Lp

HARMONIC ANALYSIS ON SL(2,R) 9

We let G' denote the collection of regular elements in G:

G' = {x£G :\tr x\ # 2} = G n [MN)C'

(c) Spherical functions. Suppose n,m€Z. A complex valued function f on G is called

spherical of type (n,m) if

/(*!**,) = '„(*l)/(*)'m(*.)

(2-3)

for all Jkx k2e K and x € G, where ^(A^) = exp(in0). Let Cc°°n

m

denote the space of all

spherical functions of type (n,m) in CC°°(G), and define a mapping Pn

m

on CC°°(G) by

for all x £ G. Then P

n m

is a continuous mapping of CC°°(G) onto Ce.n

m

. [28, p.261].

The left and right regular representations of G on Ce (G) are defined by

[L(x)f](y) =/(x-'y),

for all f 6 CC°°(G) and x,y € G. Let CC°°(G,K) denote the space of K-finite functions in

CC°°(G), i.e., those f G 7c°°(G) such that the linear span of {L{kx) R(*2)f | ibj, *2 € K} is

finite dimensional. CC°°(G,K) is then the algebraic sum of all the spaces Cc°°n

m

, and it is

dense in Cc°°(G). [28, Prop. 4.4.3.5]

(d) Representations. Let K be the set of equivalence classes of irreducible represen-

tations of K. All the representations in K are one-dimensional, and K is parametrized

by Z, where the character of K corresponding to n£Z is given by

Let ir be a continuous representation of G on a Banach space V. For each n€Z we

let Vn denote the subspace of all K-finite vectors in V transforming according to rn. We

further let V denote the subspace of all K-finite vectors in V. Then

We say that n is admissible if each Vn is finite dimensional.

If

(TT,V)

is an admissible representation, then all K-finite vectors are analytic, and V

can be made into a (a ,K)-module by defining, for each X 6 a , v£ V ,