## October 7, 2007

### Geometric Representation Theory (Lecture 1)

#### Posted by John Baez

This fall, the so-called Quantum Gravity Seminar at U. C. Riverside will actually tackle geometric representation theory — the marvelous borderland where geometry, groupoid theory and logic merge into a single subject. And there are two other new things about this seminar.

First, it will be jointly run by John Baez and James Dolan. In addition to explaining well-known stuff, we’ll report on research we’ve done with Todd Trimble over the last few years. Second, we plan to offer videos as well as written notes of the seminar. We’re still working the bugs out of the technology, so please bear with us.

As usual, the seminar will meet on Tuesdays and Thursdays, and you can ask questions and discuss things here at the $n$-Category Café.

This week, I kicked off the proceedings with a gentle introduction to a few of the main themes.

• Lecture 1 (Sept. 27) - John Baez on some of the basic ideas of geometric representation theory. Classical versus quantum; the category of sets and functions versus the category of vector spaces and linear operators. Group representations from group actions. Representations of the symmetric group $n!$ from types of structure on $n$-element sets. Representations of the general linear group $GL(n,F_q)$ from types of structure on the $n$-dimensional vector spaces over the field with $q$ elements, $F_q$. Uncombed Young diagrams $D$, and ‘$D$-flags’ as structures either on $n$-element sets or $n$-dimensional vector spaces. Irreducible representations of $n!$ versus representations coming from the actions of $n!$ on sets of $D$-flags. Counting $D$-flags: $q$-factorials and their limit as $q \to 1$. The ‘field with one element’. Projective geometry.

#### Videos

We’re offering the videos in streaming and/or downloadable form, both as .mov files. Downloading them takes a long time, but you may need to do this, since the streaming videos seem to work well only if you have a good internet connection.

.mov files can best be played using a free program called QuickTime. If you have QuickTime and your web browser has .mov files associated to this program, you should be able to click on the first “streaming video” link above and watch the video. An alternate method is to launch the QuickTime player on your computer, click on “File” and then “Open URL”, and type in the URL provided. This has the advantage that you can easily make the picture bigger.

If you can handle URL’s that begin with rtsp, you can instead go the corresponding URL of that form, e.g. rtsp://mainstream.ucr.edu/baez_9_27_stream.mov. This may also have advantages, but at present my computer gags on such URL’s.

If you encounter problems or — even better — know cool tricks to solve such problems, please let us know about them here!

#### Errata

If you catch mistakes, let me know and I’ll add them to the list of errata.

Posted at October 7, 2007 12:43 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1450

### Re: Geometric Representation Theory (Lecture 1)

This looks great, but I wonder if there’s any chance of making the video files available for download rather than just streaming? My effective bandwidth seems to be a factor of 4 too small to receive the stream – and trying to watch a lecture that plays for 5 seconds then halts for 15 is pretty painful – but if I could download the files to play smoothly I wouldn’t care how long that took.

Posted by: Greg Egan on October 7, 2007 2:34 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

The videos are being recorded by the multimedia technologies group at UCR, and being stored on their server. I’ll ask them if there’s a way to let people download them. There should be some way.

Alas, right now the second video seems really bad: the sound keeps dropping out completely. It’s a video of Jim Dolan introducing his way of thinking about this stuff. If I can’t solve the problem any other way, I may even ask him to give this class again! It won’t be the same, though — in part because there were lots of interesting questions.

We have a lot to learn about making and distributing videos. From my home, the streaming video works flawlessly. I don’t know if the folks at UCR ever tried watching these videos from farther away, e.g. Australia or Europe.

Posted by: John Baez on October 7, 2007 4:10 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Just to say that I could watch the whole video without problems from Europe, but then it’s sunday morning…
I agree with Greg that it would be nice to be able to download the videos, perhaps to use some of them as material for a course.
Nice lecture btw. (apart from the final confusion about PGL(n+1,F))

Posted by: lieven on October 7, 2007 9:56 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

The video streams fine to Canada. Very nice!

On a related topic, I’ve started looking into options for giving lectures live over the net with video. Does anyone have suggestions? One requirement is that audio be bidirectional, so the audience can ask questions.

