Complementary series representation
Encyclopedia
In mathematics, complementary series representations of a reductive real or p-adic Lie groups are certain irreducible unitary representation
s 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, 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.
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 is a unitary operator for every g ∈ G...
s that are not tempered
Tempered representation
In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the Lp spacefor any ε > 0.-Formulation:...
and do not appear in the decomposition of the regular representation
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....
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, are equivalent to saying that certain representations are not complementary. For examples see the representation theory of SL2(R)
Representation theory of SL2(R)
In mathematics, the main results concerning irreducible unitary representations of the Lie group SL are due to Gelfand and Naimark , V...
. Elias M. Stein
Elias M. Stein
Elias Menachem Stein is a mathematician and a leading figure in the field of harmonic analysis. He is the Albert Baldwin Dod Professor of Mathematics at Princeton University.-Biography:...
(1972) constructed some families of them for higher rank groups using analytic continuation, sometimes called the Stein complementary series.