  Let \Y be a derived algebraic stack satisfying some mild conditions. The
purpose of this paper is three-fold. First, we introduce and study H(\Y), a
monoidal DG category that might be regarded as a categorification of the ring
of differential operators on \Y. When \Y = \LS_G is the derived stack of
G-local systems on a smooth projective curve, we expect H(\LS_G) to act on both
sides of the geometric Langlands correspondence, compatibly with the
conjectural Langlands functor. Second, we construct a novel theory of D-modules
on derived algebraic stacks. Contrarily to usual D-modules, this new theory, to
be denoted by D^{der}, is sensitive to the derived structure. Third, we
identify the Drinfeld center of H(\Y) with D^{der}(L\Y), the DG category of
D^{der}-modules on the loop stack of \Y.