By the way, is there a place here for people to discuss meta topics, such as getting mathml to work (I’ve spent hours on this and still can’t get it to work correctly), choosing an RSS reader that handles the n-category cafe well, etc?

Posted by: Dan Christensen on October 7, 2007 2:04 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Hi, Dan. I’m glad our streaming video streams nicely right up to Canada.

“Meta topics” covers a lot of ground, even $n$-category theory itself, which is about as “meta” as you can get. But, the particular meta topics you list are perfectly suited to our perennial thread on TeXnical Issues. Despite the title, this thread is not just about TeX. Go for it!

Posted by: John Baez on October 8, 2007 1:23 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

I can stream this fine (from Nebraska, not very far), but I’d still like to download if possible! My Internet connections go in and out, and I have plenty of disk space, so I like to download anything that I’m liable to want to watch again (in case there’s no Internet when I decide to review it).

On a related note, does anybody know how to get Firefox (or Shockwave Flash) to tell me where it’s storing the temporary file behind any given display? I could download all of the Catsters’ videos from YouTube, since I found them in my operating system’s temp folder; other times, I find thing in the browser’s cache folder. But this video I can’t find anywhere!

Posted by: Toby Bartels on October 7, 2007 8:06 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

If you want to grab videos from YouTube quickly, you should try a plugin for Firefox called DownloadHelper. It doesn’t work for this, but it should help you with the rest of the flash video websites.

Posted by: Anonymous Coward on October 7, 2007 9:54 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Hey, that works great! Now I don’t even have to turn on Javascript to surf YouTube (unlike some websites …). Thanks, Anonymous Coward!

And since John’s videos are now also available for download, I’m all set!

Posted by: Toby Bartels on November 13, 2007 3:44 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Congratulations on the video! However I too have been unable to watch very much because of the streaming problem.

Posted by: Eugenia Cheng on October 8, 2007 12:30 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

I’ll tell the multimedia folks at UCR to make the video available in other forms. At the very least, they can give it to me as a file which I can put on my website, YouTube, and so on.

As tdstephens points out, we’re at the bleeding edge here.

Posted by: John Baez on October 8, 2007 3:25 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Thank you for your efforts on this project. These videos, the catsters, and several others are at the leading edge of these exciting times for undergrad through post-grad level communication.

(“several others” is vague, and I can’t actually think of any others that are this good…)

Posted by: tdstephens3 on October 8, 2007 3:07 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

I have a remark/question/observation on the groupoidification program.

One of the big messages of this program is, I gather, that in order to understand representations well we ought to be looking at the corresponding action groupoids.

So, if a group $G$ acts on a set or space $V$

$\rho(g) : V \to V$

we’d form the groupoid

$V // G$

whose objects are the elements of $V$ and which has all morphisms of the form

$v \stackrel{\rho(g)}{\to}\rho(g)(v)$

for all $v \in V$ and $g \in G$.

Now, what is an action groupoid, abstractly speaking? One striking property of $V// G$ is that it is still equipped with an action of $G$:

$\tilde \rho(g) : V//G \to V//G$

These $\tilde \rho(g)$ now are functors. This means they can have natural transformations between them.

Once could say that the action groupoid $V//G$ has precisely the right morphisms in order to make all group element actions homotopic.

Namely the action $\tilde \rho$ on $V // G$ has the property that

$\array{ & \nearrow \searrow^{\tilde \rho(g)} \\ V//G &\Downarrow^{\simeq}& V//G \\ & \searrow \nearrow_{\tilde \rho(g')} }$

any two group element actions $\tilde \rho(g)$ and $\tilde \rho(g')$ are related by a unique natural transformation.

This is of course just another aspect of the statement that the weak quotient $V // G$ is equivalent to the strict quotient $V / G$ (regarded as a discrete category).

But how can we describe the existence of these unique 2-morphisms abstractly?

