To the best of my knowledge, the conjecutres on quantum geometric Langlands and Mirror symmetry for D-modules (not in the form of quantum D-module of Givental, but in the form of Beilinson-Drinfeld chiral algebras) were first proposed by me (Makoto Sakurai) at the Strings 2005 conference in Toronto (Fields Institute), and my slides of presentations on the web were delivered to Dennis Gaitsgory next month at the Seattle 2005 summer instiute held by AMS. Ivan Mirkovic and David Ben-Zvi know my presentations, becuase I e-mailed to them at the end of the Seattle 2005 conference.
"Duality between open Gromov-Witten invariants and Beilinson-Drinfeld chiral algebras" (July 2005, Fields institute, Strings 2005)(available ppt, pdf)
Of course, I did not write the quantum groups formalism, and I did not mention about the irrational quantum deformation parameter formalism, which was originally claimed at the Kashiwara conference by Dennis Gaitsgory. Rather, I am now working on supersymmetric Poisson sigma model, as a stratification (globalization) of quantum groups (non-commutative schemes), which is a generalization of the quantum flag varieties (affine Kac-Moody algebras as the fiber) and this has the Euler obstruction classes as summation (Riemann-Roch theorem) of the 1st and 2nd cohomology class of the gerbe cohomology of chiral differential operators.
Of course, the quantum deformation by 1 parameter or 2 parameters are not yet sufficiently done in my work, although I am working on the quantum cohomology and elliptic genus (algebraic hom of cobordism due to Thom and character (supertrace of conformal blocks) of super conformal field theory, elaborating Eguchi-Sugawara-Taormina's representation of non-rational conformal field theory). I will just talk about the relation between the algebraic cobordism ("open-closed duality of cylinder amplitude") and the double loop group ("double Kac Moody" algebras) at the end of this month at the Mathematical Society of Japan.
Also, Braverman said that Dennis should mention about the work by Feigin-Stoyanovsky (while at RIMS = arXiv:math/0610974 ?) where I could not find out exacly what was going on.
Since I do not want to argue about who was the first (like Newton and Leibnitz on calculus...), but we must be more careful on which point of our works owes the idea to who.
