A Feigin-Frenkel theorem with n singularities

For a simple Lie algebra g we consider an analogue of the affine algebra gk with n singularities, defined starting from the ring of functions on the n-pointed disk. We study the center of its completed enveloping algebra and prove an analogue of the Feigin-Frenkel theorem in this setting. In particular, we first give an algebraic description of the center by providing explicit topological generators; we then characterize the center geometrically as the ring of functions on the space of LG-Opers over the n-pointed disk. Finally, we prove some factorization properties of our isomorphism, thus establishing a relation between our isomorphism and the usual isomorphism of Feigin-Frenkel.(c) 2023 Elsevier Inc. All rights reserved.