I believe that one way to do it is this:

let me write

$\Sigma \mathrm{Aut}(V)$

for the category which contains the single object $V$ and all its automorphisms. This way our representation is a morphism

$\rho : \Sigma G \to \Sigma \mathrm{Aut}(V)$

I am thinking that the action groupoid $V//G$ is the strict pushout (in $2\mathrm{Cat}$) of

$\array{ \Sigma G &\hookrightarrow& \Sigma(G // G) \\ \downarrow^\rho \\ \Sigma \mathrm{Aut}(V) }$

in that

$\array{ \Sigma G &\hookrightarrow& \Sigma ( G // G ) \\ \downarrow^\rho && \downarrow \\ \Sigma \mathrm{Aut}(V) &\stackrel{\tilde {(\cdot)}}{\to}& \Sigma \mathrm{Aut}(V // G) }$

is the universal strict completion of this cone.

Posted by: Urs Schreiber on October 8, 2007 10:07 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Completely irrelevant, but this just drives me crazy. $\begin{matrix} &\overset{\tilde{\rho}(g)}{↷}&\\ V//G&\Downarrow\simeq&V//G\\ &\underset{\tilde{\rho}(g')}{\curvearrowbotright}& \end{matrix}$

Well, OK, that still looks a little goofy (the clockwise top arrow is rather ugly, at least on my system — when, oh when will the Stix fonts arrive?). But it’s better than what you had.

Carry on …

Posted by: Jacques Distler on October 8, 2007 3:45 PM | Permalink | PGP Sig | Reply to this

### TeXnical Issues

I have moved the ensuing discussion of mathematical graphics and fonts to the thread on TeXnical Issues. I think it’s best if we discuss such matters there, so people who come along later can easily find what was said.

Posted by: John Baez on October 8, 2007 8:45 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

There is now a discussion of this point (the action groupoid as the $\mathrm{INN}(G)$-rep induced from a $G$-rep) on slides 161 and following here.

Posted by: Urs Schreiber on October 9, 2007 2:05 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Urs, at least on my computer your slides do not appear to be numbered.

Just wondering out loud about action groupoids and 2-morphisms: What would it mean (if anything) if we were to assume that all 2-morphisms are isomorphisms?

Posted by: Charlie Stromeyer Jr on October 9, 2007 9:10 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Urs, at least on my computer your slides do not appear to be numbered.

The slides don’t carry explicit numbers themselves, but the pdf reader with which you view them should be able to tell you which page of the pdf file you are viewing. The pdf readers that I use do that automatically.

And, by the way, the idea is that you use the hypertext navigation tools provided by your pdf reader to navigate these slides. I have provided various hyperlinks. You should follow them as desired and then use the odf readers BACK button to jump back.

Just wondering out loud about action groupoids and 2-morphisms: What would it mean (if anything) if we were to assume that all 2-morphisms are isomorphisms?

Which 2-morphisms do you have in mind here?

Since we are talking about groupoids, most everything in sight tends to be invertible. All natural transformations between functors between groupoids are, for instance. Those were the only 2-morphisms that appeared in my above comment.

Or maybe are you wondering how we generalize action groupoids to action 2-groupoids, when we have a 2-group acting on something?

That’s an interesting question, I think. This is in part what I tried to address in my comment: how do we define the action groupoid abstractly (instead of in components as often done), such that we would know, for instance, how it categorifies.

In as far as the characterization in terms of that pushout along

$\array{ \Sigma G &\to& \Sigma \mathrm{INN}(G) \\ \downarrow^\rho \\ \Sigma \mathrm{Aut}(V) }$

makes sense, this would indicate the right categorification.

This is at least apparently what appears when we consider non-fake-flat associated $n$-transport.

(If you cannot see any slide numbers, open the document, go to the table of contents on the second slide, follow the link “Parallel $n$-transport” and then the link “Associated $n$-vector transport”).

Posted by: Urs Schreiber on October 10, 2007 9:50 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Thank you, Urs, for answering my two inquiries. Even though I was using the Safari web browser, I could read all of your Math ML perfectly (weird, huh?), but then I had to switch to Windows to get the PDF reader to show numbers at the bottom with your slides.

I had meant my question in a general categorified sense, i.e., what happens if a 2-group is acting on something. Although I know something, e.g., about groupoids and n-gerbes, the concept of n-groupoids is new to me so I will read your slides and your previous posts on this subject.

