Lp

HARMONIC ANALYSIS ON SL(2,R) 11

It is a fundamental result of Harish-Chandra [14, Theorem 2] that an invariant Z-

finite distribution 0 on G can be represented as integration against a locally summable

function which is analytic on the regular set G'. We shall use 0 to denote both the dis-

tribution and its associated function.

(e) Universal enveloping algebra. Let a

e

denote the complexification of a . Then

U will denote the universal enveloping algebra of a e, and Z will denote the center of

U. The infinitesimal representation (w, V) associated to an admissible representation

(TT, V) then extends uniquely to an associative algebra representation of U onto V.

Moreover, if (7r, V) is irreducible, then there exists a homomorphism X :Z - C such that

TT(Z) = X ( Z ) I IbrallZeZ,

where I is the identity operator on V . X is called the infinitesimal character of

(TT,V).

Define the Casimir element of U to be

ft = H2 + H - YY. (2.6)

Then Z is generated by n , and the infinitesimal character of an irreducible, admissible

representation is determined by its value on ft.

Each -X"€n gives rise to a left invariant vector field Lx and a right invariant vector

field Rx by the formulas

Lx /(*) = /(«;*) = (djdt)f(x «p(tt))|

=0

Rx f(x) - f(X*) = {d/dt)f( «p(tf)*)|

tm0

These identifications give rise to an isomorphism between U and the algebra of left

invariant differential operators on G, and an anti-isomorphism between U and the alge-

bra of right invariant differential operators. If gvg2£U are considered as right invariant

and left invariant differential operators respectively, then their action at any x E G will

be denoted by f(gx s;02)- The elements of Z correspond to the bi-invariant differential

operators on G.

(f) Operator-valued derivatives. Suppose B(H) is the space of bounded linear opera-

tors on a Hilbert space H, U an open subset of C, and F a mapping from U into B(H).