|Location:||MSRI: Baker Board Room|
Abstract: I'll describe a new perspective on the category O over symplectic reflections algebras for the groups $Z/\ell Z \wr S_n$, using ideas from the theory of categorical actions of Lie algebras. This builds on earlier work of Rouquier, Shan, Vasserot, Losev and others, but takes a more explicit diagrammatic perspective. In particular, I'll give an explicitly presented finite dimensional algebra (generalizing KLR algebras) whose representation category is the same as a block of the SRA category O mentioned above.
This algebra is quite interesting in its own right (for example, it is cellular), but it also allows hands-on proofs of several facts which are hard to see from the SRA perspective:
* it is manifestly graded, and thus provides a grading on category O
* the derived equivalences between blocks predicted by Rouquier are
given by easily guessed bimodules
* the identification of decomposition numbers with coefficients of a
canonical basis follows from the application of well-established
techniques from geometric representation theory, for example, from
Varagnolo and Vasserot's proof that projective modules over KLR
algebras give the canonical basis.
I'll try to convince you that this is a fruitful perspective and indicate avenues for better understanding its ties to other approaches to these categories.