# Lusztig's completion for universal enveloping algebra

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?

• Are you assuming that the ground field has characteristic zero? is algebraically closed? are you assuming that $\mathfrak{g}$ is semisimple? – YCor Apr 24 at 21:36
• @YCor I have edited as the setting in their paper. Though I am curious whether such construction exists for general field or not. – userabc Apr 25 at 1:16
• in which page of the cited article where is this mentioned ? – Konstantinos Kanakoglou Apr 25 at 1:45
• Section 2.6 Frobenius functor – userabc Apr 25 at 3:09
• 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 $n$. – YCor Apr 25 at 6:11