cotangent to flags as a quiver variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T12:12:33Z http://mathoverflow.net/feeds/question/114557 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114557/cotangent-to-flags-as-a-quiver-variety cotangent to flags as a quiver variety Roman 2012-11-26T17:14:53Z 2013-03-19T12:43:26Z <p>It is easy to realize cotangent space to the flag variety \$Fl=SL_n/B\$ as a Nakajima quiver variety: consider the finite quiver of type A, the dimension vectors v=(1,2,...,n-1), w=(0,...,0,n); an appropriate stability condition (polarization) amounts to the condition that the arrow from the i-dimensional space to the (i+1)-dimensional one is injective, and we end up with a complete flag in the n-space, the arrows in the opposite direction giving a cotangent vector.</p> <p>Now, if I understand correctly, the other stability conditions (of which there is n!) should produce quiver varieties which are also isomorphic to \$T^*(Fl)\$. How to see this, preferably using equally explicit linear algebra? Is it explained in the literature?</p> http://mathoverflow.net/questions/114557/cotangent-to-flags-as-a-quiver-variety/124953#124953 Answer by Hiraku Nakajima for cotangent to flags as a quiver variety Hiraku Nakajima 2013-03-19T12:43:26Z 2013-03-19T12:43:26Z <p>Consider the stability parameter lives in the Cartan subalgebra. The corresponding variety is smooth if it lies in a chamber, i.e., outside of any root hyperplanes. It is clear that the variety remains the same if the parameter stay in the same chamber. If one cross the wall, the variety is changed.</p> <p>On the other hand, one can change the stability condition by what I call reflection functors. They are quiver varieties analog of reflection functors for quiver reprsentations, but behave much nicer than the original ones.</p> <p>They are compatible with the Weyl group action on the space of stability parameters, i.e., the Cartan subalgebra. The dimension vector \$v\$ is also changed compatible with the Weyl group action, if we think \$w - Cv\$ as a weight.</p> <p>The reflection functors give an isomorphism between quiver varieties. Therefore, for a finite type quiver varieties, as in this question. One can take any stability parameter. It can be moved to the standard one by successive applications of reflection functors.</p>