III.1. - Hilbertian operators

Summary

A family of operators

Fix an equivalence class of pairs (M, ℬ) where M is a standard Levi of G and ℬ an orbit under XG_M of irreducible cuspidal automorphic representations of M, for the equivalence relation defined in II.2.1. We denote by the set of pairs (M,π) where π is an irreducible cuspidal automorphic representation of M such that if ℬ is the orbit of π, then (M, ℬ) ∈. Denote by ξ the character of Z_G which is the restriction to Z_G of the central character of π for any pair (M,π) ∈. Fix a real number R such that if (M, ℬ) ∈ ξ and if P is the standard parabolic subgroup of G of Levi M then R Ⅱ ρP Ⅱ; the norm Ⅱρp Ⅱ is the same for all pairs (M, ℬ) ∈ 3E. Denote by Θ _R the space generated by the functions θ_φ for φ ∈P_R(_{M ℬ}) and (M, ℬ) ∈ In the case where ξ is unitary, we write L for its closure in L²(G(k)\G) ξ (the space L² is independent of R).

Let us introduce the space whose elements are the functions ƒ defined on∊that associate with (M,π) an element ƒ (M,π) ∈ End_M (A₀(M(k)\M)π) (see 1.3.3; this last space is finite-dimensional), and that satisfy: ƒ is holomorphic on ^R.