What is this pretty picture?
When studying the asymptotics of matrix coefficients attached to an
admissible representation, everything comes down to a study of
the "leading exponents". These exponents will be vectors
in a*, the dual
of the Lie algebra of the split portion of a maximally split torus
in the group G. Measuring the "distance" between
the exponents and the green negative
chamber gives a rough notion of size. If all of these exponents lie
in the pretty purple tempered cone, you can conclude your representation
was tempered to start with. At the opposite extreme, the
leading exponents for
a finite dimensional representation will all lie inside the green
negative chamber.