Representations of differential operators on a Lie group

AbstractIn this paper we apply the theory of second-order partial differential operators with nonnegative characteristic form to representations of Lie groups. We are concerned with a continuous representation U of a Lie group G in a Banach space B. Let E be the enveloping algebra of G, and let dU be the infinitesimal homomorphism of E into operators with the Gårding vectors as a common invariant domain. We study elements in E of the form P=∑1rX2j |X0 with the Xj,'s in the Lie algebra G.If the elements X0, X1,…, Xr generate G as a Lie algebra then we show that the space of C∞-vectors for U is precisely equal to the C∞-vectors for the closure dU(P), of dU(P). This result is applied to obtain estimates for differential operators.The operator dU(P) is the infinitesimal generator of a strongly continuous semigroup of operators in B. If X0 = 0 we show that this semigroup can be analytically continued to complex time ζ with Re ζ > 0. The generalized heat kernels of these semigroups are computed. A space of rapidly decreasing functions on G is introduced in order to treat the heat kernels.For unitary representations we show essential self-adjointness of all operators dU(Σ1r Xj2 + (−1)12X0 with X0 in the real linear span of the Xj's. An application to quantum field theory is given.Finally, the new characterization of the C∞-vectors is applied to a construction of a counterexample to a conjecture on exponentiation of operator Lie algebras.Our results on semigroups of exponential growth, and on the space of C∞ vectors for a group representation can be viewed as generalizations of various results due to Nelson-Stinespring [18], and Poulsen [19], who prove essential self-adjointness and a priori estimates, respectively, for the sum of the squares of elements in a basis for G (the Laplace operator). The work of Hörmander [11] and Bony [3] on degenerate-elliptic (hypoelliptic) operators supplies the technical basis for this generalization. The important feature is that elliptic regularity is too crude a tool for controlling commutators. With the aid of the above-mentioned hypoellipticity results we are able to “control” the (finite dimensional) Lie algebra generated by a given set of differential operators