In mathematics, the principal series representations of certain kinds of

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...

''G'' occur in the case where ''G'' is not a

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 ...

. There, by analogy with

spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...

, one expects that the

regular representation In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation. One distinguishes the left regular rep ...

of ''G'' will decompose according to some kind of continuous spectrum, of representations involving a continuous parameter, as well as a

discrete spectrum A observable, physical quantity is said to have a discrete spectrum if it takes only distinct values, with gaps between one value and the next. The classical example of discrete spectrum (for which the term was first used) is the characterist ...

. The ''principal series'' representations are some

induced representation In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" represe ...

s constructed in a uniform way, in order to fill out the continuous part of the spectrum. In more detail, the

unitary dual 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 ''G'' ...

is the space of all representations relevant to decomposing the regular representation. The

discrete series In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group ''G'' that is a subrepresentation of the left regular representation of ''G'' on L²(''G''). In the Plancherel meas ...

consists of 'atoms' of the unitary dual (points carrying a Plancherel measure > 0). In the earliest examples studied, the rest (or most) of the unitary dual could be parametrised by starting with a subgroup ''H'' of ''G'', simpler but not compact, and building up induced representations using representations of ''H'' which were accessible, in the sense of being easy to write down, and involving a parameter. (Such an induction process may produce representations that are not unitary.) For the case of a

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 ...

''G'', the subgroup ''H'' is constructed starting from the

Iwasawa decomposition In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a cons ...

:''G'' = ''KAN'' with ''K'' a

maximal compact subgroup In mathematics, a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the class ...

. Then ''H'' is chosen to contain ''AN'' (which is a non-compact

solvable Lie group In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted : mathfrak,\mathfrak/math> that consist ...

), being taken as :''MAN'' with ''M'' the

centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...

in ''K'' of ''A''. Representations ρ of ''H'' are considered that are irreducible, and unitary, and are the

trivial representation In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a ...

on the subgroup ''N''. (Assuming the case ''M'' a trivial group, such ρ are analogues of the representations of the group of

diagonal matrices In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...

inside the

special linear group In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the ge ...

.) The induced representations of such ρ make up the principal series. The spherical principal series consists of representations induced from 1-dimensional representations of ''MAN'' obtained by extending characters of ''A'' using the homomorphism of ''MAN'' onto ''A''. There may be other continuous series of representations relevant to the unitary dual: as their name implies, the principal series are the 'main' contribution. This type of construction has been found to have application to groups ''G'' that are not Lie groups (for example,

finite groups of Lie type In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phra ...

, groups over

p-adic field In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensio ...

s).

Examples

For examples, see the representation theory of SL₂(R). For the

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

GL₂ over a

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...

, the dimension of the Jacquet module of a principal series representation is two.

References

External links

*{{springer, id=C/c025750, title=Continuous series of representations, author=A.I. Shtern
''Computing the unitary dual'' (PDF)
Unitary representation theory Representation theory of Lie groups