LECTURE 1. BASIC CONCEPTS 305This is the right regular action of Gv on the space of smooth functions f from Gv
to H(7rv), satisfying:
(1.10)If v = oo, smooth would mean that f is infinitely many times differentiable. On the
other hand for p < oo, it simply means that f is locally constant, i.e., f(gk) = f(g)
fork in some open compact subgroup depending on f (cf. [Cal,Car]).
One can now define a Whittaker functional >-x,, (s, 1rv) for I(s, 1rv) as follows:
(1.11) Ax,,(S,1rv)(f) = r (f(wo^1 n'),A)X(n')dn^1 ,
jN~
where >. is a fixed Whittaker functional on H(7rv)· The integral converges as a
principal value integral and stabilizes as N~ is approached by an increasing sequence
of open compact subgroups (cf. [CS,Sh3]), i.e., we may replace N~ with a sufficiently
large open compact subgroup depending on f.