Whittaker coefficients in quantum geometric Langlands program

Abstract

We study Whittaker coefficients in the context of quantum geometric Langlands program. In the first part of this work, we prove that for any semisimple group $G$ cuspidal automorphic twisted $D$-modules have non-vanishing quantum Whittaker coefficients. The argument provides a microlocal interpretation of the Whittaker coefficients functor for any $\check{\Lambda}^+$-valued divisor under some hypothesis on singular support. In the second part, we use quantum Whittaker coefficients functor to construct the quantum geometric Langlands functor in the Betti setting. We show that the functor is compatible with the 2-Fourier-Mukai equivalence between sheaves of categories over 2-stacks $\ger_{Z_G}$ and $\ger_{\pi_1(\check{G})}$, which classify gerbes on $X$ with respect to the center $Z_G$ of $G$ and algebraic fundamental group $\pi_1(\check{G})$ of the Langlands dual group $\check{G}$.