2. NOTATION AND PRELIMINARIES
(a) General Notation, the symbols N, Z and Z are used for the non-negative
integers, the positive integers, and the non-zero integers respectively. If T is a subset of
S, and f is a function on S, then f\
denotes the restriction of f to T.
The space of continuous functions from a topological space S into C is denoted by
C(S), with CC(S) denoting the subset of functions with compact support. The support of
f 6 C(S) is denoted by supp f. Int A denotes the interior of the set A C S.
For M a C00 manifold countable at infinity, C^°(M) denotes the space of complex-
functions on M of compact support. C^°(M) is equipped with the usual
inductive limit topology.
If V is a finite dimensional vector space over R or C, then S (V) denotes the space
of rapidly decreasing functions on V with the usual Schwartz topology.
(b) The group G. Let G denote the 2 x 2 real special linear group SL(2,R), i.e,
G= SL(2,R) =i I a I : ad - 6c = 1, a,6,e,feR [
The Lie algebra of G, denoted by & , can be realized as
Important elements in a are
(2,R) = | ( ° J :a+ d = 0, a,6,c,d€R \
Corresponding elements in the group to the first three of these algebra elements are
/ cos0 sin0\
sin^ cos0 /
= exp(2tH), teR