The Cox rings of del Pezzo surfaces are closely related to the Lie groups En. In this paper, we generalize the definition of Cox rings to G-surfaces defined by us earlier, where the Lie groups G = An,Dn or En. We show that the Cox ring of a G-surface S is closely related to an irreducible representation V of G, and is generated by degree one elements. The Proj of the Cox ring of S is a sub-variety of the orbit of the highest weight vector in V , and both are closed sub-varieties of P(V ) defined by quadratic equations. The GIT quotient of the Spec of such a Cox ring by a natural torus action is considered.

Given any Kodaira curve C in a complex surface X, we construct a simply-laced affine Lie algebra bundle E over X. When pg(X) = 0, we construct deformations of holomorphic structures on E such that the new bundle is trivial over any ADE curve C inside C and therefore descends to the singular surface obtained by contracting C.

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We describe recent progress on QH*(G/P) with special emphasis on its functoriality property, quantum Pieri rule and their applications.

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