Representations of differential operators on a Lie group

Palle E.T Jørgensen

doi:10.1016/0022-1236(75)90045-2

In 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= ∑ 1 r X 2 j |X 0 with the X j ,'s in the Lie algebra G . If the elements X 0, X 1,…, X r 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 X 0 = 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(Σ 1 r X j 2 + (−1) 1 2 X 0 with X 0 in the real linear span of the X j '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.

Representations of differential operators on a Lie group

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