A while back, David wrote a post describing how to produce the Hecke algebra, and I described in comments (very tersely) how David’s description can be categorified. I thought I would expand on that a bit for selfish reasons that will soon become apparent. Not only is some beautiful geometry involved, there’s also a bonus connection to knot homology.
The way David was thinking of the Hecke algebra was as some operators on the set of flags in an n-dimensional vector space over for some prime power q, which I’ll denote . These operators actually have a very natural description: Let and let be the upper triangular matrices. Then the set of functions on is a -module in the obvious way, and the operators David described are exactly those that commute with the -action (incidentally, this is why the Hecke algebra has Hecke’s name attached to it).
I leave it as an exercise to the reader to show that this endomorphism algebra is just functions on that are invariant under the action on the left and right of , thought of as elements of the group algebra (this just uses that these are finite groups). You can also calculate that if is the characteristic function of , this endomorphism algebra is just .
So, the Hecke algebra is really bi-invariant functions on . I like this description a lot, since it is easy to categorify (at least if you’ve read SGA 4 1/2). This is a general rule (you could replace by any group and by any -space in the sentence below), so it gets it’s own paragraph:
The most obvious categorification of -invariant functions on is -equivariant sheaves on .
Other places this principle shows up are:
- replacing the usual Satake correspondence with the geometric Satake correspondence.
- replacing the Hall algebra construction of with Lusztig’s categorification.
Now, by “-equivariant sheaves on ,” I really mean the -equivariant derived category of , usually denoted . I don’t want to get too far into the weeds of what this is: if you asked someone comfortable with stacks, s/he would say “that’s just the derived category with constructible cohomology on .” If you’re really determined not to think about stacks, you could also read the book of Bernstein and Lunts on the subject, or (more briefly, but more obscurely) read the note that Geordie and wrote to make sure we had all our ducks in a row on some recent papers of ours. One point which is actually important here, though I’ll try to hide it as much as possible is that I’m working over positive characteristic, so all my varieties have a Frobenius acting on them, and I really want to take the category that consists of sheaves with a choice of how the Frobenius acts on them after I extend scalars to the algebraic closure (“mixed sheaves”). That way, I get to change how the Frobenius acts on my sheaf, which is what multiplying by corresponds to in the Grothendieck group. Lack of care about this point will make some of my theorems below lies, but that’s not an issue for our purposes.
The rough point here is that this is a category where any reasonable complex of sheaves with constructible cohomology you can think of that acts on is an object, and Homs are computed in such a way that Ext from the constant sheaf to itself is equivariant cohomology.
What? You can’t think of any complex of sheaves with constructible cohomology? Well, there’s really only one we’ll need, which is that for any space , and subspace , you can think of the sheaf on of singular cochains supported on . This is a perfectly good complex, and its homology (as a complex on ; we haven’t done any sheaf cohomology yet) is construcible (in fact, the stalk at a point is just the cohomology of a small neighborhood of , intersected with ). (This is one of these points where you should ignore that we’re working over characteristic and should think about doing this over the complex numbers and using the magical equivalence of categories I mentioned in this post).
So, we get a nice supply of objects in : for any permutation , look at the double coset , and consider the sheaf .
Theorem: The Grothendieck group of is the Hecke algebra (where multiplication by q is “Tate twist”). Under this isomorphism, the classes are sent to the classes given by the characteristic functions on the orbits .
This isomorphism is given roughly by sending each sheaf to function given by the supertrace of the Frobenius acting on stalks.
Note to those of you who understand Verdier duality: I’m really using the stalks of the Verdier dual of the sheaves above to make things a little easier. You’ll just have to live with it.
Of course, this theorem only refers to the vector space structure on the Hecke algebra. What about the multiplication? Of course, we just look at multiplication in the group algebra, and categorify that: we have maps given by multiplication and projection to the two factors (we’re thinking of these as finite sets; is the quotient of by the action ), and multiplication of two classes is
where is the “sum over fibers with weights for stabilizers” operator on functions. We can categorify this directly to the formula for convolution for sheaves
Theorem: This categorifies the multiplication on the Hecke algebra.
Now, there’s a map from the braid group to the Hecke algebra, sending the positive twists of adjacent strands to the class corresponding to an adjacent transposition, and extending to arbitrary braids by multiplying. So now, we know how to associate an object in to each twist: the complex . To get a complex associated to a general positive braid by convolving these. This gives a sheaf for any positive braid . We can extend this to all braids by declaring that should be the Verdier dual of the pull back of by the inverse map. It’s a nontrivial but nonterrible calculation to show that this gives a unique assignment to each braid; that is
Theorem: The assignment is a categorification of the map of the braid group to the Hecke algebra.
Well, that’s a cool result, but let me give you an even cooler one which gives you a hint of where this is going: There’s a theorem of Olaf Schnürrer which says roughly that is equivalent to the category of homotopy complexes of Soergel bimodules. Can you guess where goes?
Theorem: Up to some normalizations, is sent by this equivalence to the complex of Soergel bimodules assigned by Khovanov to the braid !