In mathematics, the orbit method (also known as the Kirillov theory, the method of coadjoint orbits and by a few similar names) establishes a correspondence between irreducible

unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ' ...

s of a

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

and its

coadjoint orbit In mathematics, the coadjoint representation K of a Lie group G is the dual of the adjoint representation. If \mathfrak denotes the Lie algebra of G, the corresponding action of G on \mathfrak^*, the dual space to \mathfrak, is called the coadjoint ...

s: orbits of the action of the group on the dual space of its

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

. The theory was introduced by for

nilpotent group In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . In ...

s and later extended by

Bertram Kostant Bertram Kostant (May 24, 1928 – February 2, 2017) was an American mathematician who worked in representation theory, differential geometry, and mathematical physics. Early life and education Kostant grew up in New York City, where he gradu ...

, Louis Auslander, Lajos Pukánszky and others to the case of

solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group (mathematics), group that can be constructed from abelian groups using Group extension, extensions. Equivalently, a solvable group is a ...

s. Roger Howe found a version of the orbit method that applies to ''p''-adic Lie groups.

David Vogan David Alexander Vogan, Jr. (born September 8, 1954) is a mathematician at the Massachusetts Institute of Technology who works on unitary representations of simple Lie groups. While studying at the University of Chicago, he became a Putnam Fellow ...

proposed that the orbit method should serve as a unifying principle in the description of the unitary duals of real reductive Lie groups.

Relation with symplectic geometry

One of the key observations of Kirillov was that coadjoint orbits of a Lie group ''G'' have natural structure of

symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called s ...

s whose symplectic structure is invariant under ''G''. If an orbit is the

phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...

of a ''G''-invariant classical mechanical system then the corresponding quantum mechanical system ought to be described via an irreducible unitary representation of ''G''. Geometric invariants of the orbit translate into algebraic invariants of the corresponding representation. In this way the orbit method may be viewed as a precise mathematical manifestation of a vague physical principle of quantization. In the case of a nilpotent group ''G'' the correspondence involves all orbits, but for a general ''G'' additional restrictions on the orbit are necessary (polarizability, integrality, Pukánszky condition). This point of view has been significantly advanced by Kostant in his theory of

geometric quantization In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way ...

of coadjoint orbits.

Kirillov character formula

For a

G

, the

Kirillov orbit method In mathematics, the orbit method (also known as the Kirillov theory, the method of coadjoint orbits and by a few similar names) establishes a correspondence between irreducible unitary representations of a Lie group and its coadjoint orbits: orbit ...

gives a heuristic method in

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

. It connects the

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

s of

s, which lie in the

dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...

of the

of ''G'', to the

infinitesimal character In mathematics, the infinitesimal character of an irreducible representation ρ of a semisimple Lie group ''G'' on a vector space ''V'' is, roughly speaking, a mapping to scalars that encodes the process of first differentiating and then diagonal ...

s of the irreducible representations. The method got its name after the

Russia Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eigh ...

n mathematician

Alexandre Kirillov Alexandre Aleksandrovich Kirillov (russian: Алекса́ндр Алекса́ндрович Кири́ллов, born 1936) is a Soviet and Russian mathematician, known for his works in the fields of representation theory, topological groups a ...

. At its simplest, it states that a character of a Lie group may be given by the

of the

Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...

support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...

ed on the coadjoint orbits, weighted by the square-root of the

Jacobian In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: *Jacobian matrix and determinant *Jacobian elliptic functions *Jacobian variety *Intermediate Jacobian In mathematics, the intermediate Jacobian of a compact Kähler m ...

of the exponential map, denoted by

j

. It does not apply to all Lie groups, but works for a number of classes of

connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...

Lie groups, including

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...

, some

semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...

groups, and

compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural ge ...

Special cases

Nilpotent group case

Let ''G'' be a

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...

. Kirillov proved that the equivalence classes of

irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergen ...

s of ''G'' are parametrized by the ''coadjoint orbits'' of ''G'', that is the orbits of the action ''G'' on the dual space

\mathfrak^*

of its Lie algebra. The Kirillov character formula expresses the

Harish-Chandra character In mathematics, the Harish-Chandra character, named after Harish-Chandra, of a representation of a semisimple Lie group ''G'' on a Hilbert space ''H'' is a distribution on the group ''G'' that is analogous to the character of a finite-dimensional re ...

of the representation as a certain integral over the corresponding orbit.

Compact Lie group case

Complex irreducible representations of

compact Lie group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...

s have been completely classified. They are always finite-dimensional, unitarizable (i.e. admit an invariant positive definite

Hermitian form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows ...

) and are parametrized by their

highest weight In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multipli ...

s, which are precisely the dominant integral weights for the group. If ''G'' is a compact

semisimple Lie group In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...

with a

Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by � ...

''h'' then its coadjoint orbits are

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

and each of them intersects the positive Weyl chamber ''h''^*₊ in a single point. An orbit is integral if this point belongs to the weight lattice of ''G''. The highest weight theory can be restated in the form of a bijection between the set of integral coadjoint orbits and the set of equivalence classes of irreducible unitary representations of ''G'': the highest weight representation ''L''(''λ'') with highest weight ''λ''∈''h''^*₊ corresponds to the integral coadjoint orbit ''G''·''λ''. The Kirillov character formula amounts to the character formula earlier proved by

Harish-Chandra Harish-Chandra FRS (11 October 1923 – 16 October 1983) was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups. Early life Harish-Chandra wa ...

References