Components of Springer fibers, category O, and Khovanov’s “functor valued invariant of tangles”

I’ll just note at the beginning, this post is a bit of an experiment. At this point, it is about a semi-finished research thought of mine (which I’m not 100% sure is original, but I’m putting it out on the internet at least in part hoping that the internet will be able to tell me whether it’s original or not), and will consequently probably be a bit more technical than the average post on this blog, but hopefully, at least a few of you will be able to follow me.

As many of you know, my co-blogger Joel recently posted a preprint (with Sabin Cautis), which constructs a knot homology theory using the geometry of coherent s heaves and Fourier-Mukai transforms on convolutions of minuscule orbits in the affine Grasmannian of .

On the other hand, last year, Catharina Stroppel published a couple of papers on the relationship between Khovanov’s original construction of “a functor valued invariant of tangles” and various flavors of category O. From what I understand, underlying this is a philosophy that the version of Khovanov-Rozansky will be related a block to category O that lies on a dimension -dimensional wall of the Weyl chamber of (where is a number relating to the number of strands in your tangle diagram).

One natural question leaps to mind: how are these related?

Hochschild homology

I finally got around to writing a bit about my paper with Geordie, and realized that this was considerably more than a one post story. I figured I would start by saying a bit about Hochschild homology. This is a pretty standard bit of homological algebra, but has suddenly bursted out of the deformation theory ghetto into link homology.

So, what is Hochschild homology? Unfortunately, in most books, you seem to get a horrifying and useless definition (a bit like what happens with group cohomology). So let me attempt to explain what it really is.