Some regular symmetric pairs
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- by Avraham Aizenbud and Dmitry Gourevitch PDF
- Trans. Amer. Math. Soc. 362 (2010), 3757-3777 Request permission
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
- MR Author ID: 843930
- ORCID: 0000-0001-6436-2092
- Email: guredim@yahoo.com
- 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.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 3757-3777
- MSC (2010): Primary 20G05, 20G25, 22E45
- DOI: https://doi.org/10.1090/S0002-9947-10-05008-7
- MathSciNet review: 2601608