Abstract

A Hermitian formq on the dual space, g , of the Lie algebra, g; of a Lie group,G; de- termines a sub-Laplacian,1; onG: It will be shown that Hcondition for hypoellipticity of the sub-Laplacian holds if and only if the associated Hermitian form, induced by q on the dual of the universal enveloping algebra, U 0 , is non-degenerate. The subelliptic heat semigroup,e t1=4 ; is given by convolution by aC 1 probability density t: WhenG is complex andu : G! C is a holomorphic function, the collection of derivatives ofu at the identity inG gives rise to an element, O u.e/2 U 0 . We will show that ifG is complex, connected, and simply connected then the Taylor map,u7! O u.e/, defines a unitary map from the space of holomorphic functions inL 2 .G; t/ onto a natural Hilbert space lying in U 0 .

Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call