Hypermaximality of Certain Operators on Lie Groups1

We show that any polynomial in the infinitesimal generators of a Lie group is essentially hypermaximal symmetric (and so has a spectral resolution), when suitably interpreted as an operator on a Hubert space, provided it is formally symmetric and commutes with all the infinitesimal generators. Such operators are frequently of use in investigations of the unitary representations of Lie groups. The requirement that the polynomial commute with the infinitesimal generators cannot be eliminated, unless some alternative restriction is put on the polynomial (for example, on its degree, as in [2]2); this is shown by an unpublished example due to von Neumann. r Let G be a connected Lie group with Lie algebra £ (of right-invariant infinitesimal transformations) and let £ be the enveloping algebra of <£ (see for example [l]). If U is a (strongly) continuous unitary representation of G on a Hilbert space 3C, there exists (see [2]) an associated representation dU of £ by operators on 5C with domain D consisting of all sums of vectors of the form fU(a)xf(a)da, with xE5C, f a function of class CM on G and vanishing outside a compact set (the integral being taken in the strong sense relative to the left-invariant Haar measure on G with element da), and characterized by the property that if XE°C< then idU(X) is essentially hypermaximal symmetric (T is such an operator if T* = (T**), and exp (it(dU(X))*) = U(exp (tX)) for all real t. An element of £ will be called "symmetric" if it is invariant under the unique real linear operaton on £ which takes any monomial aXxX2 • • • Xr(a complex, the Xi in jQ into (-\)räXr ■ • • ZÄ.