Lie theory and separation of variables. 4. The groups SO(2,1) and SO(3)

Winternitz and coworkers have shown that the eigenfunction equation for the Laplacian on the hyperboloid x₀² - x^{1 2}-x₂² = 1 separates in nine orthogonal coordinate systems, associated with nine symmetric quadratic operators L in the enveloping algebra of SO (2,1). Corresponding to each of the operators L, we employ the standard one-variable model for the principal series of representations of SO (2,1) and compute explicitly an L basis for the Hilbert space as well as the unitary transformations relating different bases. We also compute the associated results for realizations of these representations on the hyperboloid. Three of our bases are related to well-known subgroup reductions of SO (2,1). Of the remaining six, one is related to Bessel functions, two to Legendre functions, and three to Lamé functions. We show that there is virtually a perfect correspondence between the known theory of the Lamé functions and the representation theory of SO (2,1) and SO (3).