Volume 2010, Issue 645

In this article we define and study a Harder-Narasimhan filtration of finite flat group schemes over an unequal characteristic valuation ring. We also study that type of filtrations for families of finite flat group schemes parametrised by formal schemes.

Let E/F be a finite cyclic extension of p -adic fields, of degree d , and let κ be a character of F × with kernel N E/F ( E × ). Automorphic induction corresponds, via the Langlands correspondence, to inducing Galois representations from E to F . To a smooth irreducible representation τ of GL m ( E ) automorphic induction attaches a smooth irreducible representation π of GL md ( F ) which is equivalent to ( κ ○ det) ⊗ π . When π is generic the relation between τ and π is expressed by saying that a certain character function attached to τ is proportional to another character function attached to π and the choice of an intertwining operator A of ( κ ○ det) ⊗ π onto π . Here we normalize A through Whittaker models so that the proportionality constant—we prove—does not depend on τ . This is used in current work of C. J. Bushnell and the first author to give an explicit description of the Langlands correspondence for cuspidal smooth irreducible representations of GL n ( F ) when n is prime to p . In the present paper we also give a proof of the fundamental lemma for automorphic induction when p is at most n , thus completing J.-L. Waldspurger's result when p > n .

The aim of this paper is to provide a characterization of the dual of the ℝ n -module of slice monogenic functions on a class of compact sets in the Euclidean space . Despite the fact that the Cauchy formulas which are essential to such a characterization are based on different kernels, depending on whether one considers right or left slice monogenic functions, we are still able to establish a duality theorem which, since holomorphic functions are a very special case of slice monogenic functions, is the generalization of Köthe's theorem. The duality results are also obtained in the setting of quaternionic valued slice regular functions.

Let X be an algebraic variety over a finite field 𝔽 q , 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 𝔽 q n is a polynomial function of q n with integer coefficients. Moreover, the shifted polynomials, where q n is formally replaced with q n + 1, have non-negative coefficients.

We prove that if ( M, g, X ) is a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then ( M, g ) is isometric to or a finite quotient of or S 3 × ℝ. In the process we also show that a complete shrinking soliton ( M, g, X ) with bounded curvature is gradient and κ -noncollapsed and the dilation of a Type I singularity is a shrinking soliton. Further in dimension three we show shrinking solitons with bounded curvature can be classified under only the assumption of Rc ≧ 0. The proofs rely on the technical construction of a singular reduced length function, a function which behaves as the reduced length function but can be extended to singular times.

The fully matricial generalization in part I, of the difference quotient derivation on holomorphic functions, in which ℂ is replaced by a Banach algebra B , is extended from the affine case to a Grassmannian completion. The infinitesimal bialgebra duality, the duality transform generalizing the Stieltjes transform and the spectral theory with non-commuting scalars all extend to this completion. The series expansions of fully matricial analytic functions are characterized, providing a new way to generate fully matricial functions.