Consider a pseudogroup on $( \mathbb{C} , 0)$ generated by two local diffeomorphisms having analytic conjugacy classes a priori fixed in $\mathrm{Diff} \hspace{0.167em} ( \mathbb{C} , 0)$. We show that a generic pseudogroup as above is such that every point has a (possibly trivial) cyclic stabilizer. It also follows that these generic groups possess infinitely many hyperbolic orbits. This result possesses several applications to the topology of leaves of foliations, and we shall explicitly describe the case of nilpotent foliations associated to Arnold’s singularities of type ${A}^{2n+ 1} $.

In this paper we study a Haagerup inequality in the general case of discrete groupoids. We develop two geometrical tools, pinching and tetrahedral change of faces, based on deformation of triangles, to prove it. We show how to use these tools to find all the already known results just by manipulating triangles. We use these tools for groups acting freely and by isometries on the set of vertices of any affine building and give a first reduction of this inequality to its verification on some special triangles and prove the inequality when the building is of type $\tilde{A}_{k_1}\times\cdots\times\tilde{A}_{k_n}$, where $k_i\in\{1,2\}$, $i=1,\dots,n$.

Let $X$ be a compact surface such that $Y\hookrightarrow X$ as a separating, strictly pseudoconvex, real hypersurface;

$$ X\setminus Y=X_+\sqcup X_-, $$

where $X_{+}$ ($X_-$) is the strictly pseudoconvex (pseudoconcave) component of the complement. Suppose further that $X_-$ contains a positively embedded, compact curve $Z$. Under cohomological hypotheses on $(X_-,Z)$ we show that if $\dbarb'$ is a sufficiently small, embeddable deformation of the CR-structure on $Y$, then

$$ \mathrm{R-ind}(\bar{\partial}_{b},\bar{\partial}_{b}')\geq-[\dim H^{0,2}(X_-)+\dim H^0(Z,\mathcal{O}_Z)]. $$

This implies that the set of small, embeddable deformations of the CR-structure on $Y$ is closed, in the $\mathcal{C}^{\infty}$-topology on the set of all deformations.

AMS 2000 Mathematics subject classification: Primary 32V15; 32V30. Secondary 32G07; 32W10; 32W05

A key ingredient in the Taylor–Wiles proof of Fermat’s last theorem is the classical Ihara lemma, which is used to raise the modularity property between some congruent Galois representations. In their work on Sato and Tate, Clozel, Harris and Taylor proposed a generalisation of the Ihara lemma in higher dimension for some similitude groups. The main aim of this paper is to prove some new instances of this generalised Ihara lemma by considering some particular non-pseudo-Eisenstein maximal ideals of unramified Hecke algebras. As a consequence, we prove a level-raising statement.

We construct the free fusion of two geometric thories over a common ω-categorical strongly minimal reduct. If the two theories are supersimple of rank 1 (and satisfy an additional hypothesis true in particular for stable theories or trivial reduct), the completions of the free fusion are supersimple of rank at most ω.

The main purpose of this article is to define a quadratic analogue of the Chern character, the so-called Borel character, that identifies rational higher Grothendieck-Witt groups with a sum of rational Milnor-Witt (MW)-motivic cohomologies and rational motivic cohomologies. We also discuss the notion of ternary laws due to Walter, a quadratic analogue of formal group laws, and compute what we call the additive ternary laws, associated with MW-motivic cohomology. Finally, we provide an application of the Borel character by showing that the Milnor-Witt K-theory of a field F embeds into suitable higher Grothendieck-Witt groups of F modulo explicit torsion.

We study behaviours of the ‘equianharmonic’ parameter of the Grothendieck–Teichmüller group introduced by Lochak and Schneps. Using geometric construction of a certain one-parameter family of quartics, we realize the Galois action on the fundamental group of a punctured Mordell elliptic curve in the standard Galois action on a specific subgroup of the braid group . A consequence is to represent a matrix specialization of the ‘equianharmonic’ parameter in terms of special values of the adelic beta function introduced and studied by Anderson and Ihara.

