*(English)*Zbl 0809.22001

This is volume 2 of a three volume updating of Vilenkin’s classic 1965 book “Special Functions and the Theory of Group Representations” [1965; Zbl 0144.380; for the reviews of Vol. I and III see Zbl 0742.22001 and Zbl 0778.22001]. In all these books 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 group 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. Also, 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.

There are five chapters in this volume. The first, Chapter 9, is concerned with special functions that are connected with the orthogonal groups $SO\left(n\right)$, the Lorentz groups $SO(n-1,1)$, the Euclidean groups $ISO(n-1)$, the Poincaré groups $IS{O}_{o}(n-2,1)$, and the pseudo- orthogonal groups $S{O}_{o}(p,q)$. It and the following chapter are an expansion and updating of the final three chapters in Vilenkin’s 1965 book. Here, $S{O}_{o}(p,q)$ with $p+q=n$, is characterized as the connected component of the identity of the linear group of transformations preserving the bilinear form ${[x,y]}_{pq}=-{x}_{1}{y}_{1}-\cdots -{x}_{p}{y}_{p}+{x}_{p+1}{y}_{p+1}+\cdots +{x}_{p+q}{y}_{p+q}$. Thus they have a natural action on the spaces of hyperboloids ${[x,x]}_{pq}=c$, (spheres if $q=0$, $c<0$), and cones ${[x,x]}_{pq}=0$. Type I representations of these groups are typically constructed in terms of the group action in the space of functions on the cone that are homogeneous of degree $\sigma $. The representations of the Euclidean and Poincaré groups are obtained as contractions of representations of pseudo-orthogonal groups. Various parametrizations of the groups are studied (Iwasawa, Gauss and Bruhat decompositions, etc.) and these yield orthogonal coordinate systems for the hyperboloids and cones on which the groups act. For each representation, bases for the representation space are chosen which have simple transformation properties on restriction to the subgroup chain $SO(n-1)\supset SO(n-2)\supset \cdots \supset SO\left(1\right)$. The appropriate zonal spherical functions are computed and found to be Gegenbauer polynomials for the orthogonal and pseudo-orthogonal groups and Bessel functions for the inhomogeneous groups. General matrix elements are computed in these bases and shown to be expressible as products of Jacobi polynomials and Bessel functions, and the corresponding addition theorems are derived.

There is a second major theme in this chapter: the study of harmonic functions. Each of the groups studied is associated with a Laplace operator. In the case of $S{O}_{o}(p,q)$ the operator is ${L}_{pq}=-\frac{{\partial}^{2}}{\partial {x}_{1}^{2}}-\cdots -\frac{{\partial}^{2}}{\partial {x}_{p}^{2}}+\frac{{\partial}^{2}}{\partial {x}_{p+1}^{2}}+\cdots +\frac{{\partial}^{2}}{\partial {x}_{p+q}^{2}}$. The harmonic functions are those that are in the kernel of the Laplace operator. Since the Laplace operator commutes with the group action one can decompose the space of harmonic functions into irreducible subspaces.

Finally, the authors study a limiting case as $n\to \infty $, related to probability theory, to obtain the groups $O\left(\infty \right)$, $IO\left(\infty \right)$ whose action commutes with the infinite dimensional Laplace operator. The corresponding harmonic basis functions can be expressed as infinite products of Hermite polynomials.

Chapter 10 concerns the same representations of the same groups, but now the basis vectors in the representation spaces are allowed to have simple transformation properties with respect to subgroup chains other than just the “canonical” chain $SO(n-1)\supset SO(n-2)\supset \cdots \supset SO\left(1\right)$. (In particular, some of the subgroups in the chain may be non- compact.) A great variety of results is obtained. In particular, continuum bases now arise and the addition theorems may be expressible as integral identities rather than pure summation identities. Products of Gegenbauer type functions are typical basis elements and transition coefficients relating different bases involve the Meijer $G$-function.

Chapter 11 is concerned with special functions connected with the unitary groups $U\left(n\right)$, $U(n-1,1)$, and the inhomogeneous unitary groups $IU(n-1)$. The approach and results are very similar to that of Chapter 9, except that the bilinear form is now replaced with a Hermitian form which, in the case of $U(n-1,1)$, is $[z,w]=-{z}_{1}\overline{{w}_{1}}-\cdots -{z}_{n-1}\overline{{w}_{n-1}}+{z}_{n}\overline{{w}_{n}}$. Thus these groups have a natural action on the spaces of complex hyperboloids $[z,z]=c$, and cones $[z,z]=0$. Type I representations of these groups are typically constructed in terms of the group action on the space of functions on the complex cone that are homogeneous of degree $\sigma $. For each representation, bases for the representation space are chosen which have simple transformation properties on restriction to the subgroup chain $U(n-1)\supset U(n-2)\supset \cdots \supset U\left(1\right)$. The appropriate zonal spherical functions are computed and found to be Gegenbauer polynomials for the unitary groups and Bessel functions for the inhomogeneous groups. General matrix elements are computed in these bases and found to be expressible as products of hypergeometric functions of higher order and Bessel functions, and the corresponding addition theorems are derived. If we write ${z}_{j}={x}_{j}+i{y}_{j}$, ${x}_{j},{y}_{j}$ real, then the complex spheres, hyperboloids and cones can be considered as real spheres, hyperboloids and cones, and the unitary groups can be considered as subgroups of $SO\left(2n\right)$, $SO(2n-2,2)$, etc. This point of view allows the derivation of new identities. The chapter closes with brief consideration of special functions associated with the symplectic groups $Sp\left(n\right)$ and $Sp(n-1,1)$. Here the point of view is that these groups preserve Hermitian forms over the field of quaternions.

