Compactifications of reductive groups as moduli stacks of bundles

We will introduce a class of moduli problems for any reductive group G, whose moduli stacks provide us with (toroidal) equivariant compactifications of G. Morally speaking, the objects in the moduli problem could be thought of as stable maps of a twice-punctured sphere into the classifying stack BG. More precisely, they consist of G_m equivariant G-principal bundles on chains of projective lines, framed at the extremal poles. The choice of a fan determines a stability condition.
All toric orbifolds are special cases of these, as are the wonderful compactifications of semi-simple groups of adjoint type constructed by De Concini - Procesi. Our construction further provides a canonical orbifold compactification for any semi-simple group.
This is joint work with Michael Thaddeus (Columbia).