View source for Complementary series (of representations)

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.

====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) {{MR|}} {{ZBL|}} </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) {{MR|0251549}} {{ZBL|0174.19001}} </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) {{MR|0170977}} {{ZBL|0100.12001}} {{ZBL|0084.33904}} {{ZBL|0077.03602}} {{ZBL|0057.02104}} {{ZBL|0056.33802}} </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 {{MR|0245725}} {{ZBL|0229.22026}} </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 {{MR|}} {{ZBL|}} </TD></TR></table>



====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|Continuous series of representations]]) that is (infinitesimally) unitary.

====References====
<table><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></table>