Representations of differential operators on a Lie group

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 $\text{B}$. Let $\text{E}$ be the enveloping algebra of G, and let dU be the infinitesimal homomorphism of $\text{E}$ into operators with the Gårding vectors as a common invariant domain. We study elements in $\text{E}$ of the form

$\text{P=}\text{∑}\text{1}\text{r}\text{X}{}^2{}_{\mathrm{j}}\text{|X}{}_0$

with the X_j,'s in the Lie algebra $\text{G}$.If the elements X₀, X₁,…, X_r generate $\text{G}$ 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 $\text{dU(P)}\text{,}\text{of}\text{dU(P)}$. This result is applied to obtain estimates for differential operators.The operator $\text{dU(P)}$ is the infinitesimal generator of a strongly continuous semigroup of operators in $\text{B}$. 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 $\text{dU(Σ}{}_1{}^{\mathrm{r}}\text{X}{}_{\mathrm{j}}{}^2\text{+ (?1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{X}{}_0$ with X₀ 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 $\text{G}$ (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.