Posted by: Charlie Stromeyer Jr. on October 10, 2007 4:49 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

what happens if a 2-group is acting on something

Okay, good, that’s a very good question. I tried to reply to it above, but maybe it’s worth amplifying this a little more:

Given a 2-group $G_{(2)}$ and a representation of it on some object $V$ in some 2-category $C$, i.e. a 2-functor

$\rho : \Sigma G_{(2)} \to \Sigma \mathrm{Aut}_C(V)$

from the one-object 2-groupoid given by $G_{(2)}$ to the 2-groupoid whose single object is $V$ and whose groupoid of morphisms is that of automorphisms of $V$ in $C$.

Then: what is the corresponding action 2-groupoid?

The answer I proposed was based on the following observation:

if $\rho$ is used to associate an associated 2-bundle to a principal $G_{(2)}$ 2-bundle, then we are lead to find that the $3$-curvature of any 2-connection on that 2-bundles takes values in an induced 3-representation of $\mathrm{INN}_0(G_{(2)})$ – and that this 3-representation lives on the action 2-groupoid of $\rho$.

I am not entirely sure yet how much weakening to allow here (for $n=1$ it seemed we wanted everything to be strict), but it seems that we want to define the action 2-groupoid $V // G_{(2)}$ by the pushout

$\array{ \Sigma G_{(2)} &\hookrightarrow& \Sigma \mathrm{INN}_0(G_{(2)}) \\ \downarrow^\rho && \downarrow \\ \Sigma \mathrm{Aut}(V) &\to& \Sigma \mathrm{Aut}(V // G_{(2)}) } \,.$

If this holds water, I might at some point be so obnoxious as to start referring to it as groupoidification from $n$-transport.

To me it seems that this should clarify a couple of important issues, like the relation between $n$-curvature and quantization.

In any case, it makes me await all further details on the groupoidification program with great suspense.

Posted by: Urs Schreiber on October 10, 2007 6:11 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

«This is of course just another aspect of the statement that the weak quotient $V//G$ is equivalent to the strict quotient $V/G$ (regarded as a discrete category).»

Why are they equivalent? In general, there can be more than one morphism between two objects of $V//G$ (for example, if $G$ acts in the evident way on the one-element set $1$, $1//G$ will have one object and and the group of endomorphisms of this object will be $G$). So $1//G$ cannot be equivalent to a discrete category. Do you assume that the action is free?

Posted by: Mathieu Dupont on October 10, 2007 11:26 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Why are they equivalent?

Right, they are not in general. My mistake. Rather, the isomorphism classes of $V//G$ yield $V / G$, so $\pi_0(V//G) = V/G \,.$

Thanks for catching that.

Posted by: Urs Schreiber on October 10, 2007 1:07 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Here’s a comment by Apoorva Khare on the homework exercise in Lecture 1:

Dear Prof. Baez,

Hi, while doing the homework, in order to get the ($q$ or usual) multinomial coefficient — as the answer for the number of $D$-flags on $F_q^n$ (for $q$ a prime power or $q= 1$) — I realised:

When you write a $D$-flag on a set — in the form of an UNCOMBED Young diagram, do you want to specify if the integers/subsets in each row are written in INCREASING (or DECREASING) order? Because the subsets

$X_0 \subseteq X_1 \subseteq \dots$

are the only data given in a $D$-flag, i.e. $X_{i+1} - X_i$ is NOT given in a specific order, but just as a set.

The reason this was left out was because you only drew $D$-flags on sets with $n = 1+1+ \cdots +1$, so this case never arose.

Thanks,
Apoorva.

You just answered your question, but I’ll do it more slowly.

A D-flag on an $n$-element set $X$ is a bunch of nested subsets

$\emptyset = X_0 \subseteq X_1 \subseteq \dots \subseteq X_k = X$

where the cardinality of $X_{i+1} - X_i$ is the number of boxes in the $i$th row of the uncombed Young diagram $D$.

So, for example, if $D$ looks like this:

XX
X

there are 3 $D$-flags on the set $\{1,2,3\}$, namely:

$X_1 = \{1,2\} \qquad X_2 = \{1,2,3\}$ $X_1 = \{2,3\} \qquad X_2 = \{1,2,3\}$ $X_3 = \{1,3\} \qquad X_2 = \{1,2,3\}$

You’re suggesting that we cleverly keep track of these by putting numbers in the boxes of our Young diagram. We can do that:

12
3

23
1

13
2

Now to the point: the order of the numbers within each row doesn’t matter, since it’s just a notation for a set $X_i - X_{i+1}$. So, we can without loss of generality write them in increasing order, as I’ve done.

