This work was largely inspired by a paper of Shustin, in which he proves that for a plane curve of given degree n whose singularities are not too complicated the singularities are versally unfolded by embedding the curve in the space of all curves of degree n; however, our methods are very different. The main result gives fairly explicit lower bounds on the sum of the Tjurina numbers at the singularities of a deformation of a weighted-homogeneous hypersurface, when the deformation is the fibre over an unstable point of an appropriate unfolding. The result is sufficiently flexible to cover a variety of applications, some of which we describe. In particular, we will deduce a generalisation of Shustin‘s result. Properties of discriminant matrices of unfoldings of weighted-homogeneous functions are crucial to the arguments; the parts of the theory needed are described.

In this paper we construct examples of commutative rings of difference operators with matrix coefficients from representation theory of quantum groups, generalizing the results of our previous paper [ES] to the q-deformed case. A generalized Baker–Akhiezer function Ψ is realized as a matrix character of a Verma module and is a common eigenfunction for a commutative ring of difference operators. In particular, we obtain the following result in Macdonald theory: at integer values of the Macdonald parameter k, there exist difference operators commuting with Macdonald operators which are not polynomials of Macdonald operators. This result generalizes an analogous result of Chalyh and Veselov for the case q=1, to arbitrary q. As a by-product, we prove a generalized Weyl character formula for Macdonald polynomials (= Conjecture 8.2 from [FV]), the duality for the Ψ-function, and the existence of shift operators.

In this paper, we study an exceptional theta correspondence, obtained by restricting the minimal automorphic representation of the adjoint group of type E$_7$ and rank 3 over Q to the dual pair GxPGSp$_6$. Here G is the anisotropic form of G$_2$ over Q; using the correspondence, we lift certain automorphic forms on G to holomorphic cusp forms on PGSp$_6$. This lifting provides the first step in a project to construct motives of rank 7 and weight O over Q with Galois group of type G$_2$.