Eisenstein integral - Alchetron, The Free Social Encyclopedia

In mathematical representation theory, the Eisenstein integral is an integral introduced by Harish-Chandra (1970, 1972) in the representation theory of semisimple Lie groups, analogous to Eisenstein series in the theory of automorphic forms. Harish-Chandra (1975, 1976a, 1976b) used Eisenstein integrals to decompose the regular representation of a semisimple Lie group into representations induced from parabolic subgroups. Trombi (1989) gave a survey of Harish-Chandra's work on this.

Definition

Harish-Chandra (1970, section 10) defined the Eisenstein integral by

E ( P : ψ : ν : x ) = ∫ K ψ ( x k ) τ ( k − 1 ) exp ⁡ ( ( i ν − ρ P ) H P ( x k ) ) d k

where:

x is an element of a semisimple group G

P = MAN is a cuspidal parabolic subgroup of G

ν is an element of the complexification of a

a is the Lie algebra of A in the Langlands decomposition P = MAN.

K is a maximal compact subgroup of G, with G = KP.

ψ is a cuspidal function on M, satisfying some extra conditions

τ is a finite-dimensional unitary double representation of K

H_P(x) = log a where x = kman is the decomposition of x in G = KMAN.

References

Eisenstein integral Wikipedia

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