We show that continuous group homomorphisms between unitary groups of unital C*-algebras induce maps between spaces of continuous real-valued affine functions on the trace simplices. Under certain $K$-theoretic regularity conditions, these maps can be seen to commute with the pairing between $K_0$ and traces. If the homomorphism is contractive and sends the unit circle to the unit circle, the map between spaces of continuous real-valued affine functions can further be shown to be unital and positive (up to a minus sign).

In a series of three earlier papers, we considered a family of restriction problems for classical groups (over local and global fields) and proposed precise answers to these problems using the local and global Langlands correspondence. These restriction problems were formulated in terms of a pair $W \subset V$ of orthogonal, Hermitian, symplectic, or skew-Hermitian spaces. In this paper, we consider a twisted variant of these conjectures in one particular case: that of a pair of skew-Hermitian spaces $W = V$.

In this paper, we prove a conjecture of Wei Zhang on comparison of certain local relative characters from which we draw some consequences for the Ichino–Ikeda conjecture for unitary groups.

Under endoscopic assumptions about $L$-packets of unitary groups, we prove the local Gan–Gross–Prasad conjecture for tempered representations of unitary groups over $p$-adic fields. Roughly, this conjecture says that branching laws for $U(n-1)\subset U(n)$ can be computed using epsilon factors.

We prove the injective stability theorem for unitary K1 under the usual stable range condition on the ground ring. This improves the stability theorem of A. Bak, V. Petrov and G. Tang where a stronger Λ-stable range condition was used.

Let A be a ring with anti-involution and F a nice functor (tensor or symmetric power, for example) from finitely-generated projective A-modules to abelian groups. We show that the homology of the hyperbolic unitary groups Un,n(A) with coefficients in F(A2n) can be expressed stably (i.e. after taking the colimit over n) by the homology of these groups with untwisted coefficients and functor homology groups that we can compute in suitable cases (for example, when A is a field of characteristic 0 or a ring without ℤ-torsion and F a tensor power). This extends the result where A is a finite field, which was dealt with previously by C. Vespa and the author (Ann. Sci. ENS, 2010).

The proof begins by relating, without any assumption on F, our homology groups to the homology of a category of hermitian spaces with coefficients twisted by F. Then, when F is polynomial, we establish — following a method due to Scorichenko — an isomorphism between this homology and the homology of another category of (possibly degenerate) hermitian spaces, which is computable (in good cases) by standard methods of homological algebra in functor categories (using adjunctions, Künneth formula…). We give some examples.

Finally, we deal with the analogous problem for non-hyperbolic unitary groups in some special cases, for example euclidean orthogonal groups On (A) (the ring A being here commutative). The isomorphism between functor homology and group homology with twisted coefficients does not hold in full generality; nevertheless we succeed to get it when A is a field or, for example, a subring of ℚ containing ℤ[1/2]. The method, which is similar to that in the previous case, uses a general result of symmetrisation in functor homology proved at the beginning of the article.

We prove modularity lifting theorems for ℓ-adic Galois representations of any dimension satisfying a unitary type condition and a Fontaine–Laffaille condition at ℓ. This extends the results of Clozel, Harris and Taylor, and the subsequent work by Taylor. The proof uses the Taylor–Wiles method, as improved by Diamond, Fujiwara, Kisin and Taylor, applied to Hecke algebras of unitary groups, and results of Labesse on stable base change and descent from unitary groups to GLn.

We prove that the homology of unitary groups over rings of S-integers in number fields stabilizes. Results of this kind are well known to follow from the high acyclicity of ad-hoc polyhedra. Given this, we exhibit two simple conditions on the arithmetic of hermitian forms over a ring A relatively to an anti-automorphism which, if they are satisfied, imply the stabilization of the homology of the corresponding unitary groups. When R is a ring of S-integers in a number field K, and A is a maximal R-order in an associative composition algebra F over K, we use the strong approximation theorem to show that both of these properties are satisfied. Finally we take a closer look at the case of On(ℤ[½]).

For every prime p and integer n≥3 we explicitly construct an abelian variety of dimension n such that for a suitable prime l the group of quasi-isogenies of  of l-power degree is canonically a dense subgroup of the nth Morava stabilizer group at p. We also give a variant of this result taking into account a polarization. This is motivated by the recent construction by Behrens and Lawson of topological automorphic forms which generalizes topological modular forms. For this, we prove some arithmetic results of independent interest: a result about approximation of local units in maximal orders of global skew fields which also gives a precise solution to the problem of extending automorphisms of the p-divisible group of a simple abelian variety over a finite field to quasi-isogenies of the abelian variety of degree divisible by as few primes as possible.

Let $E$ be a $\textrm{CM}$-field and $\pi$ a cuspidal representation of $\textrm{GL}_n({\mathbb{A}}_E)$ which admits a spherical vector (at all places) <formula form="inline" disc="math" id="ffm005"><formtex notation="AMSTeX">$\phi_0$. We evaluate the period of $\phi_0$ with respect to any compact unitary group. The result is consistent with a conjecture of Sarnak.

We study a theta lift from a cusp form $f$ on $U(1,1)$ to a cusp form $\mathcal{L} f$ on $U(2,1)$ (the unitary Kudla lift of $f$). We give an explicit expression of the Fourier–Jacobi expansion of $\mathcal{L} f$ in terms of periods and Hecke eigenvalues of $f$. As an application, we give a criterion for the nonvanishing of $\mathcal{L} f$.

Our aim is to study the cohomology groups of some coherent sheaves on a Griffiths–Schmid variety associated with an anisotropic Q-from of the unitary group SU(2,1). We define some transforms relating this cohomology to the coherent cohomology groups of some sheaves defined on certain threefolds, which are fibered in projective lines over Picard modular surfaces. In particular, we give a complete and explicit description, in terms of classical Picard modular forms, of the holomorphic (resp. anti-holomorphic) part of the 1-cohomology of the Griffiths-Schmid variety. From this, it results an explicit generating system for the part of the 2–cohomology which correspond to those automorphic representations whose archimedean component is a degenerate limit of discrete series.