Representation of Lie Groups and Special Functions

In the 18th and 19th centuries there appeared a great number of types of special functions to solve the differential equations of mathematical physics and to calculate integrals. Many of them turned out to be special or limiting cases of the hypergeometric function F(α, β; γ; x) introduced in 1769 by L. Euler and scrutinized at the beginning of the 19th century by Gauss. Gauss’ work triggered a flow of investigations which established different recurrence relations, differential equations, integral representations, generating functions, addition and multiplication theorems, asymptotic expansions for the hypergeometric function and its associates (Legendre, Gegenbauer, Hermite, Laguerre, Chebyshev polynomials; Bessel, Neumann, Macdonald, Whittaker functions, etc.), sought for relations between these functions, and calculated puzzling integrals involving them, etc.

This chapter presents basic concepts and results of the theory of Lie groups and Lie algebras which will be used in the sequel. As a rule, the results will be given without proofs. They can be found, for example, in the books [5, 21, 22, 33, 38, 58]. The “null” section contains the information from Algebra, Topology, and Functional Analysis which is used in the present book.

By a representation of a group G in a linear space £ over a field κ (the space of the representation) we shall mean a homomorhism T: G → GL(£, κ), where GL(£, κ) is the group of non-singular linear transformations of £. Thus, T is a mapping of G into GL(£, κ) satisfying the conditions

As we have shown in Section 2.2.8, irreducible finite dimensional representations of commutative groups are one-dimensional. In particular, irreducible representations of the additive group R are onedimensional. In order to find them it is necessary to solve the functional equation f(t + s) = f(t) f (s), where f is a scalar function. It follows from formula (4) of Section 2.1.5 that f’(t) = f’(0) f (t).

ISO(2). Transformations of the Euclidean plane, which preserve the distance between points and do not change the orientation of the plane, are called motions of this plane. Parallel shifts of the plane and rotations of the plane about some point are examples of motions. The set of all motions of the plane forms a group, denoted by ISO(2).

In section 5.1 we shall study representations of the group G of third order real triangular matrices .

The group SU(2) consists of unimodular unitary matrices of the second order, i.e. of matrices . Therefore, each element u of SU(2) is uniquely determined by a pair of complex numbers α and β such that ∣α∣²+∣β∣²=1.

In the preceding chapter we have introduced representations T_χ, x = (τ, ε), of the group SU(1,1) and have studied their matrix elements in the basis e^inθ which diagonalizes the operators T_χ(g(t)), g(t) = diag(e^it/2, e^-it/2). Now we study other realizations of these representations. It will be convenient for us to consider representations T_χ of the group SU(2, ℝ) which is isomorphic to SU(1, 1). Subgroups and decompositions, considered below, have simpler form for S L(2, ℝ).

In Section 6.2.1 we have constructed the realization of irreducible representations T_ℓ of the group SU(2) in the space ℌ_ℓ of homogeneous polynomials in two variables of degree 2ℓ.