In Arkhipov, Bezrukavnikov and Ginzburg's paper "Quantum Groups, the loop Grassmannian and the Springer resolution", they mentioned that Lusztig introduced a certain completion for universal enveloping algebra $\widehat{U\mathfrak{g}}$, so that the finite dimensional representions $Rep(\widehat{U\mathfrak{g}})$ is isomorphic to $Rep(G)$, where $G$ is the connected semisimple group of adjoint type with lie algebra $\mathfrak{g}$ over algebraically closed field of characteristic zero.

What is the construction of this completion? Any references?

  • $\begingroup$ Are you assuming that the ground field has characteristic zero? is algebraically closed? are you assuming that $\mathfrak{g}$ is semisimple? $\endgroup$ – YCor Apr 24 at 21:36
  • $\begingroup$ @YCor I have edited as the setting in their paper. Though I am curious whether such construction exists for general field or not. $\endgroup$ – userabc Apr 25 at 1:16
  • $\begingroup$ in which page of the cited article where is this mentioned ? $\endgroup$ – Konstantinos Kanakoglou Apr 25 at 1:45
  • $\begingroup$ Section 2.6 Frobenius functor $\endgroup$ – userabc Apr 25 at 3:09
  • $\begingroup$ If $(V_n)$ is the list of finite-dim irreducibles that integrate to $G$, it's tempting to take this completion as the product of matrix algebras $\prod_nEnd(V_n)$. We just need it to only have the $V_n$ as finite-dim irreducibles. This can be forced by endowing the product with the product topology (say the field is discrete, so $End(V_n)$ is discrete). But I guess that in this setting it's unnecessary, in the sense that every finite-dim rep of $\prod_nEnd(V_n)$ factors through $\prod_{k\le n}End(V_k)$ for some $n$. $\endgroup$ – YCor Apr 25 at 6:11

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