Principal series representation

In mathematics, the principal series representations of certain kinds of topological groupG 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.

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 groupG, 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

con M el centralizador en K de A. Se consideran representaciones ρ de H que son irreducibles y unitarias, y son la representación trivial en el subgrupo N. (Suponiendo el caso M un grupo trivial, tales ρ son análogas a las representaciones del grupo de matrices diagonales dentro del grupo lineal especial ). Las representaciones inducidas de tales ρ conforman la serie principal. La serie principal esférica consiste en representaciones inducidas a partir de representaciones unidimensionales de MAN obtenidas al extender caracteres de A usando el homomorfismo de MAN sobre A.

Puede haber otras series continuas de representaciones relevantes para el dual unitario: como su nombre lo indica, las series principales son la contribución "principal".

Se ha descubierto que este tipo de construcción tiene aplicación a grupos G que no son grupos de Lie (por ejemplo, grupos finitos de tipo Lie , grupos sobre cuerpos p-ádicos ).

Para ejemplos, véase la teoría de representación de SL ₂ (R) . Para el grupo lineal general GL ₂ sobre un cuerpo local , la dimensión del módulo de Jacquet de una representación en serie principal es dos. ^{[ 1 ]}

• AI Shtern (2001) [1994], "Series continuas de representaciones" , Enciclopedia de Matemáticas , EMS Press
• Cálculo de la función de densidad de probabilidad (PDF) unitaria dual