Category Archives: travel

In which I stop by NYU and get categorified by Kazhdan

I had the good luck to be in New York on Friday when David Kazhdan gave an unscheduled lecture at NYU about categorification and representations of finite groups.  For people like me, who spend most of our days dismally uncategorified, the talk was a beautiful advertisement for categorification.

Actually, the first twenty minutes of the talk were apparently a beautiful advertisement for the Langlands program, but I got lost coming from the train and missed these.  As a result, I don’t know whether the results described below are due to Kazhdan, Kazhdan + collaborators, or someone else entirely.  And I missed some definitions — but I think I can transmit Kazhdan’s point even without knowing them.  You be the judge.

It went something like this:

Let G be a reductive split group over a finite field k and B a Borel. Then C[G(k)/B(k)] is a representation of G(k), each of whose irreducible constituents is a unipotent representation of G(k).  (Note:  the definition of “unipotent representation” is one that I missed but it comes from Deligne-Lusztig theory.)

When G = GL_n, all unipotent representations of G appear in C[G(k)/B(k)], so this procedure gives a very clean classification of unipotent representations — they are precisely the constituents of C[G(k)/B(k)].  Equivalently, they are the direct summands of the center of the Hecke algebra C[B(k) \G(k) / B(k)].  For more general G (e.g. Sp_6, E_8) this isn’t the case.  Some unipotent representations are missing from C[G(k)/B(k)]!

Where are they?

One category-level up, naturally.

(see what I did there?)

OK, so:  instead of C[B(k)\G(k)/B(k)], which is the algebra of B(k)-invariant functions on G(k)/B(k), let’s consider H, the category of B-invariant perverse l-adic sheaves on G/B.  (Update:  Ben Webster explained that I didn’t need to say “derived” here — Kazhdan was literally talking about the abelian category of perverse sheaves.)  This is supposed to be an algebra (H is for “Hecke”) and indeed we have a convolution, which makes H into a monoidal category.

Now all we have to do is compute the center of the category H.   And what we should mean by this is the Drinfeld center Z(H).  Just as the center of an algebra has more structure than the algebra structure — it is a commutative algebra! — the Drinfeld center of a monoidal category has more structure than a monoidal category — it is a braided monoidal category.  It’s Grothendieck Group K_0(Z(H)) (if you like, its decategorification) is just a plain old commutative algebra.

Now you might think that if you categorify C[B(k)\G(k)/B(k)], and then take the (Drinfeld) center, and then decategorify, you would get back the center of C[B(k)\G(k)/B(k)].

But you don’t!  You get something bigger — and the bigger algebra breaks up into direct summands which are naturally identified with the whole set of unipotent representations of G(k).

How can we get irreducible characters of G(k) out of Z(H)?  This is the function-sheaf correspondence –  for each object F of Z(H), and each point x of G(k), you get a number by evaluating the trace of Frobenius on the stalk of F at x.  This evidently yields a map from the Grothendieck group K_0(Z(H)) to characters of G(k).

To sum up:  the natural representation C[G(k)/B(k)] sometimes sees the whole unipotent representation theory of G(k), but sometimes doesn’t. When it doesn’t, it’s somewhat confusing to understand which representations it misses, and why.  But in Kazhdan’s view this is an artifact of working in the Grothendieck group of the thing instead of the thing itself, the monoidal category H, which, from its higher categorical perch, sees everything.

(I feel like the recent paper of Ben-Zvi, Francis and Nadler must have something to do with this post — experts?)

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Always eat at restaurants with Cochon in the name

This heuristic has served me well at Au Pied de Cochon in Montreal, and at Le Cochon Dingue in Quebec.  Wednesday night it was a winner again in New Orleans.  Since some of my readers may still have a New Orleans dinner or two ahead of them, let me recommend Cochon — no more than a 15 minute walk from your special session.  If you’ve just got a lunch left, their afternoon deli Cochon Butcher is also supposed to be good (and is also covered by my heuristic.)

