Eigenfunction Asymptotics in the Complex Domain for a Compact Lie Group

Let (G, k)be a compact connected Lie group endowed with a biinvariant Riemannian metric, and let G be the complexification of G. We apply Grauert tube techniques to the near-diagonal scaling asymptotics of certain operator kernels, which are defined in terms of the matrix elements of an irreducible representation drifting to infinity along a ray in weight space. These kernels are the equivariant components of Poisson and Szegő kernels on a fixed sphere bundle in G, when the latter is identified with the tangent bundle of G in an appropriate way.