Complementary series (of representations)

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