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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 ,
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