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\

T

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-

valued,

C00

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\

tgk

.

A

^

sin^ cos0 /

= exp(2tH), teR

tX

0

t