Finite-dimensional representations of quantum affine algebras at roots of unity

We describe explicitly the canonical map Spec Spec , where is a quantum loop algebra at an odd root of unity . Here is the center of and Spec stands for the set of all finite--dimensional irreducible representations of an algebra . We show that Spec is a Poisson proalgebraic group which is essentially the group of points of over the regular adeles concentrated at and . Our main result is that the image under of Spec is the subgroup of principal adeles.

     

Lattice paths and Kazhdan-Lusztig polynomials

The purpose of this paper is to present a new non-recursive combinatorial formula for the Kazhdan-Lusztig polynomials of a Coxeter group . More precisely, we show that each directed path in the Bruhat graph of has a naturally associated set of lattice paths with the property that the Kazhdan-Lusztig polynomial of is the sum, over all the lattice paths associated to all the paths going from to , of where , and are three natural statistics on the lattice path.

     

Generalizations of Müntz's Theorem via a Remez-type inequality for Müntz spaces

The principal result of this paper is a Remez-type inequality for Müntz polynomials:

or equivalently for Dirichlet sums:

where . The most useful form of this inequality states that for every sequence satisfying , there is a constant depending only on and (and not on , , or ) so that

for every Müntz polynomial , as above, associated with , and for every set of Lebesgue measure at least . Here denotes the supremum norm on . This Remez-type inequality allows us to resolve two reasonably long-standing conjectures.

The first conjecture it lets us resolve is due to D. J. Newman and dates from 1978. It asserts that if , then the set of products is not dense in .

The second is a complete extension of Müntz's classical theorem on the denseness of Müntz spaces in to denseness in , where
is an arbitrary compact set with positive Lebesgue measure. That is, for an
arbitrary compact set with positive Lebesgue measure,
is dense in if and only if .

Several other interesting consequences are also presented.

     

Mean growth of Koenigs eigenfunctions

In 1884, G. Koenigs solved Schroeder's functional equation

in the following context: is a given holomorphic function mapping the open unit disk into itself and fixing a point , is holomorphic on , and is a complex scalar. Koenigs showed that if , then Schroeder's equation for has a unique holomorphic solution satisfying

moreover, he showed that the only other solutions are the obvious ones given by constant multiples of powers of . We call the Koenigs eigenfunction of . Motivated by fundamental issues in operator theory and function theory, we seek to understand the growth of integral means of Koenigs eigenfunctions. For , we prove a sufficient condition for the Koenigs eigenfunction of to belong to the Hardy space and show that the condition is necessary when is analytic on the closed disk. For many mappings the condition may be expressed as a relationship between and derivatives of at points on that are fixed by some iterate of . Our work depends upon a formula we establish for the essential spectral radius of any composition operator on the Hardy space .

     

tensor categories from quantum groups

Let be a semisimple Lie algebra and let be the ratio between the square of the lengths of a long and a short root. Moreover, let be the quotient category of the category of tilting modules of modulo the ideal of tilting modules with zero -dimension for . We show that for a sufficiently large integer, the morphisms of are Hilbert spaces satisfying functorial properties. As an application, we obtain a subfactor of the hyperfinite II factor for each object of .

     

Structure of a Hecke algebra quotient

Let be a Coxeter group with Coxeter graph . Let be the associated Hecke algebra. We define a certain ideal in and study the quotient algebra . We show that when is one of the infinite series of graphs of type , the quotient is semi-simple. We examine the cell structures of these algebras and construct their irreducible representations. We discuss the case where is of type , , or .

     

Potentially semi-stable deformation rings

Let be a finite extension and the absolute Galois group of . For a complete local ring with finite residue and a finite free -module equipped with an action of , we show that has a maximal quotient over which the representation is semi-stable with Hodge-Tate weights in a given range. We show an analogous result for representations which are potentially semi-stable of fixed Galois type and -adic Hodge type.

If is the universal deformation of , then we compute the dimension of and we show that these rings are sometimes smooth.

Finally we apply this theory to show, in some new cases, the compatibility of the -adic Galois representation attached to a Hilbert modular form with the local Langlands correspondence at .

     

The -method, weak Lefschetz theorems, and the topology of Kähler manifolds

A new approach to Nori's weak Lefschetz theorem is described. The new approach, which involves the -method, avoids moving arguments and gives much stronger results. In particular, it is proved that if and are connected smooth projective varieties of positive dimension and is a holomorphic immersion with ample normal bundle, then the image of in is of finite index. This result is obtained as a consequence of a direct generalization of Nori's theorem. The second part concerns a new approach to the theorem of Burns which states that a quotient of the unit ball in () by a discrete group of automorphisms which has a strongly pseudoconvex boundary component has only finitely many ends. The following generalization is obtained. If a complete Hermitian manifold of dimension has a strongly pseudoconvex end and for some positive constant , then, away from , has finite volume.

     

Local exactness in a class of differential complexes

The article studies the local exactness at level in the differential complex defined by commuting, linearly independent real-analytic complex vector fields in independent variables. Locally the system admits a first integral , i.e., a complex function such that and . The germs of the ``level sets' of , the sets , are invariants of the structure. It is proved that the vanishing of the (reduced) singular homology, in dimension , of these level sets is sufficient for local exactness at the level . The condition was already known to be necessary.

     

On the size of -fold sum and product sets of integers

In this paper, we show that for all 1$"> there is a positive integer such that if is an arbitrary finite set of integers, 2$">, then either N^{b}$"> or N^{b}$">. Here (resp. ) denotes the -fold sum (resp. product) of . This fact is deduced from the following harmonic analysis result obtained in the paper. For all 2$"> and 0$">, there is a 0$"> such that if satisfies , then the -constant of (in the sense of W. Rudin) is at most .