The aim of this article is to compute the Hochschild and cyclic homology groups of Yang-Mills algebras, that have been defined by A. Connes and M. Dubois-Violette. We proceed here the study of these algebras that we have initiated in a previous article. The computation involves the use of a spectral sequence associated to the natural filtration on the enveloping algebra of the Lie Yang-Mills algebra. This filtration in provided by a Lie ideal which is free as Lie algebra.
Let X be an algebraic variety over a finite field 𝔽q, homogeneous under a linear algebraic group. We show that there exists an integer N such that for any positive integer n in a fixed residue class mod N, the number of rational points of X over 𝔽qn is a polynomial function of qn with integer coefficients. Moreover, the shifted polynomials, where qn is formally replaced with qn + 1, have non-negative coefficients.
We give various equivalent formulations to the (partially) open problem about Lp-boundedness of Bergman projections in tubes over cones. Namely, we show that such boundedness is equivalent to the duality identity between Bergman spaces, Ap′ = (Ap)*, and also to a Hardy type inequality related to the wave operator. We introduce analytic Besov spaces in tubes over cones, for which such Hardy inequalities play an important role. For p ≧ 2 we identify as a Besov space the range of the Bergman projection acting on Lp, and also the dual of Ap′. For the Bloch space we give in addition new necessary conditions on the number of derivatives required in its definition.
We propose a definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple Lie groups of Hermitian type. This definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting: assuming that the rank of the target Lie group is not greater than two, we prove that the Toledo invariant satisfies a Milnor-Wood type inequality and we characterize the corresponding maximal representations.
Let F be a non-Archimedean locally compact field and let D be a central division algebra over F. Let π1 and π2 be respectively two smooth irreducible representations of GL(n1, D) and GL(n2, D), n1, n2 ≧ 0. In this article, we give some sufficient conditions on π1 and π2 so that the parabolically induced representation of π1 ⊗ π2 to GL(n1 + n2, D) has a unique irreducible quotient. In the case where π1 is a cuspidal representation, we compute the Zelevinsky's parameters of such a quotient in terms of the parameters of π2. This is the key point for making explicit Howe correspondence for dual pairs of type II (cf. [Mínguez, thesis, Orsay 2006]).
