Analytic families of eigenfunctions on a reductive symmetric space

Abstract

Let X = G / H X=G/H be a reductive symmetric space, and let D ( X ) \mathbb D(X) denote the algebra of G G -invariant differential operators on X X . The asymptotic behavior of certain families f λ f_\lambda of generalized eigenfunctions for D ( X ) \mathbb D(X) is studied. The family parameter λ \lambda is a complex character on the split component of a parabolic subgroup. It is shown that the family is uniquely determined by the coefficient of a particular exponent in the expansion. This property is used to obtain a method by means of which linear relations among partial Eisenstein integrals can be deduced from similar relations on parabolic subgroups. In the special case of a semisimple Lie group considered as a symmetric space, this result is closely related to a lifting principle introduced by Casselman. The induction of relations will be applied in forthcoming work on the Plancherel and the Paley-Wiener theorem.

The Plancherel decomposition for a reductive symmetric space. I. Spherical functions

“…The proof of the Paley-Wiener theorem has now appeared in [16], which is independent of the present paper and [15]. The present paper as well as [15] and [16] rely on [12] and [14].…”

“…The present paper finishes at this point, where it is not yet clear that the residual Eisenstein integrals are related to induced representations. This fact will be established in the second paper [15] by using the vanishing theorem of [14]. The contributions of all K-types can then be collected and lead to the representation theoretic Plancherel formula.…”

“…We shall briefly recall the structure of the parabolic subgroups from P σ , meanwhile fixing notation. For details we refer to [14], §2. If P ∈ P σ , then M P and A P are σ-invariant, and σN P = θN P = NP .…”

“…In particular, in Section 5 we specify the normalizations of the residual operators and the measures used in the rest of the paper. In Section 6 we give a formulation of the vanishing theorem of [14], in a form suitable for this paper. Let a q be a maximal abelian subspace of p ∩ q, the intersection of the −1 eigenspaces in g for θ and σ, respectively.…”

“…In [12], Def. 10.7, see also [14], Remark 16.12, this Eisenstein integral was defined by means of a linear combination of residual operators (a so called Laurent functional) applied to the Eisenstein integral E • (P 0 : λ) with respect to the variable λ. As a family in the parameter ν, the generalized Eisenstein integral belongs to E hyp PF (X : τ ).…”