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 space
L2+ε(G) for any ε > 0.
Formulation This condition, as just given, is slightly weaker than the condition that the matrix coefficients are square-integrable, in other words lie in
L2(G), which would be the definition of a discrete series representation. If G is a linear semisimple Lie group with a maximal compact subgroup K, an admissible representation ρ of G is tempered if the above condition holds for the K-finite matrix coefficients of ρ. The definition above is also used for more general groups, such as p-adic Lie groups and finite central extensions of semisimple real algebraic groups. The definition of "tempered representation" makes sense for arbitrary unimodular locally compact groups, but on groups with infinite centers such as infinite central extensions of semisimple Lie groups it does not behave well and is usually replaced by a slightly different definition. More precisely, an irreducible representation is called tempered if it is unitary when restricted to the center Z, and the absolute values of the matrix coefficients are in L2+ε(G/Z). Tempered representations on semisimple Lie groups were first defined and studied by Harish-Chandra (using a different but equivalent definition), who showed that they are exactly the representations needed for the Plancherel theorem. They were classified by Knapp and Zuckerman, and used by Langlands in the Langlands classification of irreducible representations of a reductive Lie group G in terms of the tempered representations of smaller groups.
History Irreducible tempered representations were identified by Harish-Chandra in his work on harmonic analysis on a semisimple Lie group as those representations that contribute to the Plancherel measure. The original definition of a tempered representation, which has certain technical advantages, is that its Harish-Chandra character should be a "tempered distribution" (see the section about this below). It follows from Harish-Chandra's results that it is equivalent to the more elementary definition given above. Tempered representations also seem to play a fundamental role in the theory of automorphic forms. This connection was probably first realized by Satake (in the context of the Ramanujan-Petersson conjecture) and Robert Langlands and served as a motivation for Langlands to develop his classification scheme for irreducible admissible representations of real and p-adic reductive algebraic groups in terms of the tempered representations of smaller groups. The precise conjectures identifying the place of tempered representations in the automorphic spectrum were formulated later by James Arthur and constitute one of the most actively developing parts of the modern theory of automorphic forms.
Harmonic analysis Tempered representations play an important role in the harmonic analysis on semisimple Lie groups. An irreducible unitary representation of a semisimple Lie group G is tempered if and only if it is in the support of the Plancherel measure of G. In other words, tempered representations are precisely the class of representations of G appearing in the spectral decomposition of L2 functions on the group (while discrete series representations have a stronger property that an individual representation has a positive spectral measure). This stands in contrast with the situation for abelian and more general solvable Lie groups, where a different class of representations is needed to fully account for the spectral decomposition. This can be seen already in the simplest example of the additive group R of the real numbers, for which the matrix elements of the irreducible representations do not fall off to 0 at infinity. In the Langlands program, tempered representations of real Lie groups are those coming from unitary characters of tori by Langlands functoriality.
Examples The Plancherel theorem for a semisimple Lie group involves representations that are not the discrete series. This becomes clear already in the case of the group SL2(R). The principal series representations of SL2(R) are tempered and account for the spectral decomposition of functions supported on the hyperbolic elements of the group. However, they do not occur discretely in the regular representation of SL2(R). The two limit of discrete series representations of SL2(R) are tempered but not discrete series (even though they occur "discretely" in the list of irreducible unitary representations). For non-semisimple Lie groups, representations with matrix coefficients in L2+ε do not always suffice for the Plancherel theorem, as shown by the example of the additive group R of real numbers and the Fourier integral; in fact, all irreducible unitary representations of R contribute to the Plancherel measure, but none of them have matrix coefficients in L2+ε. The complementary series representations of SL2(R) are irreducible unitary representations that are not tempered. The trivial representation of a group G is an irreducible unitary representation that is not tempered unless G is compact.
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