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 In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

are certain irreducible

unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G'' ...

s that are not tempered and do not appear in the decomposition of the

regular representation In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation. One distinguishes the left regular rep ...

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) In mathematics, the main results concerning irreducible unitary representations of the Lie group SL(2,R) are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952). Structure of the complexified Lie algebra We choo ...

Elias M. Stein Elias Menachem Stein (January 13, 1931 – December 23, 2018) was an American mathematician who was a leading figure in the field of harmonic analysis. He was the Albert Baldwin Dod Professor of Mathematics, Emeritus, at Princeton University, w ...

(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