Geometric Representation Theory (Lecture 19)
Posted by John Baez
In the penultimate lecture of last fall’s Geometric Representation Theory seminar, James Dolan lays the last pieces of groundwork for the Fundamental Theorem of Hecke Operators.
I’ll actually state the Fundamental Theorem in the next lecture. But for those of you who can’t stand the suspense, there’s some supplementary reading below, where the theorem is actually stated — and in a much more detailed way! Right now this is just a rough draft, not containing proofs. It’ll seem awfully dry if you haven’t been following the seminar so far, because it lacks all the fun examples we’ve been talking about all along. I’ll fix that later.

Lecture 19 (December 4)  James Dolan on the Fundamental
Theorem of Hecke Operators. Answer to last week’s puzzle.
A new puzzle: find an interesting groupoid with cardinality $e^e$.
A harder one: find an interesting groupoid with cardinality $\pi$.
Degroupoidification turns a finite groupoid $G$ into a finitedimensional vector space, its zeroth homology $H_0(G)$. It turns a span of finite groupoids
$G \stackrel{j}{\leftarrow} S \stackrel{k}{\rightarrow} H$ into the linear operator defined as the composite
$G \stackrel{j^!}{\rightarrow} S \stackrel{k_*}{\rightarrow} H$
where $k_*$ is the pushforward (defined in an obvious way) and $j^!$ is the transfer (defined in a clever way using groupoid cardinality, as explained here).
Degroupoidification is a weak monoidal 2functor
$D: FinSpan \to FinVect$
where
$FinSpan = [finite groupoids, spans of finite groupoids, equivalences between spans ]$
and
$FinVect = [finitedimensional vector spaces, linear operators, equations between linear operators]$
The latter is really just a category in disguise. So, we can use degroupoidification to obtain a weak 3functor
$\overline{D}: [bicategories enriched over FinSpan] \to [categories enriched over FinVect]$
For us, the key example of a bicategory enriched over FinSpan is the Hecke bicategory of a finite group $G$, $Hecke(G)$. This has finite Gsets as objects, and for any pair of finite Gsets A and B it has
$hom(A,B) = (A × B)//G$
Composition in the Hecke bicategory involves a “trispan”.
Future directions: following the plan outlined on page 400 of Daniel Bump’s book Lie Groups, in the chapter “The Philosophy of Cusp Forms”.
 Supplementary reading: John Baez, HigherDimensional Algebra VII: Groupoidification (draft version).

Streaming
video in QuickTime format; the URL is
http://mainstream.ucr.edu/baez_12_04_stream.mov  Downloadable video
 Lecture notes by Alex Hoffnung
 Lecture notes by Apoorva Khare
Re: Geometric Representation Theory (Lecture 19)
Hold on  I’m confused here. I have the same confusion with the supplementary reading. Don’t you have to tensor the homsets with the ground field $k$? In the category of permutation representations of $G$, the homsets are $k$vector spaces.