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 principal series representations of certain kinds of

topological group In mathematics, topological groups are 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 structures ...

''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 operator (mathematics), operators in a variety of mathematical ...

, 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 re ...

of ''G'' will decompose according to some kind of

continuous spectrum In the physical sciences, the term ''spectrum'' was introduced first into optics by Isaac Newton in the 17th century, referring to the range of colors observed when white light was dispersion (optics), dispersed through a prism (optics), prism. ...

, of representations involving a continuous parameter, as well as a

discrete spectrum In the physical sciences, the term ''spectrum'' was introduced first into optics by Isaac Newton in the 17th century, referring to the range of colors observed when white light was dispersion (optics), dispersed through a prism (optics), prism. ...

. The ''principal series'' representations are some

induced representation In group theory, the induced representation is a group representation, 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 "m ...

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 (mathematics), 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-d ...

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 measur ...

consists of 'atoms' of the unitary dual (points carrying a

Plancherel measure In mathematics, Plancherel measure is a measure defined on the set of irreducible unitary representations of a locally compact group G, that describes how the regular representation breaks up into irreducible unitary representations. In some cas ...

> 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 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 ...

''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 conse ...

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

maximal compact subgroup In 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. T ...

. 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 :H := ''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 \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...

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 ...

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 diagona ...

inside the

special linear group In mathematics, the special linear group \operatorname(n,R) of degree n over a commutative ring R is the set of n\times n Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix ...

.) 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 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 ...

s (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 number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infin ...

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 n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...

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 metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact t ...

, 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