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).

Let <i>G</i> be a semisimple adjoint group over <b>C</b> and <i></i> be the De ConciniProcesi completion of <i>G</i>. In this paper, we define a Lagrangian subvariety of the cotangent bundle of <i></i> such that the singular support of any character sheaf on <i></i> is contained in .

We will use the combinatorics of the G-stable pieces to describe the closure relation of the partition of partial flag varieties in [L3, section 4].

Let G be a semisimple group over an algebraically closed field. Steinberg has associated to a Coxeter element G of minimal length G a subvariety G of G isomorphic to an affine space of dimension G which meets the regular unipotent class G in exactly one point. In this paper this is generalized to the case where G is replaced by any elliptic element in the Weyl group of minimal length G in its conjugacy class, G is replaced by a subvariety G of G isomorphic to an affine space of dimension G , and G is replaced by a unipotent class G of codimension G in such a way that the intersection of G and G is finite.

We prove that the DeligneLusztig variety associated to minimal length elements in any <i></i>-conjugacy class of the Weyl group is affine, which was conjectured by Orlik and Rapoport in [S. Orlik, M. Rapoport, DeligneLusztig varieties and period domains over finite fields, arXiv: 0705.1646].