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In Chapter 17 of [371] we have studied symmetric polynomials called zonal polynomials. In this chapter we consider other types of symmetric polynomials as well as their generalizations called symmetric functions (symmetric “polynomials” of an infinite number of indeterminates).

Irreducible representations \({T_{{m_n}}}\) of the unitary group U(n) are given (up to an equivalence) by highest weights m n = (m 1n ,... ,m nn ), where m in ∈ℤ and m 1n ≥ m 2n ≥ ... ≥ m nn . The tensor product \({T_{{m_n}}} \otimes {T_{\left( {p,0} \right)}}\) of the irreducible representations of the group U(n) with highest weights m n and (p,0) ≡ (p, 0,... , 0) decomposes into irreducible representation and every of them is contained in the decomposition not more that once.

The aim of this chapter is to give Dunkl’s results on h-harmonic polynomials and the h-Hankel transform. They are related to symmetries with respect to Coxeter groups. The theory of h-harmonic polynomials is an analogue of the theory of usual harmonic polynomials. h-Harmonic polynomials are polynomials on the Euclidean space E n vanishing under action of the h-Laplacian. The analogue of majority of results from the theory of harmonic polynomials is valid for h-harmonic polynomials. The theory of h-harmonic polynomials leads to the h-Bessel function and to the h-Hankel transform.

In Chapter 8 of the book [371] we exposed the theory of Clebsch-Gordan coefficients (CGC’s) and Racah coefficients (RC’s) of finite dimensional representations of the group SU(2). These coefficients led us to orthogonal polynomials of a discrete variable. In this chapter we represent a foundation of CGC’s and RC’s for finite dimensional representations of compact groups. These coefficients form complete orthonormal systems of functions of many discrete variables. CGC’s and RC’s of the unitary group U(n) lead to generalized multivariate hypergeometric functions. The theory of these hypergeometric functions will be given in Chapter 5.

Let us fix a positive number α determining Jack polynomials J λ(x (r); α). Let d = 2/α. For every partition λ = (λ 1, λ 2,...,.λ n ) we put $$\left( a \right)_\lambda ^{\left( \alpha \right)} \equiv {\left( a \right)_\lambda } = \sum\limits_{i = 1}^{l\left( \lambda \right)} {{{\left( {a - \frac{d}{2}\left( {i - 1} \right)} \right)}_{{\lambda _i}}}} ,$$ (1) where (b) k = b(b+1)...(b+k−1), (b)0=1.

I. M. Gel’fand and his collaborators created the theory of general hypergeometric functions (multivariate hypergeometric functions on lattices and hypergeometric functions on Grassmannians). These functions are multivariate generalizations of hypergeometric functions of one variable. They are solutions of certain systems of differential equations (general hypergeometric systems) and are related to the Radon transform. At the present time, a deep connection of these hypergeometric functions with representations of groups is absent. But it is clear that this connection exists. Remark that the theory of q-analogues of Grassmannians and of related generalizations of Gel’fand hypergeometric functions is under elaboration (see, for example, [432]).

In the preceding chapter we have considered spherical functions of irreducible representations of SO(n) and of related groups with respect to the canonical basis \(\{ \tilde \Xi _M^{n - 1,m}\} \) in \({\xi ^2}({S^{n - 2}})\).

SO(n), SO 0(n − 1,1) and corresponding spaces. Let E n be the n-dimensional real Euclidean space with the scalar product \((x,y) = \sum\limits_{k = 1}^n {xkyk}\) Let us recall that O(n) denotes the group of orthogonal transformations in E n , that is, linear transformations in E n preserving (x, y), and SO(n) denotes the subgroup of unimodular transformations from O(n). The groups O(n) and SO(n) are compact, SO(n) is connected and O(n)consists of two connected components SO(n)and g 0 SO(n), where g 0 = diag(1,...,1, −1).

The maximal nilpotent group N in U(n − 1,1) consists of matrices (6) of Section 11.1.1. Replace in these matrices a by −2c.

