Functional Calculus and Coadjoint Orbits

The Kirillov character formula gives a striking
correspondence between the unitary irreducible
representations of a compact semisimple Lie group and
its set of integral orbits on the dual of its Lie
algebra. In this thesis, the same correspondence is
derived without the use of character theory. It is
shown to be related to the convexity properties of
the support of the Weyl functional calculus of the
infinitesimal generators of the representation. This
result uses Edward Nelson's theory of "operants" in a
fundamental way. This had
been developed to put Feynman's operator calculus on
a rigorous basis. In particular, a beautiful explicit
formula of Nelson for the Weyl calculus facilitates
the extension of the Kirillov formula to the matrix
coefficients of the representation, thus giving a
"non-commutative" Kirillov-type formula for compact
Lie groups.

Autorentext

Raed Raffoul completed a B.Sc.(Hons) at the University of Sydneyand a Ph.D. in harmonic analysis on Lie groups at the Universityof New South Wales.

Klappentext

The Kirillov character formula gives a strikingcorrespondence between the unitary irreduciblerepresentations of a compact semisimple Lie group andits set of integral orbits on the dual of its Liealgebra. In this thesis, the same correspondence isderived without the use of character theory. It isshown to be related to the convexity properties ofthe support of the Weyl functional calculus of theinfinitesimal generators of the representation. Thisresult uses Edward Nelson's theory of "operants" in afundamental way. This hadbeen developed to put Feynman's operator calculus ona rigorous basis. In particular, a beautiful explicitformula of Nelson for the Weyl calculus facilitatesthe extension of the Kirillov formula to the matrixcoefficients of the representation, thus giving a"non-commutative" Kirillov-type formula for compactLie groups.