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