Mathematics and Its Applications. Soviet Series. 75. Dordrecht: Kluwer Academic Publishers Group. xix, 629 p. (1992).

This is volume 3 of a three volume updating of the first (and recently deceased) author’s classic 1965 book “Special functions and the theory of group representations” [

Zbl 0144.380; for a review of volume 1 see

Zbl 0742.22001]. The point of view is that special functions are spherical functions on groups, i.e., they are matrix elements of irreducible group representations with respect to appropriate bases and in appropriate coordinates. If the basis is orthonormal then the group homomorphism property of the matrix elements leads to addition theorems for the associated special functions. Corresponding to a continuum basis, these addition theorems can be interpreted as integral transforms. The authors also consider Clebsch-Gordan coefficients and Racah coefficients for tensor products of group representations as special functions of discrete arguments. Then the unitarity conditions for matrix elements, Clebsch-Gordan and Racah coefficients can (typically) be interpreted as orthogonality relations for families of polynomials of a discrete variable. Finally, the authors may study as `special’ those elements $f$ belonging to a functional representation space of some group $G$ that transform in a simple manner under the action of a subgroup $H$ of $G$, e.g., $f$ is invariant under this action. Not all tabulated special functions (or even special functions associated with groups) arise in these ways, but enough of them do so that this uniform treatment by group representation methods is justified.
Most of the results in this hefty volume were obtained after the publication of Vilenkin’s 1965 book and there is almost no overlap in contents. The amount of material covered in huge and the pace of coverage is rapid. Most of the background results in representation theory needed for the introduction and analysis of the special functions are stated but not proved, and attribution for the results is usually not given in the text. (The three volumes contain an extensive bibliography and bibliographic notes so that the original sources can be identified.) A significant number of new results by Klimyk and his collaborators are included without special mention.
There are five chapters in this volume. The first is concerned with the relationship between the theory of quantum groups and algebras and the properties of $q$-orthogonal polynomials and basic hypergeometric functions. Jimbo’s definition of the quantum algebra $U\sb q({\germ {sl}}\sb 2)$ is introduced and its representation theory and associated Clebsch-Gordan and Racah coefficients are worked out in analogy with the corresponding computations for the Lie algebra ${\germ {sl}}\sb 2$ in volume 1. Standard unitarity conditions for these coefficients and other properties obeyed by them lead to identities for various basic hypergeometric functions expressible in the form $\sb{n+1}\varphi\sb n$ for $1\leq n\leq 7$. For the Lie algebra ${\germ {sl}}\sb 2$ and its Lie group $SL\sb 2$ it is well known that a product of matrix elements corresponding to two irreducible representations of $SL\sb 2$ can be written as a linear combination of matrix elements in the tensor product decomposition, where the coefficients are products of Clebsch-Gordan coefficients. For the quantum algebra $U\sb q({\germ {sl}}\sb 2)$ the authors require that the analogous expression be valid, even though, since $U\sb q({\germ {sl}}\sb 2)$ is not a Lie algebra for $q\ne 1$, there exists no corresponding Lie group. Analyzing this expression they obtain the general matrix elements in terms of little $q$-Jacobi polynomials in the non-commuting matrix elements of the 2-dimensional irreducible representation, where these latter matrix elements define the “quantum group” $SL\sb q(2,C)$. This provides an entry into the theory of quantum groups, whose relation to Hopf algebras is sketched. The addition formula for the matrix elements is worked out and the quantum 3- sphere, related to big $q$-Jacobi polynomials, is studied. The Askey- Wilson polynomials are presented, but not linked with group theory.
The next two chapters concern first the classification and analysis of semisimple Lie groups and related homogeneous spaces, and then the representation theory of these groups and computation of the matrix elements. In volume 1 only the lowest dimensional Lie groups plus degenerate representations of higher dimensional Lie groups were considered. Now the general case is addressed. The classification theory (Cartan subalgebras, roots, Dynkin diagrams, invariant integration, etc.) and the group decompositions (Iwasawa, Bruhat, Gauss, Cartan) are rather standard and few proofs are given. The material on the finite dimensional representations, on principal series representations and on the computation of matrix elements for classical groups is also mostly standard. There is a section showing how the Lauricella functions (hypergeometric functions of many variables) arise as matrix elements in a continuum basis of the most degenerate series representations of $SL(n,R)$.
The next topic considered is the relationship between group representations and special functions of a matrix argument. The authors consider functions defined over the real field, the complex field and the field of quaternions, but I will limit my remarks to the complexes. The functions in question are initially restricted to the space of all $m\times m$ positive definite matrices $\Lambda$ and later extended to $m\times m$ hermitian matrices. The associated group theory concerns zonal spherical functions and character formulas for the group $GL(m,C)$. The group action is given by $\Lambda\to g\sp*\Lambda g$, $g\in GL(m,C)$ and the space of positive definite matrices can be identified with the homogeneous space $U(m,C)/GL(m,C)$. Zonal spherical polynomials (polynomials in the matrix elements of hermitian matrices, invariant under the restriction of $g$ to $U(m,C)$) can be identified with the zonal spherical functions of the pair $GL(m,C)$, $U(m,C)$ and then with the simple characters of $U(m,C)$, denoted $Z\sb{k\sb 1,\dots,k\sb m}$, $k\sb 1\geq k\sb 2\geq\dots\geq k\sb m$. Roughly speaking, the hypergeometric functions of a matrix argument are obtained by replacing the monomials $z\sb 1\sp{k\sb 1}\cdots z\sb m\sp{k\sb m}$ in the power series expansion of a hypergeometric function of $m$ scalar variables by the zonal spherical polynomials $Z\sb{k\sb 1,\dots,k\sb m}$. Group theoretic properties of these spherical functions enable one to derive integral representations and integral identities for the hypergeometric functions of a matrix argument that are analogies to those known for the scalar case.
The next chapter concerns the computation of matrix elements and Clebsch- Gordan coefficients of irreducible representations of $GL(n,C)$, $U(n,C)$ and $SO(n)$ in a Gel’fand-Tsetlin basis. The fundamental idea here is to assign and label a basis of vectors in the space of a finite dimensional irreducible representation of $GL(n,C)$ according to the transformation properties of the vectors under successive restrictions of $GL(n,C)$ through a subgroup chain $GL(n,C)\supset GL(n-1,C)\supset\ldots\supset GL(1,C)$. This, together with an appropriate decomposition of the group, yields a multiplicity free labeling and one convenient for computation. Various forms of hypergeometric functions arise. A novelty is the introduction of matroms, special functions with matrix indices, in the computation of matrix elements for irreducible representations of $SO(n)$, $n>3$.
The last chapter is devoted to modular forms, theta functions and representations of affine Lie algebras. Modular forms are treated in the standard way, associated with their transformation properties under subgroups of $SL(2,Z)$. Theta functions of one and several variables arise in a variety of manners but, typically, are defined as functions in the representation space of a higher order Heisenberg group that are invariant under the action of a discrete subgroup. $Sp(n,R)$ also appears, essentially as an automorphism group.
The representation theory of affine Lie algebras is treated in detail but usually without proof. (Much of the material is taken from the book “Infinite dimensional Lie algebras” (1983;