Chapter 12 is concerned with special functions associated with the Heisenberg group ${H}_{n}$. This is the connected $2n+1$ dimensional group whose Lie algebra has the basis ${Q}_{j},{P}_{k},H$, $1\le j,k\le n$ and nonzero commutation relations $[{Q}_{j},{P}_{k}]={\delta}_{jk}H$. This group admits a family of irreducible unitary representations ${R}^{\lambda}$, parametrized by the nonzero real number $\lambda $, and known as the Schrödinger representations. These representations have a simple realization on the Hilbert space of square integrable functions of $n$ real variables, and are ubiquitous in quantum mechanics. There is a standard basis for this space, expressed in terms of products of Hermite polynomials, and the corresponding matrix elements are products of associated Laguerre polynomials. However, there is a second realization of the representations ${R}^{\lambda}$ in terms of a Hilbert space of entire functions in $n$ complex variables, the Fock space realization. Here the basis functions are simple monomials. The intertwining between these two realizations is the source of many interesting identities. Finally, there is a third realization of ${R}^{2\pi}$ in terms of a space of functions on a compact set, with certain periodicity properties. The intertwining mapping between the Schrödinger representation of ${R}^{2\pi}$ and this third realization is known as the Weil-Brezin-Zak operator and is closely associated with the Poisson summation formula.

Another noteworthy property of the Heisenberg group is that it has a nontrivial outer automorphism group. Indeed, the automorphism group contains the symplectic group $Sp(n,R)$, which itself contains $SO\left(n\right)\times SL(2,R)$ as a subgroup. One can then show that the Schrödinger representation of ${H}_{n}$ can be extended to a representation of the semidirect product group $Sp(n,R)\times {H}_{n}$ on ${L}_{2}\left({R}^{n}\right)$. This is called the Weyl representation and the action of $Sp(n,R)$ is via integral operators. This interplay between group realizations allows the authors to relate special functions associated with the Heisenberg group to special functions associated with the orthogonal and symplectic groups.

Chapter 13 discusses representations of discrete groups and their relation to special functions of a discrete argument. This chapter has no direct counterpart in Vilenkin’s original book and is somewhat sketchy. There is a brief discussion of the representation theory of the symmetric group ${S}_{n}$ and a characterization of the homogeneous space ${X}_{mn}\equiv {S}_{m+n}/{S}_{m}\times {S}_{n}$ as the collection of all $m$-element subsets of a set with $m+n$ elements. The natural action of ${S}_{m+n}$ on the space of functions on ${X}_{mn}$ defines a representation of ${S}_{m+n}$, and this can be decomposed into irreducible representations. The zonal spherical functions, with respect to the subgroup ${S}_{m}\times {S}_{n}$ turn out to be expressible as Hahn polynomials. More complicated cases lead to Racah polynomials. There is a brief review of a connection with combinatorics which subsumes the previous examples: the association between homogeneous graphs and orthogonal polynomials.

Next there is a brief assessment of the relations between special function theory and the representation theory of matrix groups whose elements belong to finite fields (Chevalley groups). For finite fields with $q={p}^{s}$ elements the related spherical functions are expressible in terms of $q$-Hahn polynomials. This leads to preliminary results about $q$-hypergeometric functions which, from a different point of view, are studied more deeply in Volume 3.

Finally, there is a section concerning matrix groups with matrix elements belonging to locally compact totally disconnected fields of characteristic 0 (rational $p$-adic numbers). The authors introduce $p$- adic versions of Gamma, Beta, Bessel and hypergeometric functions and relate them to representations of the Euclidean, general linear and special linear groups over the $p$-adics.

In contrast with Vilenkin’s original volume, this book, though quite useful, is not very stimulating to read. It is written in a linear order (and clearly written) but “it just keeps going”, without any digressions to explain what is important and what is not. There is no philosophy and no speculation. Nevertheless, it is a most impressive achievement.

##### MSC:

22-02 | Research monographs (topological groups) |

22E45 | Analytic representations of Lie and linear algebraic groups over real fields |

43A05 | Measures on groups and semigroups, etc. |

33C80 | Connections of hypergeometric functions with groups and algebras |

33D80 | Connections of basic hypergeometric functions with groups, algebras and related topics |