In Chapters 6 and 8 we have studied connections of Krawtchouk and Hahn polynomials with representations of the groups SU(2) and SL(2,ℝ).Matrix elements of representations, considered as functions of row number, and Clebsch-Gordan coefficients are expressed respectively in terms of Krawtchouk and Hahn polynomials. But there is another connection of these poly­nomials with the theory of group representations, namely, with irreducible represen­tations of the symmetric group S n and of the wreath symmetric group S(n + 1, k).Spherical functions of these representations with respect to corresponding subgroups are expressed in terms of Krawtchouk and Hahn polynomials.

One of fundamental results of the theory of finite dimensional representations is the following theorem (see, for example, reference [58] of the first volume).

A product of two positive definite matrices can be not a positive definite matrix. Therefore, another product operation is defined in P m (F). To define it let us note that every matrix Λ ∈ P m (F) is representable in the form Λ = TT*, where T ∈ N_(m, F)A and A is the group of diagonal matrices diag (a 1,... , a m ) with a 1 > 0. We transfer the operation of group multiplication, defined in N_(m, F)A, into the set P m (F). Namely, for Λ = T 1 T 1*, M = T 2 T 2 we put $$\Lambda \Delta M = {T_1}{T_2}({T_1}{T_2})* = {T_1}M{T_1}*$$ (1) .

As we have seen in Chapter 13, representations of Chevalley groups lead to q-orthogonal polynomials with positive integral q. Basic hypergeometric functions and q-orthogonal polynomials for arbitrary (including complex) values of q are connected with quantum algebras and groups. They are considered in the present Chapter.

In the first two volumes of this book we have considered special functions related to most degenerate series of irreducible representations of classical simple Lie groups and of corresponding inhomogeneous groups. In the next chapters we study special functions related to non-degenerate series of representations. These special functions depend on many variables and in some cases it is convenient to consider them as functions of matrix argument or as functions with matrix indices. In this chapter we give necessary information from the theory of semisimple Lie groups and related homogeneous spaces. We describe also invariant measures and invariant differential operators on these groups and spaces.

Let \(g = \left( {\begin{array}{*{20}{c}} a & b \\ c & d \\ \end{array} } \right) \in GL\left( {2,\mathbb{C}} \right)\,and\,let\,\bar{\mathbb{C}} = \mathbb{C} \cup \left\{ \infty \right\}\) be the extended (compactified) complex plane.

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

In section 5.1 we shall study representations of the group G of third order real triangular matrices $$ g \equiv g\left( {a,b,d,\tau } \right) = \left( {\begin{array}{*{20}c} 1 & a & b \\ 0 & {e^\tau } & d \\ 0 & 0 & 1 \\ \end{array} } \right),a,b,d,\tau \in \mathbb{R} $$ (1) .

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 a) T(g1g2) = T(g1)T(g2), (1) b) T(e) = E, (2) where E is the identity operator in £.

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.

The group SU(2) consists of unimodular unitary matrices of the second order, i.e. of matrices $$ u = (\begin{array}{*{20}c} \alpha & \beta \\ {\bar \beta } & {\bar \alpha } \\ \end{array} ),\left| \alpha \right|^2 + \left| \beta \right|^2 = 1 $$ (1) . 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.

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).

In the preceding chapter we have introduced representations Tχ, x = (τ, e), of the group SU(1,1) and have studied their matrix elements in the basis einθ which diagonalizes the operators Tχ(g(t)), g(t) = diag(eit/2, e-it/2). Now we study other realizations of these representations. It will be convenient for us to consider representations Tχ 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).

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).

The associated spherical functions of the homogeneous spaces SO(n)/SO(n−1), U(n)/U(n−1), and Sp(n)/Sp(n−1) are found in different coordinate systems: They are matrix elements of representation operators, right invariant with respect to the subgroups SO(n−1), U(n−1), and Sp(n−1), respectively. The left index of these matrix elements corresponds to the reductions onto the subgroups SO(p)×SO(q); U(p)×U(q); and Sp(p)×Sp(q), p+q=n. The relations between these spherical functions are derived. These relations lead to the formulas connecting the Clebsch–Gordan coefficients for the groups SO(n), U(n), and Sp(n).

The group-theoretical methods give a unified approach to main classes of special functions [1,2]. Special functions are connected with integral transforms. This is why the theory of group representations is closely related to integral transforms. The group-theoretical description is given for many integral transforms.