Spherical Functions—The General Theory

Abstract

Historically the theory of spherical functions dates to the classical papers of É. Cartan and H. Weyl; they showed that spherical harmonics arise in a natural way from a study of functions on G/K where G is the orthogonal group in n-space and where K consists of those transformations in G which leave a given vector invariant — this study is carried out by the methods of group representations. However in order to get a theory applying to larger classes of ‘special functions’ it is necessary to drop the assumption that G is compact and also to consider functions not just on G/K but also on G. In 1947 Bargmann studied the pair (G, K) where G = SL(2, R) and K = SO(2); there functions O on G were considered which, for a given character X of K, verify the relation

$$\Phi ({k_1}x{k_2}) = \chi ({k_1})\Phi (x)\chi ({k_2})({k_1},{k_2} \in K;x \in G)$$

such functions arise upon considering finite or infinite dimensional irreducible representations of G; these representations have coefficients satisfying the above relation and, when G is suitably ‘parameterized’, it turns out that these functions can be identified with hypergeometric functions; in particular when χ is the trivial character of K one is led to Legendre functions of arbitrary index and to group theoretical explanations of three important properties of these functions, namely their differential equation, their representation by integral formulas and their functional equation.