In what may be a first in the math blogosphere I will be liveblogging Ben’s talk. Until the appearance of math vlogging I think this puts us at the cutting edge of math blogging technology. In the absense of heckling I will be keeping track of how often Ben says “perverse sheaves” and “equavariant cohomology of a point.”

(3:40) Some excitement, Ben’s computer threatens to die during the talk. Dylan’s power adapter saves the day. We also have our first laugh line: “Soergel bimodules have 3 definitions, some of you may like some more than the others, some of you may not like any of them.”

(3:50) We’re all the way through two slides and we’ve only had one mention each of “perverse sheaves” and “derived category.” There have been 5 questions, as long as the number of questions beats out the number of mentions of “perverse sheaves” things are good.

(3:55) Ben’s prepared some fancy technology: a little button saying “what?” that let’s him insert a slide on equivarient cohomology. We’re definitely taking the cohomology of a point now. I must admit I’ve always found that very confusing.

(4:02) We’re back in “Soergel-land” and the audience is still peppering Ben with questions.

(4:05) Ben is always saying he forgets what mistakes people pointed out during the talk, so, on slide 8 fix the gradings. Update: also add the “t-invariants” somewhere in the midteens.

(4:08) Urs asks in comments why we care about Soergel bimodules. Some people care because they like the BGG category O. At this conference they care because of Khovanov-Rozansky’s HOMFLY homology. The basic idea is that the Jones polynomial comes from certain nice traces on the tower of Hecke algebras. HOMFLY homology comes from a categorification of this picture. Soergel bimodules categorify the Hecke algebra (that is they are a category whose split Grothendieck group is the Hecke algebra) and Hochschild homology categorifies the Jones-Ocneanu trace. Using Rouquier’s categorification of the map from the braid group to the Hecke algebra this categorified trace gives HOMFLY homology.

(4:25) This talk is much more exciting than the last time I saw it. Apparently Jake Rasmussen’s talk included a seemingly innocuous result that suddenly makes Ben and Geordie’s results hold far more generally.

(4:31) Apparently when Ben gets an email it appears on the screen. Maybe I’ll send him one during the questions.

(4:33) Ben hits the summary right on time at 4:29 on the lecture hall clock (my computer is 4 minutes faster).

(4:37) Nice question: What is the categorification of the Jones-Wenzl idempotents?

(4:43) Signing off, great conference! I may have some posts on some earlier talks that we haven’t dicussed yet. There was a lot of interesting stuff. Jorgen is telling us to publicise the open position at CTQM. Tell your topologist friends!

Then here is the first live comment to a real-time entry on a math talk:

In a sentence or two: can you make me excited about wanting to learn about Soergel bimodules? Why should I care? Why do you all care?

Drat I missed it! Damn you, time zones!

Of course, it was video-taped….

I have a terminology question. It seems to me (looking at Ben’s talk) that Soergel bimodules refers to a specific

collection of bimodules, not the full category?

The whole category (the categorical form of the finite Hecke algebra or finite braid group) as far as I understand is what is usually known as Harish-Chandra bimodules — and these again are a form of the category of (admissible) representations of the complex group G [i.e. the algebraic form of the category of G-reps given by Harish-Chandra —

with fixed generalized infinitesimal character]. Of course our understanding of this category is profoundly different since Soergel’s work, but the category itself has been a fundamental object in representation theory for decades, so we should probably keep the names H-C bimodules or (more suggestively I think) finite Hecke category?

Not quite the same as liveblogging, but four or five years ago a friend gave me live updates via IM during a talk by Mark Kisin at Nick Katz’s 60th birthday conference. (Mark was announcing an important result.)

From what I understand, Harish-Chandra bimodules and Soergel bimodules are rather different. Harish-Chandra bimodules are bimodules over the universal enveloping algebra, whereas Soergel bimodules are bimodules over a polynomial ring, which are constructed from deformed category O. It might be there’s some closer connection between them than I know about, but they are legitimately different objects, and I’m talking about stuff that is hard to see on the category O side of things.

The term “Soergel bimodule” does seem to be rather new (the only online reference I could find is Khovanov’s paper), but I know at least two former or current Soergel students who use the term, so it’s not as though I made it up.