In this paper, we consider how to express an Iwahori–Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman–Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a $p$-adic group; this corrects a result of Bump–Nakasuji.

By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface over an algebraically closed field, a variety whose geometric generic fiber is a projective homogeneous space admits a rational point if and only if the elementary obstruction vanishes.

We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\unicode[STIX]{x2202}M$ has dimension $n$ even. Its definition depends on the choice of metric $h_{0}$ on $\unicode[STIX]{x2202}M$ in the conformal class at infinity determined by $g$, we denote it by $\text{Vol}_{R}(M,g;h_{0})$. We show that $\text{Vol}_{R}(M,g;\cdot )$ is a functional admitting a ‘Polyakov type’ formula in the conformal class $[h_{0}]$ and we describe the critical points as solutions of some non-linear equation $v_{n}(h_{0})=\text{constant}$, satisfied in particular by Einstein metrics. When $n=2$, choosing extremizers in the conformal class amounts to uniformizing the surface, while if $n=4$ this amounts to solving the $\unicode[STIX]{x1D70E}_{2}$-Yamabe problem. Next, we consider the variation of $\text{Vol}_{R}(M,\cdot ;\cdot )$ along a curve of AHE metrics $g^{t}$ with boundary metric $h_{0}^{t}$ and we use this to show that, provided conformal classes can be (locally) parametrized by metrics $h$ solving $v_{n}(h)=\text{constant}$ and $\text{Vol}(\unicode[STIX]{x2202}M,h)=1$, the set of ends of AHE manifolds (up to diffeomorphisms isotopic to the identity) can be viewed as a Lagrangian submanifold in the cotangent space to the space ${\mathcal{T}}(\unicode[STIX]{x2202}M)$ of conformal structures on $\unicode[STIX]{x2202}M$. We obtain, as a consequence, a higher-dimensional version of McMullen’s quasi-Fuchsian reciprocity. We finally show that conformal classes admitting negatively curved Einstein metrics are local minima for the renormalized volume for a warped product type filling.

We study the compatibility of the action of the DAHA of type GL with two inverse systems of polynomial rings obtained from the standard Laurent polynomial representations. In both cases, the crucial analysis is that of the compatibility of the action of the Cherednik operators. Each case leads to a representation of a limit structure (the +/– stable limit DAHA) on a space of almost symmetric polynomials in infinitely many variables (the standard representation). As an application, we show that the defining representation of the double Dyck path algebra arises from the standard representation of the +stable limit DAHA.

Let k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack Ĝ over k, the metaplectic extension of the Greenberg realization of . We also construct a geometric analogue of the Weil representation of Ĝ, this is a triangulated category on which Ĝ acts by functors. This triangulated category and the action are geometric in a suitable sense.

The main result is that the ellipticity and the Fredholm property of a $\Psi $ DO acting on Sobolev spaces in the Weyl-Hörmander calculus are equivalent when the Hörmander metric is geodesically temperate and its associated Planck function vanishes at infinity. The proof is essentially related to the following result that we prove for geodesically temperate Hörmander metrics: If $\lambda \mapsto a_{\lambda }\in S(1,g)$ is a $\mathcal {C}^N$ , $0\leq N\leq \infty $ , map such that each $a_{\lambda }^w$ is invertible on $L^2$ , then the mapping $\lambda \mapsto b_{\lambda }\in S(1,g)$ , where $b_{\lambda }^w$ is the inverse of $a_{\lambda }^w$ , is again of class $\mathcal {C}^N$ . Additionally, assuming also the strong uncertainty principle for the metric, we obtain a Fedosov-Hörmander formula for the index of an elliptic operator. At the very end, we give an example to illustrate our main result.

We establish a transfer of (twisted) orbital integrals in the context of twisted endoscopy between a real reductive algebraic group $G$ and a reductive quasi-split real group $H_1$ associated to an endoscopic datum.

