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.
Read more about this topic: Tempered Representation
Famous quotes containing the word tempered:
“The generation of women before us who rushed to fill the corporate ranks altered our expectations of what working motherhood could be, tempered our ambition, and exploded the supermom myth many of us held dear.”
—Melinda M. Marshall (20th century)