Principal series representation
In
continuous spectrum, of representations involving a continuous parameter, as well as a discrete spectrum. The principal series representations are some induced representations
constructed in a uniform way, in order to fill out the continuous part of the spectrum.
In more detail, 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
- G = KAN
with K a
solvable Lie group
), being taken as
- H := MAN
with M the
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
p-adic fields
).
Examples
For examples, see the representation theory of SL2(R). For the general linear group GL2 over a local field, the dimension of the Jacquet module of a principal series representation is two.[1]
References
- MR 1431508
External links
- A.I. Shtern (2001) [1994], "Continuous series of representations", Encyclopedia of Mathematics, EMS Press
- Computing the unitary dual (PDF)