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Representation Theory

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Analytic families of eigenfunctions on a reductive symmetric space

Authors: E. P. van den Ban and H. Schlichtkrull
Journal: Represent. Theory 5 (2001), 615-712
MSC (2000): Primary 22E30, 22E45
Published electronically: December 12, 2001
MathSciNet review: 1870604
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Abstract: Let $X=G/H$ be a reductive symmetric space, and let $\mathbb D(X)$ denote the algebra of $G$-invariant differential operators on $X$. The asymptotic behavior of certain families $f_\lambda$ of generalized eigenfunctions for $\mathbb D(X)$ is studied. The family parameter $\lambda$ 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.

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Additional Information

E. P. van den Ban
Affiliation: Mathematisch Institut, Universiteit Utrecht, PO Box 80 010, 3508 TA Utrecht, The Netherlands
MR Author ID: 30285

H. Schlichtkrull
Affiliation: Matematisk Institut, Københavns Universitet, Universitetsparken 5, 2100 København Ø, Denmark
MR Author ID: 156155
ORCID: 0000-0002-4681-3563

Received by editor(s): February 20, 2001
Received by editor(s) in revised form: September 6, 2001
Published electronically: December 12, 2001
Article copyright: © Copyright 2001 American Mathematical Society