Fix a quiver $Q$ without loop. Denote the set of vectices of $Q$ by $I$.
Let $\Lambda\_V$ be the Lusztig nilpotent scheme with associated vector space $V$ over $I$. Briefly speaking, when $Q$ is a $ADE$ quiver, $\Lambda\_V$ is just the scheme of $\Lambda$-module of dimension $|V|$ where $\Lambda$ is the preprojective algebra of $Q$.


Lusztig's theorem:
Let $\frak{g}$ be the Kac-Moody algebra constructed from $Q$. Let $\frak{n\_+}$ be the postive part of $\frak{g}$. Then there is an algebra embedding $$Un\_+\to \sum\_{V} M(\Lambda\_V)^{G\_V}$$,where $V$ runs through all isomorphic type vector spaces over $I$.
$M(\Lambda\_V)^{G\_V}$ denotes the set of constructible function over $\Lambda\_V$ which are $G\_V$ invariant.


For more details (like the algebra structure on $\sum\_V M(\Lambda\_V)^{G\_V}$), you can have a look on [[1](http://webpages.math.luc.edu/%7Eptingley/oldseminars/QuantumGroupsSpring2011/lecture7a.pdf)] , section 5.2 of [[2](http://arxiv.org/abs/math/0402448)] and section 12 of [[3](http://www.ams.org/journals/jams/1991-04-02/S0894-0347-1991-1088333-2/)].


Lusztig constructed nilpotent scheme and considered algebra of construcible function on them so that we get an embedding from $U\frak{n}\_+$ to $\sum\_VM\_V(\Lambda\_V)^{G\_V}$, a convolution algebra of construcible function.


**My question is**: Is there an analog of nilpotent scheme such that there is an embedding from $U\frak{g}$ to the convolution algebra of construcible function on that analogous space? In this case, we have a geometric interpretation of the whole universal enveloping algebra but not just its positive part.


**EDIT:** I am aware of the construction of representation of modified enveloping algebra via Nakajima variety. I am more interested in a construction via Euler characteristic measure (like the construction of $U\frak{n}\_+$ by construcutible function the convolution of which involves Euler characteristic measure) .


I am mainly interested in Dynkin case and more specifically type A. Any references on this aspect is appreciated. It would be great to know if people have considered this before or not. Thank you so much!


Reference:


1.[Note from Quantum groups, combinatorics and geometry seminar Spring 2011](http://webpages.math.luc.edu/%7Eptingley/oldseminars/QuantumGroupsSpring2011/lecture7a.pdf)


2.[Christof Geiss, Bernard Leclerc, Jan Schröer's paper on semicanonical basis](http://arxiv.org/abs/math/0402448)


3.[Lusztig's original paper](http://www.ams.org/journals/jams/1991-04-02/S0894-0347-1991-1088333-2/)



analog of Lusztig nilpotent scheme
One thing that fits the role you're looking for is the Nakajima quiver varieties. These are discussed [later in the seminar you linked to](http://webpages.math.luc.edu/~ptingley/oldseminars/QuantumGroupsSpring2011/lecture10.pdf). There's a natural map of $U(\mathfrak g)$ to functions on these spaces (defined by [Nakajima](https://scholar.google.com/scholar?cluster=13975301444238828647&hl=en&as_sdt=0,47)), but rather than being injective, the spaces depend on a highest weight and the intersection of the kernels over all weights is trivial. See also the [notes of Ginzburg](http://arxiv.org/abs/0905.0686).


**EDIT:** Lusztig wrote to me in an email that there is a (partial) answer to that
question in his paper "Constructible functions on varieties attached to
quivers", in "Studies in memory of I. Schur." I haven't had a chance to look at that paper myself, but it's another place to look.


