(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 $(\pi,E)$ 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

Universal low-temperature behavior of two-dimensional lattice scalar chromodynamics

We study the role that global and local non-Abelian symmetries play in two-dimensional (2D) lattice gauge theories with multicomponent scalar fields. We start from a maximally O(M)-symmetric multicomponent scalar model. Its symmetry is partially gauged to obtain an SU(N-c) gauge theory (scalar chromodynamics) with global U(N-f) (for N-c >= 3) or Sp(N-f) symmetry (for N-c = 2), where N-f > 1 is the number of flavors. Correspondingly, the fields belong to the coset SM/SU(N-c) where S-M is the M-dimensional sphere and M = 2N(f) N-c. In agreement with the Mermin-Wagner theorem, the system is always disordered at finite temperature and a critical behavior only develops in the zero-temperature limit. its universal features are investigated by numerical finite-size scaling methods. The results show that the asymptotic low-temperature behavior belongs to the universality class of the 2D CP>N-f-1 field theory for N-c > 2 and to that of the 2D Sp(N-f ) field theory for N-c = 2. These universality classes correspond to 2D statistical field theories associated with symmetric spaces that are invariant under Sp(N-f ) transformations for N-c = 2 and under SU(N-f ) for N-c > 2. These symmetry groups are the same invariance groups of scalar chromodynamics, apart from a U(1) flavor symmetry that is present for N-f >= N-c > 2, which does not play any role in determining the asymptotic behavior of the model

Solvable Lie algebras with triangular nilradicals

All finite-dimensional indecomposable solvable Lie algebras $L(n,f)$, having the triangular algebra T(n) as their nilradical, are constructed. The number of nonnilpotent elements $f$ in $L(n,f)$ satisfies $1\leq f\leq n-1$ and the dimension of the Lie algebra is $\dim L(n,f)=f+{1/2}n(n-1)$

From p-adic to real Grassmannians via the quantum

Let F be a local field. The action of GL(n,F) on the Grassmann variety Gr(m,n,F) induces a continuous representation of the maximal compact subgroup of GL(n,F) on the space of L^2-functions on Gr(m,n,F). The irreducible constituents of this representation are parameterized by the same underlying set both for Archimedean and non-Archimedean fields. This paper connects the Archimedean and non-Archimedean theories using the quantum Grassmannian. In particular, idempotents in the Hecke algebra associated to this representation are the image of the quantum zonal spherical functions after taking appropriate limits. Consequently, a correspondence is established between some irreducible representations with Archimedean and non-Archimedean origin.Comment: 24 pages, final version, to appear in Advances in Mathematic