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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.
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.