Principal series representation
Encyclopedia
In mathematics
, the principal series representations of certain kinds of topological group
G occur in the case where G is not a compact group
. There, by analogy with spectral theory
, one expects that the regular representation
of G will decompose according to some kind of continuous spectrum
, of representations involving a continuous parameter, as well as a discrete spectrum
. The principal series representations are some induced representation
s constructed in a uniform way, in order to fill out the continuous part of the spectrum.
In more detail, the unitary dual is the space of all representations relevant to decomposing the regular representation. The discrete series 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 G, the subgroup H is constructed starting from the Iwasawa decomposition
with K a maximal compact subgroup
. Then H is chosen to contain AN (which is a non-compact solvable Lie group), being taken as
with M the centralizer in K of A. Representations ρ of H are considered that are irreducible, and unitary, and are the trivial representation
on the subgroup N. (Assuming the case M a trivial group, such ρ are analogues of the representations of the group of diagonal matrices inside the special linear group
.) 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, groups over p-adic fields).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the principal series representations of certain kinds of topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
G occur in the case where G is not a compact group
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
. There, by analogy with spectral theory
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 of studies of linear algebra and the solutions of...
, one expects that the regular representation
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....
of G will decompose according to some kind of continuous spectrum
Continuous spectrum
The spectrum of a linear operator is commonly divided into three parts: point spectrum, continuous spectrum, and residual spectrum.If H is a topological vector space and A:H \to H is a linear map, the spectrum of A is the set of complex numbers \lambda such that A - \lambda I : H \to H is not...
, of representations involving a continuous parameter, as well as a discrete spectrum
Discrete spectrum
In physics, an elementary explanation of a discrete spectrum is that it is an emission spectrum or absorption spectrum for which there is only an integer number of intensities. Atomic electronic absorption and emission spectrum are discrete, as contrasted with, for example, the emission spectrum...
. The principal series representations are some induced representation
Induced representation
In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...
s constructed in a uniform way, in order to fill out the continuous part of the spectrum.
In more detail, the unitary dual is the space of all representations relevant to decomposing the regular representation. The discrete series consists of 'atoms' of the unitary dual (points carrying a Plancherel measure
Plancherel measure
In mathematics, Plancherel measure is a probability measure defined on the set of irreducible representations of a finite group G. In some cases the term Plancherel measure is applied specifically in the context of the group G being the finite symmetric group S_n – see below...
> 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 G, the subgroup H is constructed starting from the Iwasawa decomposition
Iwasawa decomposition
In mathematics, the Iwasawa decomposition KAN 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...
- G = KAN
with K a maximal compact subgroup
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....
. Then H is chosen to contain AN (which is a non-compact solvable Lie group), being taken as
- MAN
with M the centralizer in K of A. Representations ρ of H are considered that are irreducible, and unitary, and are the trivial representation
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...
on the subgroup N. (Assuming the case M a trivial group, such ρ are analogues of the representations of the group of diagonal matrices inside the special linear group
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
.) 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, groups over p-adic fields).