mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 the ca ...

s of a

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

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 coadjoin ...

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 ident ...

. 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, it has a central series of finite length or its lower central series terminates with . I ...

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 gradua ...

, 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 that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminat ...

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 i ...

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 sy ...

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

phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...

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 w ...

of coadjoint orbits.

Kirillov character formula

For a

G

, the Kirillov orbit method gives a heuristic method in

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

. It connects the

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

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 cons ...

of the

of ''G'', to the

infinitesimal character In mathematics, the infinitesimal character of an irreducible representation \rho 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 diagon ...

s of the

irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...

s. The method got its name after the

Russia Russia, or the Russian Federation, is a country spanning Eastern Europe and North Asia. It is the list of countries and dependencies by area, largest country in the world, and extends across Time in Russia, eleven time zones, sharing Borders ...

n mathematician

Alexandre Kirillov Alexandre Aleksandrovich Kirilloff (, born 1936) is a Soviet and Russian mathematician, known for his works in the fields of representation theory, topological groups and Lie groups. In particular he introduced the orbit method into representa ...

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

of the

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

support Support may refer to: Arts, entertainment, and media * Supporting character * Support (art), a solid surface upon which a painting is executed Business and finance * Support (technical analysis) * Child support * Customer support * Income Su ...

ed on the coadjoint orbits, weighted by the square-root of the Jacobian 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 (mathematics), 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, along with its sister Idempotent (ring theory), idem ...

, 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 gen ...

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 (topology), path between two points can be continuously transformed into any other such path while preserving ...

. Kirillov proved that the equivalence classes of irreducible

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 r ...

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 space, compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are ...

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 map, linear in each of its arguments, but a sesquilinear f ...

) and are parametrized by their highest weights, which are precisely the dominant integral weights for the group. If ''G'' is a compact

semisimple Lie group In mathematics, a simple Lie group is a connected space, connected nonabelian group, non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple ...

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 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 (né Harishchandra) 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 ...

References

* * * * * . * * {{citation, last=Kirillov, first=A. A., title=Lectures on the orbit method, series=

Graduate Studies in Mathematics Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books *1 ''The General T ...