Let G be a split semi-simple adjoint group, and S an oriented surface with
punctures and special boundary points. We introduce a moduli space P(G,S)
parametrizing G-local system on S with some boundary data, and prove that it
carries a cluster Poisson structure, equivariant under the action of the
cluster modular group M(G,S), containing the mapping class group of S, the
group of outer automorphisms of G, and the product of Weyl / braid groups over
punctures / boundary components. We prove that the dual moduli space A(G,S)
carries a M(G,S)-equivariant cluster structure, and the pair (A(G,S), P(G,S))
is a cluster ensemble. These results generalize the works of V. Fock & the
first author, and of I. Le.
We quantize cluster Poisson varieties X for any Planck constant h s.t. h>0 or
|h|=1. First, we define a *-algebra structure on the Langlands modular double
A(h; X) of the algebra of functions on X. We construct a principal series of
representations of the *-algebra A(h; X), equivariant under a unitary
projective representation of the cluster modular group M(X). This extends works
of V. Fock and the first author when h>0.
Combining this, we get a M(G,S)-equivariant quantization of the moduli space
P(G,S), given by the *-algebra A(h; P(G,S)) and its principal series
representations. We construct realizations of the principal series
*-representations. In particular, when S is punctured disc with two special
points, we get a principal series *-representations of the Langlands modular
double of the quantum group Uq(g).
We conjecture that there is a nondegenerate pairing between the local system
of coinvariants of oscillatory representations of the W-algebra and the one
provided by the projective representation of the mapping class group of S.Comment: 199 pages. Minor correction