Uniformization of semistable bundles on elliptic curves

Let G be a connected reductive complex algebraic group, and E elliptic curve. GE denote the component of trivial bundle in stack semistable G-bundles on E. We introduce analytic uniformization by adjoint quotients subgroups loop group G. This can viewed as nonabelian version classical E≃C⁎/qZ. similarly construct itself via exponential map, providing standard isomorphism C⁎≃C/Z, generalizing presentation E≃C/(Z⊕Zτ). Finally, we apply these results to study sheaves with nilpotent singular support. As an application Betti geometric Langlands conjecture genus 1, define functor from ShN(GE) (the part automorphic category) IndCohNˇ(LocsysGˇ(E)) spectral category).