 

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Ankylosaurus, on the other hand, was totally in earnest

At the American Museum of Natural History with CJ and AB, I learned that the best modern thinking gives Tyrannosaurus Rex just two fingers on each hand, not three.

It must have been very demeaning to be killed and eaten by a creature making the ironic quotation gesture.

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Facts about Kyoto

  • The outskirts of Kyoto look more like the outskirts of an American city than they do like the outskirts of a European city, but downtown Kyoto looks more like a European downtown than it looks like an American downtown.
  • There are a lot of soda machines in Kyoto.  Soda machines in untrafficked corners where you would never expect to find a soda machine.  You can buy little cans of cold coffee out of them.
  • Eel livers are delicious.
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In which Tucson, AZ is the miniature golf capital of these United States

At least that’s how I felt as a kid, when every visit here included a trip to Green Acres or Magic Carpet.  Miniature golf courses are just bigger and better and awesomer here than anywhere else.  Today I took CJ to Golf n Stuff, where they not only have a solid golf course but go-carts, a batting cage, and bumper boats.  You might think bumper boats would just be a slower, less fun version of bumper cars.  But you would quickly change your tune when I told you that bumper boats have water cannons mounted on them.

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Two great tastes that taste great together

Why yes, this is a pastrami sandwich with two potato pancakes in place of the bread. Courtesy of Kaufman’s Bagel and Delicatessen in Skokie, IL.

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Deforming Galois representations in Rwanda

I just now learned that my friend Ravi Ramakrishna from Cornell spent a sabbatical term last spring at the Kigali Institute of Science and Technology in Rwanda.  And he blogged his semester.  Good reading for anyone interested in math in the developing world, or who likes awesome pictures of gorillas and volcanoes.  Ravi made a side trip to Uganda with Teach and Tour Sojourners; seems like a nice program, though note that you pay your own way to the continent.

See also:  Dino Lorenzini’s notes on visiting math departments in Africa.

Note:  I don’t know if Ravi actually deformed any Galois representations while in Rwanda.  But come on, if you know the guy, you know he probably did.  He can’t leave those things alone.

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Airport barbecue

You can never get good pizza in the airport, or deli, or burritos, or Chinese. But I just ate yet another satisfying airport meat-and-two-sides, this one at the Speedway Grill in Charlotte-Douglas, where we’re paused on our long way home from Thanksgiving. It was good enough that CJ, to my great surprise, demanded more than his share of green beans; and it was only the second best airport BBQ I’ve had this month. (Brookwood Farms BBQ in Raleigh-Durham takes the crown here.)

Why does barbecue translate so well to the airport setting?

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In which Athens, GA is great

Blogging from the airport shuttle on my way from Athens to ATL. “Radio Free Europe” on the radio, on a station whose bumpers bill it as “the college football voice of the south.”

Things I liked about Athens:

Grits at Five Star Cafe.

Steak and gravy (roughly equivalent to chicken fried steak) at Weaver D’s (“Automatic For The People.”) White, very peppery gravy, as it should be. After lunch, coffee next door at Jittery Joe’s, a coffee roaster/bike shop/used book store in a converted barn.

The great young group in number theory and algebraic geometry at UGA, almost all hired in the last three years.

Cheesecake with green tomato relish at Five and Ten. I am less enthusiastic about dessert than most people but this was the best thing I ate yesterday.

R.E.M tourism. Had the shuttle driver take us by the abandoned church where they used to practice in 1981. And I went to the 40 Watt to see an Athens band, Twin Tigers, play. It’s true what they say about music in Athens; the place was much bigger than the High Noon and was packed, even for the first band. And not just with tragic indie kids with ironic mustaches. I think regular kids in Athens go to indie rock shows, kids in fraternities who listen to the college football voice of the south. Maybe there’s a counterculture of tragic country and western kids who run the college radio station!

As for Twin Tigers: loud washes of sound which was boring at first but which won me over. Given the style of guitar-playing, surprisingly non-deadpan vocals, heavily reverbed a la Simple Minds. “Automatic” was the standout track. Don’t know how to include links when blogging by phone, but I think you can download this free.

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