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Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbf{C}$, and $\widehat{\mathfrak{g}}_\kappa$ the associated affine Lie algebra. It is the central extension of the loop algebra $\mathfrak{g}((t))$ using the invariant bilinear form $\kappa$ on $\mathfrak{g}$. Write $V_\kappa(\mathfrak{g})=\text{Ind}^{\widehat{\mathfrak{g}}_\kappa}_{\mathfrak{g}[[t]]\oplus \mathbf{C}c}\mathbf{C}$ for the Verma module of $\widehat{\mathfrak{g}}_\kappa$, where on $\mathbf{C}$, $\mathfrak{g}[[t]]$ acts by $0$ and the central element $c$ acts by $1$.

Is there a space (topological space, stack, etc.) $X$ whose cohomology is $H^*(X)=V_\kappa(\mathfrak{g})$?

For instance, if $Y$ is a smooth algebraic surface then $X=\coprod_{n\ge 0}\text{Hilb}^nY$ has cohomology of the form $V_\kappa(\mathfrak{h})$, where the vector space $\mathfrak{h}=H^*(Y)$ is viewed as a commutative Lie algebra with form $\kappa(\alpha,\beta)=\int_Y\alpha\wedge\beta$. I am curious if there is a construction for non-commutative Lie algebras.

If the natural thing is not to consider cohomology but some other functor from spaces to vector spaces, that is fine too (though ideally it would generalise the above example).

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    $\begingroup$ A couple queries. What is the grading u propose to put on the affine verma? also as an algebra isn't it a rather silly object, doesn't remember anything about the lie algebra except it's dimension for example. I think this question could use reforumlating so that the natural structures on the two sides match up better. Maybe u want to see the kac moody action or the vertex algebra structure motivically on X, like in the Grojnowski Nakajima picture $\endgroup$ – EBz Dec 28 '19 at 22:44
  • $\begingroup$ @Ebz The grading should be whatever it is in the Grojnowski-Nakajima construction, which I can't remember offhand, an element in $H^\star(X^{[n]})$ probably has grading $\star+f(n)$ for some function $f(n)$ (probably linear or quadratic). $\endgroup$ – Meow Mar 21 at 12:24
  • $\begingroup$ @Ebz I think it's hard enough to come up with a natural example with $H^*(X)=V_\kappa(\mathfrak{g})$ as vector spaces, I am happy to worry about the extra structure later. $\endgroup$ – Meow Mar 21 at 12:27

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