Groups and Analysis

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.

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.

The present article covers topics of life, scientific, pedagogical and social activities of the famous Romanian mathematician Simoin Stoilov (1887-1961), professor of Chernivtsi and Bucharest universities. Stoilov was working at Chernivtsi University during 1923-1939 (at this interwar period Chernivtsi region was a part of royal Romania. The article is aimed on the occasion of honoring professors’ memory and his managerial abilities in the selection of scientific and pedagogical staff to ensure the educational process and research in Chernivtsi University in the interwar period. In addition, it is noted that Simoin Stoilov has made a significant contribution to the development of mathematical science, in particular he is the founder of the Romanian school of complex analysis and the theory of topological analysis of analytic functions; the main directions of his research are: partial differential equation; set theory; general theory of real functions and topology; topological theory of analytic functions; issues of philosophy and foundation of mathematics, scientific research methods, Lenin’s theory of cognition. The article focuses on the active socio-political and state activities of Simoin Stoilov in terms of restoring scientific and cultural ties after the Second World War.

The paper presents new developments in harmonic analysis associated with the motion transformations embedded in digital signals. In this context, harmonic analysis provides motion analysis with a complete theoretical construction of perfectly matching concepts and a related toolbox leading to fast algorithms. This theory can be built from only two assumptions: an associative structure for the local motion transformations expressed as a Lie group and a principle of optimality for the global evolution expressed as a variational extremal. Motion analysis means not only detection, estimation, interpolation, and tracking but also propagator's motion-compensated filtering, signal decomposition, and selective reconstruction. The optimality principle defines the trajectory and provides the appropriate equations of motion, the selective tracking equations, the selective constants of motion to be tracked, and all the symmetries to be imposed on the system. The harmonic analysis provides new special functions, orthogonal bases, partial differential equations (PDE), ordinary differential equations (ODE) and integral transforms. The tools to be developed rely on group representations, continuous and discrete wavelets, the estimation theory (prediction, smoothing and interpolation) and filtering theory (Kalman filters, motion-based convolutions, integral transforms). All the algorithms are supported by fast and parallelizable implementations based on the fast Fourier transform (FFT) and dynamic programming.

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.

In this paper, we study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of [P. Hall and C. R. Kulatilaka, A property of locally finite groups, J. Lond. Math. Soc. 39 1964, 235–239] and a characterization of a certain class of Lie groups, due to [S. K. Grosser and W. N. Herfort, Abelian subgroups of topological groups, Trans. Amer. Math. Soc. 283 1984, 1, 211–223], we prove that a c-minimal locally solvable Lie group is compact. It is shown that a topological group G is c-(totally) minimal if and only if G has a compact normal subgroup N such that G / N G/N is c-(totally) minimal. Applying this result, we prove that a locally compact group G is c-totally minimal if and only if its connected component c ⁢ ( G ) c(G) is compact and G / c ⁢ ( G ) G/c(G) is c-totally minimal. Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering [D. Dikranjan and M. Megrelishvili, Minimality conditions in topological groups, Recent Progress in General Topology. III, Atlantis Press, Paris 2014, 229–327, Question 3.10 (b)], we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact.

Over 100 years ago Harald Bohr identified a deep problem about the convergence of Dirichlet series, and introduced an ingenious idea relating Dirichlet series and holomorphic functions in high dimensions. Elaborating on this work, almost twnety years later Bohnenblust and Hille solved the problem posed by Bohr. In recent years there has been a substantial revival of interest in the research area opened up by these early contributions. This involves the intertwining of the classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. New challenging research problems have crystallized and been solved in recent decades. The goal of this book is to describe in detail some of the key elements of this new research area to a wide audience. The approach is based on three pillars: Dirichlet series, infinite dimensional holomorphy and harmonic analysis.

Invariant theory is the great romantic story of mathematics. For one hundred and fifty years, from its beginnings with Boole to the time, around the middle of this century, when it branched off into several independent disciplines, mathematicians of all countries were brought together by their common faith in invariants: in England, Cayley, MacMahon, Sylvester and Salmon, and later, Alfred Young, Aitken, Littlewood and Turnbull. In Germany, Clebsch, Gordan, Grassmann, Sophus Lie, Study; in France, Hermite, Jordan and Laguerre; in Italy, Capelli, d’Ovidio, Brioschi, Trudi and Corrado Segre; in America, Glenn, Dickson, Carus (of the Carus Monographs), Eric Temple Bell and later Hermann Weyl. Seldom in history has an international community of scholars felt so united by a common scientific ideal for so long a stretch of time. In our century, Lie theory and algebraic geometry, differential algebra and algebraic combinatorics are offsprings of invariant theory. No other mathematical theory, with the exception of the theory of functions of a complex variable, has had as deep and lasting an influence on the development of mathematics.

The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker–Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

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.

Since the solution of the celebrated Hilbert’s Fifth Problem, one knows that a locally euclidean topological group is a Lie group. In accordance with Hilbert’s original idea of getting rid of the differentiability hypothesis in Lie theory, one can look for a homotopy analogue of this result. Unfortunately the most straightforward idea is false: Hilton, Roitberg and Stasheff have exhibited a topological group having the homotopy type of a compact manifold and which is not homotopy equivalent to any Lie group (see [Sta69]). Nevertheless, the problem of finding a homotopy characterisation of Lie groups remains open and has been intensively investigated for the last three decades. Obviously, the first step of such a programme is a better understanding of the homotopy properties of Lie groups. One has to restrict this study to the class of compact connected Lie groups: the Iwasawa decomposition tells us that any connected Lie group has the homotopy type of a compact one.

Our objective in this paper is to prove an Implicit Function Theorem for general topological spaces. As a consequence, we show that, under certain conditions, the set of the invertible elements of a topological monoid X is an open topological group in X and we use the classical topological group theory to conclude that this set is a Lie group.

This third chapter presents a brief and somewhat sketchy introduction to the theory of unitary representations of reductive and semisimple Lie groups G. The basic fact for an irreducible unitary representation π of G on a Hilbert space ℋ, is that every irreducible representation к of a maximal compact subgroup K ⊂ G has multiplicity m(к, π| к) ≤ dim к. This yields up the infinitesimal character χπ: L(g)→ ℂ and the distribution character θπ: C c ∞ (G) → ℂ, and consequently the differential equations $$z({\theta _\pi }) = \,{\chi _\pi }(z){\theta _\pi }$$ for $$z \in L{\text{(g)}}$$ which are the starting point for serious harmonic analysis on G.

It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and the structure theory of locally compact abelian groups.[...]

Efficient gluing of numerical continuation and a multiple solution method for elliptic PDEs