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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Analytic families of eigenfunctions on a reductive symmetric space
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by E. P. van den Ban and H. Schlichtkrull PDF
Represent. Theory 5 (2001), 615-712 Request permission

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
  • Email: ban@math.uu.nl
  • H. Schlichtkrull
  • Affiliation: Matematisk Institut, Københavns Universitet, Universitetsparken 5, 2100 København Ø, Denmark
  • MR Author ID: 156155
  • ORCID: 0000-0002-4681-3563
  • Email: schlicht@math.ku.dk
  • Received by editor(s): February 20, 2001
  • Received by editor(s) in revised form: September 6, 2001
  • Published electronically: December 12, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Represent. Theory 5 (2001), 615-712
  • MSC (2000): Primary 22E30, 22E45
  • DOI: https://doi.org/10.1090/S1088-4165-01-00146-7
  • MathSciNet review: 1870604