AMS 2000 Mathematics subject classification: Primary 22E45; 11F70

The automorphism group $\operatorname {Aut}(F_n)$ of the free group $F_n$ acts on a space $A_d(n)$ of Jacobi diagrams of degree d on n oriented arcs. We study the $\operatorname {Aut}(F_n)$-module structure of $A_d(n)$ by using two actions on the associated graded vector space of $A_d(n)$: an action of the general linear group $\operatorname {GL}(n,\mathbb {Z})$ and an action of the graded Lie algebra $\mathrm {gr}(\operatorname {IA}(n))$ of the IA-automorphism group $\operatorname {IA}(n)$ of $F_n$ associated with its lower central series. We extend the action of $\mathrm {gr}(\operatorname {IA}(n))$ to an action of the associated graded Lie algebra of the Andreadakis filtration of the endomorphism monoid of $F_n$. By using this action, we study the $\operatorname {Aut}(F_n)$-module structure of $A_d(n)$. We obtain an indecomposable decomposition of $A_d(n)$ as $\operatorname {Aut}(F_n)$-modules for $n\geq 2d$. Moreover, we obtain the radical filtration of $A_d(n)$ for $n\geq 2d$ and the socle of $A_3(n)$.

In a recent paper Moshe Jarden (Diamonds in torsion of Abelian varieties, J. Inst. Math. Jussieu9(3) (2010), 477–480) proposed a conjecture, later named the Kuykian conjecture, which states that if $A$ is an Abelian variety defined over a Hilbertian field $K$, then every intermediate field of $K({A}_{\mathrm{tor} } )/ K$ is Hilbertian. We prove that the conjecture holds for Galois extensions of $K$ in $K({A}_{\mathrm{tor} } )$.

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

In this paper, for each $n\geqslant g\geqslant 0$ we consider the moduli stack $\widetilde{{\mathcal{U}}}_{g,n}^{ns}$ of curves $(C,p_{1},\ldots ,p_{n},v_{1},\ldots ,v_{n})$ of arithmetic genus $g$ with $n$ smooth marked points $p_{i}$ and nonzero tangent vectors $v_{i}$ at them, such that the divisor $p_{1}+\cdots +p_{n}$ is nonspecial (has $h^{1}=0$) and ample. With some mild restrictions on the characteristic we show that it is a scheme, affine over the Grassmannian $G(n-g,n)$. We also construct an isomorphism of $\widetilde{{\mathcal{U}}}_{g,n}^{ns}$ with a certain relative moduli of $A_{\infty }$-structures (up to an equivalence) over a family of graded associative algebras parametrized by $G(n-g,n)$.

Nous donnons une nouvelle démonstration d’un théorème de Vogan qui classifie les représentations cohomologiques isolées dans le dual unitaire d’un groupe de Lie semisimple réel. Nous étudions la question analogue dans le dual automorphe et obtenons des résultats partiels. Nous appliquons finalement ces résultats (et conjectures) à l’étude de la cohomologie des variétés localement symétriques arithmétiques.

We give a new proof of a theorem of Vogan which classifies the cohomological representations of a real semisimple Lie group $G$ which are isolated in the unitary dual of $G$. We investigate the same question in the automorphic dual, and obtain partial results. We finally derive applications of all these results (and conjectures) to the study of the cohomology of arithmetic locally symmetric spaces.

It is well known in Kähler geometry that the infinite-dimensional symmetric space of smooth Kähler metrics in a fixed Kähler class on a polarized Kähler manifold is well approximated by finite-dimensional submanifolds of Bergman metrics of height k. Then it is natural to ask whether geodesics in can be approximated by Bergman geodesics in . For any polarized Kähler manifold, the approximation is in the C0 topology. For some special varieties, one expects better convergence: Song and Zelditch proved the C2 convergence for the torus-invariant metrics over toric varieties. In this article, we show that some C∞ approximation exists as well as a complete asymptotic expansion for principally polarized abelian varieties.