mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, complementary series representations of a reductive real or ''p''-adic Lie groups are certain irreducible unitary representations that are not tempered and do not appear in the decomposition of the regular representation into irreducible representations. They are rather mysterious: they do not turn up very often, and seem to exist by accident. They were sometimes overlooked, in fact, in some earlier claims to have classified the irreducible unitary representations of certain groups. Several conjectures in mathematics, such as the

Selberg conjecture In mathematics, Selberg's conjecture, also known as Selberg's eigenvalue conjecture, conjectured by , states that the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least 1/4. Selberg showed that the eig ...

, are equivalent to saying that certain representations are not complementary. For examples see the representation theory of SL2(R). Elias M. Stein (1972) constructed some families of them for higher rank groups using analytic continuation, sometimes called the Stein complementary series.

References

* *, also reprinted as Representation theory of groups {{algebra-stub