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(GL(n+1,F),GL(n,F)) is a Gelfand pair for any local field F
Let F be an arbitrary local field. Consider the standard embedding of GL(n,F)
into GL(n+1,F) and the two-sided action of GL(n,F) \times GL(n,F) on GL(n+1,F).
In this paper we show that any GL(n,F) \times GL(n,F)-invariant distribution
on GL(n+1,F) is invariant with respect to transposition.
We show that this implies that the pair (GL(n+1,F),GL(n,F)) is a Gelfand
pair. Namely, for any irreducible admissible representation of
(GL(n+1,F), dimHom_{GL(n,F)}(E,\cc) \leq 1.
For the proof in the archimedean case we develop several new tools to study
invariant distributions on smooth manifolds.Comment: v3: Archimedean Localization principle excluded due to a gap in its
proof. Another version of Localization principle can be found in
arXiv:0803.3395v2 [RT]. v4: an inaccuracy with Bruhat filtration fixed. See
Theorem 4.2.1 and Appendix
The GL-l.u.st.\ constant and asymmetry of the Kalton-Peck twisted sum in finite dimensions
We prove that the Kalton-Peck twisted sum of -dimensional Hilbert
spaces has GL-l.u.st.\ constant of order and bounded GL constant. This
is the first concrete example which shows different explicit orders of growth
in the GL and GL-l.u.st.\ constants. We discuss also the asymmetry constants of
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