Description
The fundamental local equivalence for quantum geometric Langlands is
a conjecture that states that the Kazhdan-Lusztig category (for a group G
at level \kappa) is equivalent to the twisted Whittaker category on the affine
Grassmannian (for the Langlands dual group G^L and the dual level \kappa^L).
Since the relationship between G and G^L is expressed combinatorially, in
order to prove this conjecture one has to introduce a combinatorial object
that encodes (or at least approximates) both categories. In this talk we will
describe such combinatorial object. It comes in the guise of a
factorization algebra, denoted \Omega_q. This factorization algebra has
many remarkable features. On the one hand, it encodes the quantum group
(attached to G, with quantum parameter q expressible in terms of \kappa):
namely U_q(n^+)^{Lus} identifies with hyperbolic cohomology of \Omega_q.
On the other hand, \Omega_q, viewed as a geometric object, can be decsribed
explicitly in terms of the Cartan matrix and \kappa, which makes it amenable
for quantum Langlands type comparison.