Zbl 0537.17001) by {\it Victor Kac}.) A (non-twisted) affine Lie algebra takes the form $\widehat {\cal G}=L({\cal G}) \oplus Cc \oplus Cd$ where ${\cal G}$ is a simple (complex) Lie algebra, $c$ is a constant, $d=t d/dt$, $L({\cal G})=P(t,t\sp{-1})\otimes{\cal G}$ and $P(t,t\sp{-1})$ is the space of finite polynomials. Here, $[t\sp m\otimes X, t\sp n \otimes Y]=t\sp{m+n}\otimes [X,Y]+m\delta\sb{m,-n} B(X,Y)c$, $X,Y\in {\cal G}$, where $B(.,.)$ is the Killing form on ${\cal G}$. (Twisted affine Lie algebras have a similar but slightly more complicated definition.) Clearly, the properties of $\widehat {\cal G}$ are completely determined by those of ${\cal G}$. The usual tools of Cartan subalgebras, roots and weights carry over to the study of representations of the infinite dimensional Lie algebras $\widehat {\cal G}$. If one restricts to the study of (usually infinite dimensional) representations of $\widehat {\cal G}$ for which all weight spaces have finite multiplicity then the formal characters of the irreducible representations can be calculated. Such a character is the sum of exponentials of the weights of the representation, each weight occurring as many times as its multiplicity. It is shown that the character can be expressed as a quotient of sums that are of a similar form. For the 1-dimensional representation of an affine Lie algebra the character is 1, so that the numerator sum and the denominator sum are equal. This yields a formal power series identity in variables $u\sb 0,\dots,u\sb \ell$, and by $q$-specialization (setting $u\sb i=q\sp{s\sb i}$ for positive integers $s\sb i$) it yields $q$- series identities (the denominator formula). One of the simplest special cases is the Jacobi triple product identity. It is shown that the numerator and denominator sums can be expressed as sums of many variable theta functions. Moreover, the character itself can be expressed as a linear combination of multivariable theta functions with string function coefficients. The study of the transformation properties of all these objects is very complicated and takes up the remainder of the book.
The volume is by no means error free, but given the enormous amount of material covered, I found the error density to be relatively low. The English translation is clear but rather stilted in places, e.g., the use of “really” for “indeed”. Volume 3 does not stand alone; the reader is frequently referred to prior volumes for definitions, notation and results. This is a real problem given the unreasonable expense of these volumes. (In this regard, 32 pages of the review copy were double printed and partially illegible.) I was disappointed that there was no treatment of the Macdonald conjectures or of the generalized hypergeometric functions of Gel’fand and collaborators. All in all though, the complete opus, providing easy access to a large share of the known connections between groups and special functions, should prove extremely valuable to the scientific community and a fitting monument to N. Ya. Vilenkin.