Alexandre Aleksandrovich Kirillov

Born: 9 May 1936 in Moscow, USSR, (now Russia)

Alexander Aleksandrovich Kirillov was brought up and educated in Moscow. He excelled at mathematics while at school and he won many mathematical competitions. He entered Moscow State University where, while still an undergraduate, he published The representations of the group of rotations of an n-dimensional Euclidean space by spherical vector fields (1957). He was awarded a Master's degree in 1959. He continued to undertake research for his Candidate's Degree (equivalent to a Ph.D.) with Israil Moiseevic Gelfand as his thesis advisor, publishing papers such as Induced representations of nilpotent Lie groups (1959), and On unitary representation of nilpotent Lie groups (1960). In 1961 a seminar on representation theory was set up at the university and Kirillov became a member of this seminar from its initiation. He submitted his thesis Unitary representations of nilpotent Lie groups in 1962 but this was considered a work of such exceptional quality that he was awarded a doctorate (equivalent to a D.Sc. or perhaps the habilitation). J M G Fell reviewing the paper in which Kirillov published the proofs of the results from his thesis writes:-
These results give beautifully compact and complete answers in the nilpotent case to most of the questions usually asked about unitary representations of a given group, namely, the classification of its irreducible representations, the decomposition of a representation restricted to a subgroup or induced from a subgroup, the structure of Kronecker products, the Plancherel formula, and the nature of the infinitesimal and distribution characters of the irreducible representations. ...
Another indication of the remarkable quality of the mathematics he had produced while a graduate student is that he was an invited speaker at the International Congress of Mathematicians held in Stockholm in August 1962.

Kirillov continued working at Moscow State University after the award of his doctorate, being made a professor in 1965. Also in 1965 he became a member of the Keldysh Institute of Applied Mathematics. In 1966 the International Congress of Mathematicians was held in Moscow and again Kirillov was invited to lecture - he gave the lecture Theory of group representations which was published in the Proceedings. In 1967 he became a member of the editorial board of the journal Functional Analysis and Applications. The editor-in-chief of the journal (holding this role from the founding of the journal) was his former thesis advisor Israil Moiseevic Gelfand, and both Kirillov and Gelfand continued to hold these roles on the editorial board until 1988 when Kirillov took over as editor-in-chief. Kirillov published a series of important papers in Functional Analysis and Applications. In particular in the 1960s these were: On certain division algebras over the field of rational functions (1967), Plancherel measure for nilpotent Lie groups (1967), The method of orbits in the theory of unitary representations of Lie groups (1968), Characters of unitary representations of Lie groups (1968), The structure of the Lie field associated with a semisimple decomposable Lie algebra (1969), and Characters of unitary representations of Lie groups: Reduction theorems (1969).

In 1972 Kirillov published his classic textbook (written in Russian) Elements of the Theory of Representations. A reviewer writes:-

The book under review, written by a leading figure in the field, constitutes the first treatise devoted to the theory of group representations that presents not only the foundations or some special aspects of the theory but also enough material to give a fair idea of what this domain of contemporary mathematics actually is.
The book was translated into French, published in 1974, and into English, published in 1976. A second Russian edition appeared in 1978. In 1991 Topics in representation theory was published containing papers written by members of the seminar on representation theory at Moscow University. Kirillov edited the volume and wrote about the developments since his 1961 textbook [1]:-
Some 20 years ago in the book "Elements of the Theory of Representations" I wrote that representation theory enters the fourth stage of its evolution, when the three main directions of its development will presumably be the following ones:

- representations of algebraic groups over local fields and adele rings;

- representations of infinite-dimensional Lie groups and Lie algebras;

- representations of Lie supergroups and Lie superalgebras.

These directions, together with the application of the orbit method, formed the main topic of discussion at our seminar.

Of course, real life corrected this prediction. Two new areas of mathematical physical physics, namely the theory of completely integrable systems and string theory, have had a great influence on mathematics in general, and on representation theory in particular. One must also mention here the growing influence of the idea of supersymmetry. Originally this idea appeared in theoretical physics as a means to "put on an equal footing" fermions and bosons, but later it became a general mathematical principle.

