Tempered Representation - Tempered Distributions

Tempered Distributions

Fix a semisimple Lie group G with maximal compact subgroup K. Harish-Chandra (1966, section 9) defined a distribution on G to be tempered if it is defined on the Schwartz space of G. The Schwartz space is in turn defined to be the space of smooth functions f on G such that for any real r and any function g obtained from f by acting on the left or right by elements of the universal enveloping algebra of the Lie algebra of G, the function

is bounded. 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.