Feigin-Frenkel centre and opers for reductive Lie algebras

Edward Frenkel (together with Boris Feigin and others) has proven many interesting results connecting the representation theory of an affine Kac-Moody algebra at the critical level with the geometry of opers. Most of these results are collected in Frenkel's book "Langlands correspondence for loop groups". Assuming that $G$ is a connected simply-connected algebraic group with a simple Lie algebra $\mathfrak{g}$, he proves for example the following results.

(1) There is an (equivariant with respect to some group actions) isomorphism between the centre $Z(\hat{\mathfrak{g}})$ of the completed universal enveloping algebra of $\hat{\mathfrak{g}}$ at the critical level and the algebra $\mathrm{Fun}\ \mathrm{Op}_{{}^LG}(D^\times)$ of functions on the space of ${}^LG$-opers on the punctured formal disc, where ${}^LG$ denotes the Langlands dual group.

(2) There is also an isomorphism between the centre $\mathfrak{z}(\hat{\mathfrak{g}})$ of the vertex algebra $V_{\kappa_c}(\hat{\mathfrak{g}})$ (whose underlying vector space is the vacuum module) and the algebra $\mathrm{Fun}\ \mathrm{Op}_{{}^LG}(D)$ of functions on the space of ${}^LG$-opers on the formal disc.

(3) There is an isomorphism between the endomorphism ring $\mathrm{End}_{\hat{\mathfrak{g}}}(\mathbb{M}_\lambda)$ of the Verma module $\mathbb{M}_\lambda$ and the algebra $\mathrm{Fun}\ \mathrm{Op}_{{}^LG}^{RS}(D)_{\varpi(-\lambda-\rho)}$ of functions on the space of ${}^LG$-opers on the formal disc with regular singularities and residue $\varpi(-\lambda-\rho)$.

(4) Let $\lambda$ be a dominant integral weight. There is an isomorphism between the endomorphism ring $\mathrm{End}_{\hat{\mathfrak{g}}}(\mathbb{V}_\lambda)$ of the Weyl module $\mathbb{V}_\lambda$ and the algebra $\mathrm{Fun}\ \mathrm{Op}_{{}^LG}^\lambda$, where $\mathrm{Op}_{{}^LG}^\lambda \subset \mathrm{Op}_{{}^LG}^{RS}(D)_{\varpi(-\lambda-\rho)}$ is the subspace of opers with trivial monodromy.

My question is: do these results hold verbatim if we replace $\mathfrak{g}$ by a reductive Lie algebra? I am essentially interested in the case $G=GL_n, \mathfrak{g}=\mathfrak{gl}_n$. Or is some modification required?

• I guess the answer is yes to all questions. I do not have reference at hand. Center and oper can be described explicitly in GL case as we done in our papers. Moreover you can wtite "universal"-oper i.e. differential operator with coefficients in the center. Thus making the correspondence very explicit. – Alexander Chervov Jan 13 '17 at 18:47