Kirillov's character formula, the holomorphic Peter-Weyl theorem, and the Blattner-Kostant-Sternberg pairing

Details Kirillov's character formula, the holomorphic Peter-Weyl theorem, and the Blattner-Kostant-Sternberg pairing Kirillov's character formula, the holomorphic Peter-Weyl theorem, and the Blattner-Kostant-Sternberg pairing

2006-10-20

arxiv.org/abs/math/0610613v5

Journal of Geometry and Physics 58 (2008) 833-848

Mathematics

Differential Geometry

29 pages; prompted by some reactions, a number of comments have been incorporated

Scientific paper

10.1016/j.geomphys.2008.02.004

Let K be a compact Lie group, endowed with a bi-invariant Riemannian metric. The complexification G of K inherits a Kaehler structure having twice the kinetic energy of the metric as its potential, and left and right translation turn the Hilbert space of square-integrable holomorphic functions on G relative to a suitable measure into a unitary (KxK)-representation. We establish the statement of the Peter-Weyl theorem for this Hilbert space to the effect that this Hilbert space contains the vector space of representative functions on G as a dense subspace and that the assignment to a holomorphic function of its Fourier coefficients yields an isomorphism of Hilbert algebras from the convolution algebra on G onto the algebra which arises from the endomorphism algebras of the irreducible representations of K by the appropriate operation of Hilbert space sum. Consequences are (i) a holomorphic Plancherel theorem and the existence of a uniquely determined unitary isomorphism between the space of square-integrable functions on K and the Hilbert space of holomorphic functions on G, and (ii) a proof that this isomorphism coincides with the corresponding Blattner-Kostant-Sternberg pairing map, multiplied by a suitable constant. We then show that the spectral decomposition of the energy operator on the Hilbert space of holomorphic functions on G associated with the metric on K refines to the Peter-Weyl decomposition of this Hilbert space in the usual manner and thus yields the decomposition of this Hilbert space into irreducible isotypical (KxK)-representations. Among our crucial tools is Kirillov's character formula. Our methods are geometric and independent of heat kernels, which are used by B. C. Hall to obtain many of these results [Journal of Functional Analysis 122 (1994), 103--151], [Comm. in Math. Physics 226 (2002), 233--268].

Affiliated with

Also associated with

No associations

Rating

Kirillov's character formula, the holomorphic Peter-Weyl theorem, and the Blattner-Kostant-Sternberg pairing does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Kirillov's character formula, the holomorphic Peter-Weyl theorem, and the Blattner-Kostant-Sternberg pairing, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Kirillov's character formula, the holomorphic Peter-Weyl theorem, and the Blattner-Kostant-Sternberg pairing will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-362010