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).
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.
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.
More precisely, it is a direct sum of 2r irreducible tempered representations indexed by the characters of an elementary abelian group R of order 2r (called the R-group).
Here Ξ is a certain spherical function on G, invariant under left and right multiplication by K, and σ is the norm of the log of p, where an element g of G is written as : g=kp for k in K and p in P.