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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some regular symmetric pairs


Authors: Avraham Aizenbud and Dmitry Gourevitch
Journal: Trans. Amer. Math. Soc. 362 (2010), 3757-3777
MSC (2010): Primary 20G05, 20G25, 22E45
Published electronically: February 12, 2010
MathSciNet review: 2601608
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Abstract: In an earlier paper we explored the question what symmetric pairs are Gelfand pairs. We introduced the notion of regular symmetric pair and conjectured that all symmetric pairs are regular. This conjecture would imply that many symmetric pairs are Gelfand pairs, including all connected symmetric pairs over $ \mathbb{C}$.

In this paper we show that the pairs $ (GL(V),O(V)), (GL(V),U(V))$, $ (U(V),O(V)), (O(V \oplus W),O(V) \times O(W)), (U(V \oplus W),U(V) \times U(W))$ are regular, where $ V$ and $ W$ are quadratic or Hermitian spaces over an arbitrary local field of characteristic zero. We deduce from this that the pairs $ (GL_n(\mathbb{C}),O_n(\mathbb{C}))$ and $ (O_{n+m}(\mathbb{C}),O_n(\mathbb{C}) \times O_m(\mathbb{C}))$ are Gelfand pairs.


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Additional Information

Avraham Aizenbud
Affiliation: Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, POB 26, Rehovot 76100, Israel
Email: aizenr@yahoo.com

Dmitry Gourevitch
Affiliation: Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, POB 26, Rehovot 76100, Israel
Address at time of publication: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
Email: guredim@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9947-10-05008-7
PII: S 0002-9947(10)05008-7
Keywords: Uniqueness, multiplicity one, Gelfand pair, symmetric pair, unitary group, orthogonal group
Received by editor(s): September 11, 2008
Published electronically: February 12, 2010
Additional Notes: Both authors were partially supported by a BSF grant, a GIF grant, and an ISF Center of excellency grant. The first author was also supported by ISF grant No. 583/09 and the second author by NSF grant DMS-0635607. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.