The Generalized Segal–Bargmann Transform and Special Functions

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.

Weights for symmetric and general linear groups

Introduction Representations of $\mathbf {SU}(2)$ The general theory of linear representations of compact groups Lie theory of representations of compact connected Lie groups Induced representations of compact groups Spherical functions on compact groups Examples spherical harmonics The general theory of spherical functions Fourier and Plancherel transforms Extension of the Plancherel transform The subtleties of harmonic analysis Differential properties of spherical functions on Lie groups Spherical functions on semisimple Lie groups More on $\mathbf {SL}(2, \mathbf R)$ Automorphic functions Groups of isometries and Bessel functions Other special functions References.

It is shown that the general addition theorems of Gegenbauer, relating Bessel functions and Gegenbauer polynomials, are special cases of identities for special functions obtained from a study of certain local irreducible representations of the complex Euclidean group in 3-space. Among the physically interesting results generalized by this analysis are the expansion for a plane wave as a sum of spherical waves and the addition theorem for spherical waves. This paper is one of a series attempting to derive the special functions of mathematical physics and their basic properties from the representation theory of Lie symmetry groups.

Representations of Lie groups are abstractions of continuous symmetries of linear spaces.By considering their differentials (linear approximations), we obtain representations of Lie algebras.These are regarded as generalizations of the Fourier analysis and important tools not only in mathematics but also in physics.In this article, I present fundamental and important examples on representations of Lie groups and Lie algebras and present some of my latest results.

The first part of this paper deals with problems concerning the symmetric algebra of complex-valued polynomial functions on the complex vector space of n by k matrices. In this context, a generalization of the so-called “classical separation of variables theorem” for the symmetric algebra is obtained. The second part is devoted to the study of certain linear representations, on the above linear space (the symmetric algebra) and its subspaces, of the complex general linear group of order k and of its subgroups, namely, the unitary group, and the real and complex special orthogonal groups. The results of the first part lead to generalizations of several well-known theorems in the theory of group representations. The above representation, of the real special orthogonal group, which arises from the right action of this group on the underlying vector space (of the symmetric algebra) of matrices, possesses interesting properties when restricted to the Stiefel manifold. The latter is defined as the orbit (under the action of the real special orthogonal group) of the n by k matrix formed by the first n row vectors of the canonical basis of the k-dimensional real Euclidean space. Thus the last part of this paper is involved with questions in harmonic analysis on this Stiefel manifold. In particular, an interesting orthogonal decomposition of the complex Hilbert space consisting of all square-integrable functions on the Stiefel manifold is also obtained.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Schrodinger equations for many potentials are solved in terms of the special functions. Almost all of these special functions are the matrix elements of the representations of the Lie groups, with their arguments being the group parameters [1,2]. Even the simplest “special functions” namely the elementary transcendentals are related to the Lie groups, i.e., the one parameter Abelian Lie groups. The connection between the group representations and the special functions explains the mystrious properties of them, such as the recurance relations and addition theorems.

Elliptic operators arise naturally in several different mathematical settings, notably in the representation theory of Lie groups, the study of evolution equations, and the examination of Riemannian manifolds. This book develops the basic theory of elliptic operators on Lie groups and thereby extends the conventional theory of parabolic evolution equations to a natural non-commutative context. In order to achieve this goal, the author presents a synthesis of ideas from partial differential equations, harmonic analysis, functional analysis, and the theory of Lie groups. He begins by discussing the abstract theory of general operators with complex coefficients before concentrating on the central case of second-order operators with real coefficients. A full discussion of second-order subellilptic operators is also given. Prerequisites are a familiarity with basic semigroup theory, the elementary theory of Lie groups, and a firm grounding in functional analysis as might be gained from the first year of a graduate course.

As it was recognized twenty five years ago by A. A. Kirillov, in the unitary representation theory of nilpotent Lie groups a crucial role is played by orbits of the coadjoint representation. B. Kostant noted that, for any connected Lie group, these orbits admit a symplectic structure and lend themselves to an intrinsic characterization. The present author later observed, that already for the purposes of unitary representation theory of solvable Lie groups, this concept has to be enlarged and replaced by that of a generalized orbit. One objective of this paper is their intrinsic characterization. Other results prepare the way for the geometric construction of the corresponding unitary representations, to be developed later.

