Principal series representation

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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 representations constructed in a uniform way, in order to fill out the continuous part of the spectrum.

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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

G = KAN

with K a maximal compact subgroup. Then H is chosen to contain AN (which is a non-compact solvable Lie group), being taken as

H := 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 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).

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]

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References

  1. Bump, Daniel (1997), Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, doi:10.1017/CBO9780511609572, ISBN   978-0-521-55098-7, MR   1431508