The so-called tempered complex smooth representations of p-adic groups have been much studied and used, in connection with automorphic forms. Nevertheless, the smooth representations that are realized geometrically often have l-adic coefficients, so that Archimedean estimates of their matrix coefficients hardly make sense. We investigate here a notion of tempered representation with coefficients in any normed field of characteristic not equal to p. The theory turns out to be different according to the norm being Archimedean, non-Archimedean with |p| ≠ 1, or non-Archimedean with |p|= 1.

In this paper we concentrate on the last case. The main applications concern modular representation theory (i.e., on a positive characteristic field) and, in particular, the study of reducibility properties of the parabolic induction functors; one of the main results is the generic irreducibility for induced families. Thanks to a suitable theory of rational intertwining operators, this allows us to define Harish-Chandra's μ-functions and show in some special cases how they track down the cuspidal constituents of parabolically induced representations. Besides, we discuss the admissibility of parabolic restriction functors and derive some lifting properties for supercuspidal modular representations.

We study the boundedness problem for maximal operators in 3-dimensional Euclidean space associated to hypersurfaces given as the graph of c + f, where f is a mixed homogeneous function that is smooth away from the origin and c is a constant. Assuming that the Gaussian curvature of this surface nowhere vanishes of infinite order, we prove that the associated maximal operator is bounded on L^p($\mathbb{R}$³) whenever p > h ≥ 2. Here h denotes a ``height'' of the function f defined in terms of its maximum order of vanishing and the weights of homogeneity. This result generalizes corresponding theorems on mixed homogeneous functions by A. Iosevich and E. Sawyer that allowed only for critical points of f at the origin. If c ≠ 0, our result is sharp.

Let B(X,$\mathscr{L}$,σ) be the twisted homogeneous coordinate ring of an irreducible variety X over an algebraically closed field k with dim X ≥ 2. Assume that c ∈ X and σ ∈ Aut(X) are in sufficiently general position. We show that if one follows the commutative prescription for blowing up X at c, but in this noncommutative setting, one obtains a noncommutative ring R = R(X,c,$\mathscr{L}$,σ) with surprising properties.

To each generic tangle projection from the three-dimensional real vector space onto the plane, we associate a derived endofunctor on a graded parabolic version of the Bernstein-Gel'fand category $\mathcal{O}$. We show that this assignment is (up to shifts) invariant under tangle isotopies and Reidemeister moves and defines therefore invariants of tangles. The occurring functors are defined via so-called projective functors. The first part of the paper deals with the indecomposability of such functors and their connection with generalised Temperley-Lieb algebras which are known to have a realisation via decorated tangles. The second part of the paper describes a categorification of the Temperley-Lieb category and proves the main conjectures of [BFK]. Moreover, we describe a functor from the category of 2-cobordisms into a category of projective functors.