Ben – sorry for overglibness/grumpy-blogger-syndrome. I realize Soergel bimodules are indeed different objects, but they do seem to be closely related to H-C bimodules, albeit on the other side of Koszul duality, if I understand the intro to your paper with Williamson correctly: Soergel bimodules are hypercohomologies of IC sheaves on B\G/B (up to shift and sum); the finite Hecke category I was discussing is the full derived category of B\G/B, which is equivalent by Soergel’s Koszul duality to that of H-C bimodules (with trivial generalized infi character), I believe – but maybe this latter identification is not so natural?

So assuming one understands the hypercohomology functor (which is what Soergel was describing, no?) it seems the category of Soergel bimodules is a semisimplified form of the finite Hecke category, which is (up to Koszul duality) HC bimodules? but again I’m probably missing the main subtlety. Also seems both are categorifications of the braid group? Do the nonsemisimple objects have a role to play in knot homology theory so far?

I’m not so sure that the category of H-C bimodules is the same as the finite Hecke category (or the principal block of category O, as representation theorists call it), even on the derived level. My understanding was (and I may have made a mistake here, but it all sounds right) this principal block is isomorphic to the category of H-C bimodules where the right action of the center is semi-simple and the generalized eigenvalues of both the right and left actions match the trivial representation (this is a rather smaller subcategory than the full block, where the center may act “nilpotently” on the right as well as the left).

Now, it is very true that the Soergel bimodules are “just” the image of the semi-simple perverse sheaves on after applying pushforward to and then the equivalence of categories between and the derived category of dg-modules over thought of as a dg-module with trivial differential.

But considering that knot theorists tend to run for the hills at the appearance of equivariant cohomology, if I had said that at the beginning of my talk, the audience you have revolted (by which I mean “fallen asleep”). For purposes of talking to knot theorists who have very limited appetite for geometric representation theory, it is rather useful to have a friendly, non-threatening name like “Soergel bimodules.”

In terms of your final questions, I would say that Soergel bimodules are one, very explicit way of describing the action of the braid group on the finite Hecke category, or on H-C bimodules. You could just as easily talk about projective functors on this finite Hecke category, but the whole point of this paper is to explain what taking Hochschild homology of those should mean, and I have to have some terms for contrasting the two sides of this equivalence.

As far as I know, there is no reason to consider the non-semisimple objects in knot theory.

Again, I fully retract any slights towards the term Soergel bimodule or its expository use, just trying to understand its role for my edification. Also sadly we seem to have pushed the discussion beyond the level of technical detail anyone not writing papers on the subject would care about — is this appropriate use of your blog? if not, sorry, will cease and desist.

Anyway I agree with all you say about the difference

between HC bimodules and the principal block of category O – where I get confused is the identification of the latter with

the finite Hecke category, which is the derived category of

B\G/B. The principal block of category O is a subcategory

of sheaves on N\G/B, with a condition on the left H action

(monodromic with generalized trivial eigenvalue), or

the rep theoretic condition you wrote.

Some subset of Beilinson-Ginzburg-Soergel prove

a derived equivalence of B\G/B with HC bimodules,

which are sheaves on N\G/N with H-monodromic

conditions on both sides. (In particular the finite Hecke

and HC categories are both monoidal under convolution, but O is just a module category for either.) A version of this is in Soergel’s article on Langlands duality, though there’s a Langlands dual group thrown in (which doesn’t affect the combinatorics I think) — using Soergel bimodules, as I can

now recognize, thanks! OK this is silly. I’m just very fond

of the HC and Hecke categories (not so much of O). I’ll stop now.

I am very intrigued though by the appearance of the adjoint action of B in your theorem — had it reared its head before in this context or is this new? very suggestive.

David

I’m not writing any articles on the subject, but I’m still listening :P

As for any slights of your stoward Soergel bimodules, just remember that you’re not the only one who suffers from grumpy-blogger-syndrome, especially when said blogger is in Denmark.

I’ll say for the record that I’m perfectly fine with having technical discussions in the comments section, and I don’t consider it blog abuse. I think of this blog as an equivalent of coffee break/tea time conversations from conferences or math departments, which to me means no need to demand too much rigor, but no need to shy away from it either. Obviously, it’s really nice when we can keep the blog accessible, but I also want to seriously discuss research here.

As for the appearance of the conjugation action, I haven’t found any signs of it being previously considered. It pops out very naturally when you start thinking about Hochschild homology, but as far as I can tell, before Khovanov got a hold of them, no one had thought to take the Hochschild homology of these bimodules. I’m also rather curious about what else it could link into.