Zonal Spherical Function - Example: SL(2,C)

Example: SL(2,C)

See also: SL(2,C), Representations of the Lorentz group, and Spectral theory of ordinary differential equations

The group G = SL(2,C) is the complexification of the compact Lie group K = SU(2) and the double cover of the Lorentz group. The infinite-dimensional representations of the Lorentz group were first studied by Dirac in 1945, who considered the discrete series representations, which he termed expansors. A systematic study was taken up shortly afterwards by Harish-Chandra, Gelfand–Naimark and Bargmann. The irreducible representations of class one, corresponding to the zonal spherical functions, can be determined easily using the radial component of the Laplacian operator.

Indeed any unimodular complex 2×2 matrix g admits a unique polar decomposition g = pv with v unitary and p positive. In turn p = uau*, with u unitary and a a diagonal matrix with positive entries. Thus g = uaw with w = u* v, so that any K-biinvariant function on G corresponds to a function of the diagonal matrix

invariant under the Weyl group. Identifying G/K with hyperbolic 3-space, the zonal hyperbolic functions ψ correspond to radial functions that are eigenfunctions of the Laplacian. But in terms of the radial coordinate r, the Laplacian is given by

Setting f(r) = sinh (r)·ψ(r), it follows that f is an odd function of r and an eigenfunction of .

Hence

where is real.

There is a similar elementary treatment for the generalized Lorentz groups SO(N,1) in Takahashi (1963) and Faraut & Korányi (1994) (recall that SO0(3,1) = SL(2,C) / ±I).