) pp. 23–67


Comments

In the theory of semi-simple Lie groups the notion of a complementary series representation often is introduced in a different fashion, viz. as a generalized principal series representation (cf. Continuous series of representations) that is (infinitesimally) unitary.

References

[a1] A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Press (1986) MR0855239 Zbl 0604.22001
How to Cite This Entry:
Complementary series (of representations). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complementary_series_(of_representations)&oldid=14974
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article