Non-symmetric quiver varieties

Question

Given a symmetric Cartan datum $(I,\cdot)$, H. Nakajima has defined a family of varieties - known as quiver varieties - and has used them to give geometric constructions of the representation theory of the corresponding Kac-Moody algebra associated to $(I,\cdot)$ using (Borel-Moore) homology and the convolution product (discussed, for example, in 'Representation Theory and Complex Geometry', by Chriss & Ginzburg). Furthermore, Nakajima has given constructions of representations of quantum affine algebras using the equivariant K-theory of said varieties leading him to prove several results on the $q$-characters of representations of these quantum algebras. These results are excellently expounded in Nakajima's original papers (which involve the term 'quiver variety' in their title).

For a non-symmetric Cartan datum (eg, the Cartan datum associated to non-simply laced finite dimensional Lie algebras of type $B,C,F,G$) the approach of Nakajima does not extend in a simple manner to provide non-symmetric quiver varieties (to me, doubling the quiver seems to destroy any information we have on root lengths, but I don't know if this is the correct way to think about this).

Question: Has there been any progress on defining/constructing quiver varieties in the non-symmetric case? Can anyone provide a reference to this material or know of anyone working on this? Or, is there a reason why such a construction cannot be realised? I am interested in this as I am wondering if there are any connections with any 'Langlands'-y things (eg, Frenkel-Hernandez's papers on Langlands duality).

F. Xu and A. Savage have have managed to realise the crystal structure of representations in the non-simply laced case using the notion of admissable quiver automorphisms. Their approach is based on an observation by Lusztig (section 14, 'Introduction to Quantum Groups') that every Cartan datum in the affine/finite-type case can be constructed as a 'quotient' of a type A,D,E quiver with an admissable automorphism. To the ADE quiver we can define a quiver variety and they show that one can realise a crystal basis using irreducible components of quiver varieties that are invariant under the admissable automorphism. Savage explicitly constructs the crystal structure using the ADE crystal structure. Both authors mention that it seems plausible that there should be a geometric construction of this crystal structure (ie, a geometric construction of the representations of the corresponding Kac-Moody algebras). However, after a cursory search of the literature (ie, typing 'quiver varieties non symmetric' into Google), this seems to be most of what is known.

Furthermore, Nakajima (here, Problem 2.1) also states a desire to give a construction of quiver varieties in the non-symmetric case so it seems hopeful that such a construction should/might exist.

Answer 1

Here is a partial answer to your question.

With Peter Tingley, I have been working on extending Lusztig's quiver varieties to the non-symmetric (but symmetrizable) setting. However the varieties that one obtains are realized over non-algebraically closed fields (e.g. $\mathbb{Q}$ - and if one tried naively to base change to $\mathbb{C}$ the varieties obtained no longer have the correct properties). Some of the constructions continue to work in this setting: for instance Kashiwara-Saito's realization of the $B(\infty)$ crystal via the union of the irreducible components of Lusztig's quiver varieties, and some constructions involving MV polytopes. Using the same idea, one can also define Nakajima's quiver varieties in the non-symmetric setting, over non-algebraically closed fields. It remains to see which constructions would continue to work; I am fairly confident that Saito's construction of the crystal $B(\lambda)$ via irreducible components of Nakajima's quiver varieties will work using the same argument. For the constructions of irreducible representations via equivariant K-theory and homology, I think one would first have to extend this construction and construct these varieties over $\mathbb{C}$ instead - which would be interesting and non-trivial.

Here's the idea of the construction over non-algebraically closed fields - first construct the pre-projective algebra. For instance, consider the example of $C_2$; for the short root associate the field $\mathbb{Q}$, and for the long root associate the field $\mathbb{Q}[\sqrt{2}]$; to both edges associate the $(\mathbb{Q}, \mathbb{Q}[\sqrt{2}])$-bimodule $\mathbb{Q}[\sqrt{2}]$ - from which one obtains a tensor algebra. The pre-projective algebra is the quotient of this tensor algebra one quadratic relation at each vertex; the quadratic relations are constructed by choosing bilinear forms between these bi-modules and choosing dual bases with respect to these bilinear forms. This generalizes easily to arbitrary symmetrizable Kac-Moody algebras (and $\mathbb{Q}$ can be replaced by other suitable fields which have enough field extensions). This is a special case of a Ringel-Dlab construction. Now Lusztig's quiver varieties will correspond to the space of nilpotent representations with a fixed dimension vector.

Answer 2

I've already sent you this answer, but one reference is the work of Enomoto on quivers with involution: http://arxiv.org/abs/0806.3615

Answer 3

An answer is given in some sense in https://arxiv.org/abs/1907.06552, the first paragraph in p.3. A quiver variety M associated with a nonsymmetric Cartan matrix does not make sense, but the space of maps from P^1 to a hypothetical M makes sense.