Let K be a CM number field and GK its absolute Galois group. A representation of GK is said to be polarized if it is isomorphic to the contragredient of its outer complex conjugate, up to a twist by a power of the cyclotomic character. Absolutely irreducible polarized representations of GK have a sign ±1, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If Π is a regular algebraic, polarized, cuspidal automorphic representation of GLn(𝔸K), and if ρ is a p-adic Galois representation attached to Π, then ρ is polarized and we show that all of its polarized irreducible constituents have sign +1 . In particular, we determine the orthogonal/symplectic alternative for the Galois representations associated to the regular algebraic, essentially self-dual, cuspidal automorphic representations of GLn (𝔸F) when F is a totally real number field.

We prove the μ-part of the main conjecture for modular forms along the anticyclotomic Zp-extension of a quadratic imaginary field. Our proof consists of first giving an explicit formula for the algebraic μ-invariant, and then using results of Ribet and Takahashi showing that our formula agrees with Vatsal’s formula for the analytic μ-invariant.

We prove two flat descent results in the setting of Artin n-stacks. First of all, a stack for the etale topology which is an Artin n-stack (in the sense of Simpson and Toën–Vezzosi) is also a stack for the flat (fppf) topology. Moreover, an n-stack, for the fppf topology, which admits a flat (fppf) n-atlas is an Artin n-stack (i.e. possesses a smooth n-atlas). We deduce from these two results a comparison between etale and fppf cohomologies (with coefficients in non-smooth group schemes and also non-abelian). This work is written in the setting of the derived stacks of Toën and Vezzosi, and all of these results are therefore also valid for derived Artin n-stacks.

We consider a mirror symmetry between invertible weighted homogeneous polynomials in three variables. We define Dolgachev and Gabrielov numbers for them and show that we get a duality between these polynomials generalizing Arnold’s strange duality between the 14 exceptional unimodal singularities.

The main goal of this paper is to deduce (from a recent resolution of singularities result of Gabber) the following fact: (effective) Chow motives with ℤ[1/p]-coefficients over a perfect field k of characteristic p generate the category DMeffgm[1/p] (of effective geometric Voevodsky’s motives with ℤ[1/p]-coefficients). It follows that DMeffgm[1/p] can be endowed with a Chow weight structure wChow whose heart is Choweff[1/p] (weight structures were introduced in a preceding paper, where the existence of wChow for DMeffgmℚ was also proved). As shown in previous papers, this statement immediately yields the existence of a conservative weight complex functor DMeffgm[1/p]→Kb (Choweff [1/p]) (which induces an isomorphism on K0-groups), as well as the existence of canonical and functorial (Chow)-weight spectral sequences and weight filtrations for any cohomology theory on DMeffgm[1/p] . We also mention a certain Chow t-structure for DMeff−[1/p] and relate it with unramified cohomology.

Bondarko defines and studies the notion of weight structure and he shows that there exists a weight structure over the category of Voevodsky motives with rational coefficients (over a field of characteristic 0). In this paper we extend this weight structure to the category of Beilinson motives (for any scheme of finite type over a base scheme which is excellent of dimension at most two) introduced and studied by Cisinsky and Déglise. We also check the weight exactness of the Grothendieck operations.

Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno–Drinfeld Lie algebra . We show that Gaudin subalgebras form a variety isomorphic to the moduli space of stable curves of genus zero with n+1 marked points. In particular, this gives an embedding of in a Grassmannian of (n−1)-planes in an n(n−1)/2-dimensional space. We show that the sheaf of Gaudin subalgebras over is isomorphic to a sheaf of twisted first-order differential operators. For each representation of the Kohno–Drinfeld Lie algebra with fixed central character, we obtain a sheaf of commutative algebras whose spectrum is a coisotropic subscheme of a twisted version of the logarithmic cotangent bundle of .

The moduli space of holomorphic maps from Riemann surfaces to the Grassmannian is known to have two kinds of compactifications: Kontsevich’s stable map compactification and Marian–Oprea–Pandharipande’s stable quotient compactification. Over a non-singular curve, the latter moduli space is Grothendieck’s Quot scheme. In this paper, we give the notion of ‘ ϵ-stable quotients’ for a positive real number ϵ, and show that stable maps and stable quotients are related by wall-crossing phenomena. We will also discuss Gromov–Witten type invariants associated to ϵ-stable quotients, and investigate them under wall crossing.

We study 2-representations of finitary 2-categories with involution and adjunctions by functors on module categories over finite-dimensional algebras. In particular, we define, construct and describe in detail (right) cell 2-representations inspired by Kazhdan–Lusztig cell modules for Hecke algebras. Under some natural assumptions we show that cell 2-representations are strongly simple and do not depend on the choice of a right cell inside a two-sided cell. This reproves and extends the uniqueness result on categorification of Kazhdan–Lusztig cell modules for Hecke algebras of type A from [V. Mazorchuk and C. Stroppel, Categorification of (induced) cell modules and the rough structure of generalised Verma modules, Adv. Math. 219 (2008), 1363–1426].

We present geometric conditions on a metric space (Y,dY) ensuring that, almost surely, any isometric action on Y by Gromov’s expander-based random group has a common fixed point. These geometric conditions involve uniform convexity and the validity of nonlinear Poincaré inequalities, and they are stable under natural operations such as scaling, Gromov–Hausdorff limits, and Cartesian products. We use methods from metric embedding theory to establish the validity of these conditions for a variety of classes of metric spaces, thus establishing new fixed point results for actions of Gromov’s ‘wild groups’.

If G is a semisimple Lie group of real rank at least two and Γ is an irreducible lattice in G, then every homomorphism from Γ to the outer automorphism group of a finitely generated free group has finite image.

In this article a general framework for studying analytic representations of a real Lie group G is introduced. Fundamental topological properties of the representations are analyzed. A notion of temperedness for analytic representations is introduced, which indicates the existence of an action of a certain natural algebra 𝒜(G) of analytic functions of rapid decay. For reductive groups every Harish-Chandra module V is shown to admit a unique tempered analytic globalization, which is generated by V and 𝒜(G) and which embeds as the space of analytic vectors in all Banach globalizations of V.

In [Kelmer, Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms, Comm. Math. Phys. 276 (2007), 381–395] we introduced a family of symplectic maps of the torus whose quantization exhibits scarring on invariant co-isotropic submanifolds. The purpose of this note is to show that in contrast to other examples, where failure of quantum unique ergodicity is attributed to high multiplicities in the spectrum, for these examples the spectrum is (generically) simple.

We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least five. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using an existence result of E. Legendre [Toric geometry of convex quadrilaterals, J. Symplectic Geom. 9 (2011), 343–385], we show that a co-oriented compact toric contact 5-manifold whose moment cone has four facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on S2×S3 admitting two non-isometric and non-transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

We develop the Springer theory of Weyl group representations in the language of symplectic topology. Given a semisimple complex group G, we describe a Lagrangian brane in the cotangent bundle of the adjoint quotient 𝔤/G that produces the perverse sheaves of Springer theory. The main technical tool is an analysis of the Fourier transform for constructible sheaves from the perspective of the Fukaya category. Our results can be viewed as a toy model of the quantization of Hitchin fibers in the geometric Langlands program.