Multiplicity One Theorems for Generalized Gelfand-Graev Representations of Semisimple Lie Groups and Whittaker Models for the Discrete Series

This paper gives a survey on the relation between Hibi algebras and representation theory. The notion of Hodge algebras or algebras with straightening laws has been proved to be very useful to describe the structure of many important algebras in classical invariant theory and representation theory (Bruns and Herzog 1993; De Concini et al. 1982; Eisenbud 1980; Gonciulea and Lakshmibai 2001; Seshadri 2007). In particular, a special type of such algebras introduced by Hibi (1987) provides a nice bridge between combinatorics and representation theory of classical groups. We will examine certain poset structures of Young tableaux and affine monoids, Hibi algebras in toric degenerations of flag varieties, and their relations to polynomial representations of the complex general linear group.

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.

This study investigates the representation theory of the multiplicative group of a finite nearfield, with a specific focus on the group <img src=image/13440431_01.gif>, arising from Dickson nearfields. Nearfields, generalizations of fields lacking one of the two distributive laws, pose interesting challenges in algebraic structures due to their noncommutativity and asymmetry. Their multiplicative groups, particularly those derived from Dickson nearfields, have a metacyclic structure that allows for intricate analysis using representation theory. The purpose of this research is to construct and analyze the character tables of these multiplicative groups. We explore foundational concepts such as Schur's lemma, Maschke's theorem, and orthogonality relations, and apply them within the context of finite nearfields. A combination of theoretical and computational methodologies is employed. Specifically, we make use of the Magma algebra system to build and verify character tables for <img src=image/13440431_01.gif> for small values of <img src=image/13440431_02.gif> and <img src=image/13440431_03.gif>, including detailed derivations for <img src=image/13440431_04.gif> and <img src=image/13440431_05.gif>. The principal results include explicit character tables, classifications of conjugacy classes, degrees of irreducible representations, and a generalization framework for any pair (<img src=image/13440431_06.gif>). The research demonstrates that the multiplicative groups <img src=image/13440431_01.gif> are not only metacyclic but also nilpotent and solvable under specific conditions. It further establishes structural conditions that determine the conjugacy class partitioning and representation degrees. Our conclusions highlight the deep connection between nearfield algebraic properties and finite group representation theory. This work contributes to the theoretical development of group representations in non-classical algebraic settings and opens doors to applications in geometry, coding theory, and cryptography. Practically, the use of Magma facilitates automated representation analysis, and socially, this research promotes a broader understanding of non-traditional algebraic systems. While the current study is computationally verified for small instances, future work will aim to optimize algorithms for handling larger <img src=image/13440431_01.gif> structures, enhancing scalability and applicability.

Since its inception in the early seventies, the study of Enveloping Algebras has undergone a significant and continuous evolution and moreover has inspired a wide variety of developments in many areas of mathematics including Ring Theory, Differential Operators, Invariant Theory, Quantum Groups and Hecke Algebras. As indicated above, one of the main goals behind this meeting was to bring together a group of participants with a wide range of interests in and around the geometric and the combinatorial side of the representation theory of Lie groups and algebras. We strongly believe that such an approach to representation theory, in particular interaction between geometry and representation theory, will open up new avenues of thought and lead to progress in a number of areas. This diversity was well reflected in the expertize represented by the conference participants, as well as in the wide range of topics covered. They may broadly be summarized under the following three headings: (1) The study of the structure and representation theory through a) Slodowy slices and the Joseph ideal, b) Equivalence of categories of representations and certain geometric counterparts, c) Hall algebras, Quotient schemes and canonical bases, d) Richardson elements and birationality questions, e) free Lie algebras and current algebras, f) Dirac cohomology, h) Gelfand-Zeitlin theory (2) Combinatorial Aspects of Lie Systems particularly through a) Saturation problems and Buildings, b) Affine and double Affine Hecke algebras, Cherednik algebras, c) Solvable lattice models (3) Geometric Structures including a) Representation theoretic methods in enumerative geometry, b) Multiplicity free symplectic reduction, c) Monodromy actions of braid groups, d) Associated varieties for Lie super algebras, e) affine Grassmannians, f) Schubert varieties and Demazure modules The single most important development reported in this conference seems to us to be the generalization of the categorical Satake isomorphism to quantum groups at roots of unity, which allows to bring in a whole lot of new geometry into representation theory. It seems likely that it will lead to the proofs of some open conjectures of Lusztig. We were particularly happy to have quite a few very strong younger participants and some highly promising postdocs. We tried to accomodate young postdocs, as much as feasible, to present their work by devoting one afternoon to their (shorter) talks.