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).
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