At the International Congress of Mathematicians held in Helsinki in August 1978 Kirillov was an invited speaker - it was the third International Congress of Mathematicians which he had been invited to address and he gave the lecture Infinite-dimensional groups, their representations, orbits, invariants. With A D Gvishiani, he published Theorems and Problems of Functional Analysis. Russian editions appeared in 1979 and 1988, a French edition in 1982, an English edition in 1982, an Italian edition in 1983, and a Hungarian edition in 1985. Frank Smithies begins a review as follows:-
This text, based on courses and seminars at Moscow University, consists of three main parts: expository text, problems and hints for solution. The exposition is concise but very clear.
In 1993 Kirillov published the text with the interesting title What is a number? Alexander Gutman indicates its contents:-
The main goal of the book is to show the meaning of the notion(s) of number in contemporary mathematics, to inform about problems arising in connection with various approaches to the notion and about methods of solving such problems. In each case, the very beginning of the corresponding theory is outlined and suitable literature is indicated for further details. Among other subjects, the basic notions of p-adic and non-standard analysis are presented, and the concepts of a quaternion and a Cayley number are explained. The author also acquaints the reader with the notion of a von Neumann algebra and the idea of "supermathematics", the calculus of anti-commuting variables. The book is addressed to students and scientists who are interested in applications of mathematics.
Kirillov continued to hold the position of Professor of Mathematics at Moscow State University until 1995. He was also a founding member of the Independent University of Moscow being Professor of the Mathematical College there from 1991 to 1995. He was elected to the Russian Academy of Sciences in 1990. In 1995 he was appointed as Professor of Mathematics at the University of Pennsylvania, Philadelphia. He continues to hold this position. He was Francis J Carey Term Professor in Mathematics at the University of Pennsylvania 1997-2002 and Miller Professor at the University of California, Berkeley, in the spring of 1999. We should also mention the conference Orbit method in Geometry and Physics held in his honour in Marseilles, France, in December 2000.

Kirillov published Lectures on the orbit method in Russian in 2001 and in English in 2004. William McGovern writes:-

This is the first systematic and reasonably self-contained exposition of the orbit method for representations of Lie groups to appear in print. It is written by a leading pioneer and major worker in the subject, as an outgrowth of an earlier paper and lectures on the subject, and is aimed at both experts and students. It surveys a large part of the subject, omitting many supplementary details not necessary to follow the main thread. The aim of the orbit method is to unite harmonic analysis and symplectic geometry to describe the representations of a group in geometric terms. This aim has been most successfully realized for nilpotent and solvable Lie groups; most of this book is devoted to their representation theory.
A review in the European Mathematical Society Newsletter states:-
The book offers a nicely written, systematic and readable description of the orbit method for various classes of Lie groups. ... [It] should be on the shelves of mathematicians and theoretical physicists using representation theory in their work.
His latest book is A Tale of Two Fractals (Russian) published in 2009.

In 2006, Kirillov reached 70 years of age and a number of journals on whose editorial boards he had served produced volumes in his honour. We give excerpts from the editors' dedication to Kirillov in the 2006 volume of the Moscow Mathematical Journal, a publication of the Independent University of Moscow distributed by the American Mathematical Society [2]:-

The name of Kirillov is known to everyone who studies representation theory or uses it in one's research. He is a classical author of the subject, the creator and developer of its basic notions and methods. Kirillov's orbit method, the Kirillov-Kostant bracket, Kirillov's character formula, the Gelfand-Kirillov conjecture, the Gelfand-Kirillov dimension, Kirillov's model, are terms firmly established in the language of mathematics. Kirillov's orbit method is one of the most original and fruitful discoveries in representation theory in its hundred-plus year history. This method soon became widely popular and stimulated a flow of results that has not dried up to these days. Another fundamental direction was called to life by the seminal joint papers of Gelfand and Kirillov on the skew fields of fractions of universal enveloping algebras. Wonderful and as always highly original results were obtained by Kirillov in the theory of infinite-dimensional Lie groups and algebras, and their representations. This work is relatively less known and should yet be carefully read and studied. Kirillov's seminar at Moscow State University gathered for 30 years. It is with deep nostalgia that all its participants remember it. Kirillov has numerous students that were formed not only by the problems he put forward, but even more by his powerful personality. ... Alexander Kirillov has the gift of creating a specific atmosphere that stimulates research and has exquisite mathematical taste. All his students and friends warmly remember his out-of-the-city seminar sessions with their traditional soccer and especially volleyball, and charming charades.
We should mention that Andrei Yuryevich Okounkov, who was awarded a Fields medal in 2006, was a student of Kirillov and was introduced to leading edge research in Kirillov's seminar in Moscow. Finally note that Kirillov's son, Alexander Kirillov Jr, is a mathematician undertaking research on the representation theory of Lie groups at the State University of New York at Stony Brook.

Article by: J J O'Connor and E F Robertson

March 2011
MacTutor History of Mathematics