In a paper by Badulescu [Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations, Invent. Math. 172 (2008), 383–438], results on the global Jacquet–Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field were established, under the assumption that the local inner forms are split at archimedean places. In this paper, we extend the main local results of that article to archimedean places so that the above condition can be removed. Along the way, we collect several results about the unitary dual of general linear groups over ℝ, ℂ or ℍ which are of independent interest.

We investigate the distribution of real algebraic numbers of a fixed degree that have a close conjugate number, with the distance between the conjugate numbers being given as a function of their height. The main result establishes the ubiquity of such algebraic numbers in the real line and implies a sharp quantitative bound on their number. Although the main result is rather general, it implies new estimates on the least possible distance between conjugate algebraic numbers, which improve recent bounds obtained by Bugeaud and Mignotte. So far, the results à la Bugeaud and Mignotte have relied on finding explicit families of polynomials with clusters of roots. Here we suggest a different approach in which irreducible polynomials are implicitly tailored so that their derivatives assume certain values. We consider some applications of our main theorem, including generalisations of a theorem of Baker and Schmidt and a theorem of Bernik, Kleinbock and Margulis in the metric theory of Diophantine approximation.

Let V be a vector space over a p-adic field F, of finite dimension, let q be a non-degenerate quadratic form over V and let D be a non-isotropic line in V. Denote by W the hyperplane orthogonal to D, and by G and H the special orthogonal groups of V and W. Let π, respectively σ, be an irreducible admissible representation of G(F) , respectively H(F) . The representation σ appears as a quotient of the restriction of π to H(F)with a certain multiplicity m(π,σ) . We know that m(π,σ)≤1 . We assume that π is supercuspidal. Then we prove a formula that computes m(π,σ)as an integral of functions deduced from the characters of π and σ. Let Π, respectively Σ, be an L-packet of tempered irreducible representations of G(F) , respectively H(F) . Here we use the sophisticated notion of L-packet due to Vogan and we assume some usual conjectural properties of those packets. A weak form of the local Gross–Prasad conjecture says that there exists a unique pair (π,σ)∈Π×Σ such that m(π,σ)=1 . Assuming that the elements of Π are supercuspidal, we prove this assertion.

We give a construction of the moduli space of stable maps to the classifying stack Bμr of a cyclic group by a sequence of rth root constructions on . We prove a closed formula for the total Chern class of μr-eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus-zero Gromov–Witten theory of stacks of the form [ℂN/μr]. We deduce linear recursions for genus-zero Gromov–Witten invariants.

Katz’s middle convolution algorithm provides a description of rigid connections on ℙ1 with regular singularities. We extend the algorithm by adding the Fourier transform to it. The extended algorithm provides a description of rigid connections with arbitrary singularities.

This paper studies affine Deligne–Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, and extends previous conjectures concerning their dimensions. We generalize the superset method, an algorithmic approach to the questions of non-emptiness and dimension. Our non-emptiness results apply equally well to the p-adic context and therefore relate to moduli of p-divisible groups and Shimura varieties with Iwahori level structure.