We study the resonances of matrix Schrödinger operators, motivated by the Born-Oppenheimer approximation. We give a simple criterion for the potential to generate resonances. This criterion also gives the location of the resonances generated.

Let σ,θ be commuting involutions of the connected reductive algebraic group G, where σ,θ, and G are defined over a (usually algebraically closed) field k, char k≠2. We have fixed point groups H:≠G^σ and K:≠G^θ and an action (H×K)×G→G, where ((h, k), g)↦hgk⁻¹, h∈H, k∈K, g∈G. Let G//(H×K) denote Spec $\mathscr{O}$(G)^H×K (the categorical quotient).

Let A be maximal among subtori S of G such that θ(s)=σ(s)=s⁻¹ for all s∈S. There is the associated Weyl group W:=W_H×K(A). We show the following.

· The inclusion A→G induces an isomorphism A/W$\widetilde{\to}$G//(H×K). In particular, the closed (H×K)-orbits are precisely those which intersect A.

· The fibers of G→G//(H×K) are the same as those occurring in certain associated symmetric varieties. In particular, the fibers consist of finitely many orbits.

We investigate

· the structure of W and its relation to other naturally occurring Weyl groups and to the action of σθ on the A-weight spaces of $\mathfrak {g}$;

· the relation of the orbit type stratifications of A/W and G//(H×K).

Along the way we simplify some of R. Richardson's proofs for the symmetric case σ=θ, and at the end we quickly recover results of M. Berger, M. Flensted-Jensen, B. Hoogenboom, and T. Matsuki [Ber], [FJ1], [Hoo], [Mat] for the case k=ℝ.

Let X be a proper, smooth, geometrically connected curve of genus g≥2 over a p-adically complete discrete valuation field K. By the Albanese morphism withrespect to a given K-rational point, the curve X can be embedded into its Jacobian variety J. Then, assuming that J has ordinary semistable reduction, we prove that the inertia group of K acts trivially on the set of torsion points of J which lie on X, under certain mild conditions. As an application, we prove that the modular curve X₀(N) (N: a prime number greater than or equal to 23), embedded into its Jacobian variety by using a cusp, contains no torsion points other than the cusps (resp., the cusps and the Weierstrass points), if N∉{23,29,31,41,47,59,71} (resp., N∈{23,29,31,41,47,59,71}). This affirmatively answers a question posed by R. Coleman, B. Kaskel, and K. Ribet [CKR].

We compute the divisor of Selberg's zeta function for convex cocompact, torsion-free discrete groups Γ acting on a real hyperbolic space of dimension n+1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on $X = Γ \backslash \mathbb{H}^{n+1}$ together with the Euler characteristic of X compactified to a manifold with boundary. If n is even, the singularities of the zeta function associated to the Euler characteristic of X are identified using work of U. Bunke and M. Olbrich.

For n≥3, the natural map Out(F_n)→Aut(K_n) from the outer automorphism group of the free group of rank n to the group of simplicial automorphisms of the spine of outer space is an isomorphism.