We prove sharp $L\sp 2$-estimates for oscillatory integral and Fourier integral operators for which the associated canonical relation $\mathscr {C}\subset T\sp \ast\Omega\sb L\times T\sp \ast\Omega\sb R$ projects to $T\sp \ast\Omega\sb L$ and to $T\sp \ast\Omega\sb R$ with corank 1 singularities of type $\leq 2$. This includes two-sided cusp singularities. Applications are given to operators with one-sided swallowtail singularities such as restricted X-ray transforms for well-curved line complexes in five dimensions.

Let $\mathfrak {g}$ be a complex, simple Lie algebra with Cartan subalgebra $\mathfrak {h}$ and Weyl group $W$. In [MTL], we introduced a new, $W$-equivariant flat connection on $\mathfrak {h}$ with simple poles along the root hyperplanes and values in any finite-dimensional $\mathfrak {g}$-module $V$. It was conjectured in [TL] that its monodromy is equivalent to the quantum Weyl group action of the generalised braid group of type $\mathfrak {g}$ on $V$ obtained by regarding the latter as a module over the quantum group $U\sb \hbar\mathfrak {g}$. In this paper, we prove this conjecture for $\mathfrak {g}=\mathfrak {sl}\sb n$.

All factorizable Lie bialgebra structures on complex reductive Lie algebras were described by Belavin and Drinfeld. We classify the symplectic leaves of all related Poisson-Lie groups. A formula for their dimensions is also proved.

Let $\mathfrak {g}$ be a reductive Lie algebra over a field of characteristic zero. Suppose that $\mathfrak {g}$ acts on a complex of vector spaces $M\sp \bullet$ by $i\sb \lambda$ and $\mathscr {L}\sb \lambda$, which satisfy the same identities that contraction and Lie derivative do for differential forms. Out of this data one defines the cohomology of the invariants and the equivariant cohomology of $M\sp \bullet$. We establish Koszul duality between them.

In this paper we extend a conjecture of A. Ash and W. Sinnott relating niveau 1 Galois representations to the $\mod p$ cohomology of congruence subgroups of ${\rm SL}\sb n(\mathbb {Z})$ to include Galois representations of higher niveau. We then present computational evidence for our conjecture in the case $n=3$ in the form of three-dimensional Galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture. Our examples include Galois representations with nontrivial weight and level, as well as irreducible three-dimensional representations that are in no obvious way related to lower-dimensional representations. In addition, we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture.

We compute the transcendental part of the normal function corresponding to the Deligne class of a cycle in $K\sb 1$ of a mirror family of quartic $K3$ surfaces. The resulting multivalued function does not satisfy the hypergeometric differential equation of the periods, and we conclude that the cycle is indecomposable for most points in the mirror family. The occurring inhomogenous Picard-Fuchs equations are related to Painlevé VI-type differential equations.