TY - JOUR AB - Let k be a number field and X a smooth, geometrically integral quasi-projective variety over k. For any linear algebraic group G over k and any G-torsor g : Z → X, we observe that if the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S for all twists of Z by elements in H^1(k, G), then the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S for X. As an application, we show that any homogeneous space of the form G/H with G a connected linear algebraic group over k satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some compactness assumptions when k is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form G/H with G semisimple simply connected and H finite, using the theory of torsors and descent. AU - Balestrieri, Francesca ID - 12427 IS - 3 JF - Proceedings of the American Mathematical Society SN - 0002-9939 TI - Some remarks on strong approximation and applications to homogeneous spaces of linear algebraic groups VL - 151 ER - TY - JOUR AB - In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the associated operators on these graphs display similarities to elliptic operators on bounded domains in the continuum. Specifically, we prove lower bounds on the eigenvalue growth and show by examples that corresponding upper bounds cannot be established. AU - Hua, Bobo AU - Keller, Matthias AU - Schwarz, Michael AU - Wirth, Melchior ID - 13177 IS - 8 JF - Proceedings of the American Mathematical Society SN - 0002-9939 TI - Sobolev-type inequalities and eigenvalue growth on graphs with finite measure VL - 151 ER - TY - JOUR AB - Let g be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g-modules Y(χ,η) introduced by Kostant. We prove that the set of all contravariant forms on Y(χ,η) forms a vector space whose dimension is given by the cardinality of the Weyl group of g. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M(χ,η) introduced by McDowell. AU - Brown, Adam AU - Romanov, Anna ID - 8773 IS - 1 JF - Proceedings of the American Mathematical Society KW - Applied Mathematics KW - General Mathematics SN - 0002-9939 TI - Contravariant forms on Whittaker modules VL - 149 ER - TY - JOUR AB - Li-Nadler proposed a conjecture about traces of Hecke categories, which implies the semistable part of the Betti geometric Langlands conjecture of Ben-Zvi-Nadler in genus 1. We prove a Weyl group analogue of this conjecture. Our theorem holds in the natural generality of reflection groups in Euclidean or hyperbolic space. As a corollary, we give an expression of the centralizer of a finite order element in a reflection group using homotopy theory. AU - Li, Penghui ID - 6986 IS - 11 JF - Proceedings of the American Mathematical Society SN - 0002-9939 TI - A colimit of traces of reflection groups VL - 147 ER - TY - JOUR AB - In this note, we consider the dynamics associated to a perturbation of an integrable Hamiltonian system in action-angle coordinates in any number of degrees of freedom and we prove the following result of ``micro-diffusion'': under generic assumptions on $ h$ and $ f$, there exists an orbit of the system for which the drift of its action variables is at least of order $ \sqrt {\varepsilon }$, after a time of order $ \sqrt {\varepsilon }^{-1}$. The assumptions, which are essentially minimal, are that there exists a resonant point for $ h$ and that the corresponding averaged perturbation is non-constant. The conclusions, although very weak when compared to usual instability phenomena, are also essentially optimal within this setting. AU - Bounemoura, Abed AU - Kaloshin, Vadim ID - 8495 IS - 4 JF - Proceedings of the American Mathematical Society SN - 0002-9939 TI - A note on micro-instability for Hamiltonian systems close to integrable VL - 144 ER -