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.
Read more about this topic: Tempered Representation
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