Best,
jb

Posted by: John Baez on October 8, 2007 7:01 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Here’s a comment by Jagannatha Prasad Senesi on Lecture 1:

Hi,

One thing you mentioned on Thursday when you wrote down the vector space $\mathbb{C}^X$ is that we should not confuse this with the set of functions from $X$ to $\mathbb{C}$. But that’s exactly what it is, isn’t it?

In fact if we think of the freely generated vector space $\mathbb{C}^X$ as ‘Vector space valued functions on $X$’, where the vector space is just $\mathbb{C}$, then this begins to sound very similar to the construction of an induced representation (from a subgroup H to a group G), one description of which goes something like…

‘Vector valued functions on $G$ which are $H$-equivariant’.

These induced representations are also freely generated.

There are two different vector spaces, which one should not mix up:

1. The complex vector space of all complex functions on the set $X$.
2. The complex vector space having the set $X$ as basis.

These are canonically isomorphic when $X$ is finite — that’s why it’s tempting to mix them up. But, they’re NOT ISOMORPHIC AT ALL when $X$ is infinite. And, even when $X$ is finite, they’re not naturally isomorphic.

(You can ask Jim about the difference between ‘naturally’ and ‘canonically’.)

The notation $A^B$ usually means ‘all functions from $B$ to $A$’. I warned the class that I’m using $\mathbb{C}^X$ to mean ‘the complex vector space with $X$ as basis’, not ‘the set of all functions from $X$ to $\mathbb{C}$’. But I added that since $X$ for us will often be finite, we can often ignore the difference between these, as long as we keep our wits about us.

Similarly, your description of induced representations is fine when $H$ and $G$ are finite, but potentially problematic otherwise.

I’ll cc this to some other people in the class, since I bet you’re not the only one who was puzzled by my remark.

Best,
jb

Posted by: John Baez on October 8, 2007 7:14 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

If $A$ is an abelian group, there is the notation $A^{(B)}$ for the set of functions $f:B\to A$ such that $f$ is nonzero at most finitely many points of $B$. In other words, it’s the direct sum of $B$ copies of $A$, rather than the direct product. It’s somewhat common in number theory. I don’t use it too often myself, but sometimes it is really convenient. Another option would just be $\oplus_B A.$

Of course, there’s a functorial map $A^{(B)}\to A^B$, which is an isomorphism if $B$ is finite. So in what sense is this not a ‘natural’ isomorphism?

Posted by: James on October 8, 2007 11:36 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

The map you describe, $A^{(B)} \to A^B$ is not functorial. The functor $B \to A^B$ is contravariant in $B$ since it works by composition, whereas the functor $B \to A^{(B)}$ is covariant since it is given by

(1)$\sum_{i=1}^n a_i b_i \to \sum_{i=1}^n a_i f(b_i)$

with $a_i \in A$ and $b_i \in B$.

The relationship between $A^B$ and $A^{(B)}$ is that one is the dual of the other. This holds for $\mathbb{C}^X$ and $\mathbb{C}[X]$ as well with their standard linear topologies. The duality pairing is the obvious one:

(2)$f(\sum_{b \in B} a_b b) = \sum_{b \in B} a_b f(b)$

where all but finitely many of the $a_b$s are finite.

For finite sets then there is an isomorphism $A^{(B)} \to A^B$ since we can make $B \to A^{(B)}$ into a contravariant functor via

(3)$\sum_{b \in B} a_b b \to \sum_{c \in C} (\sum_{b \in f^{-1}(c)} a_b) c$

with this construction, the isomorphism $A^{(B)} \to A^B$ given by sending $b$ to the delta function at $b$ is natural in the categorical sense.

So in fact there is a difference between $\mathbb{C}[n]$ and $\mathbb{C}^n$ but it’s not usually anything worth worrying about.

By the way, the notations $k^{(X)}$ and $k^X$ seem fairly standard usage for the coproduct and product in the category of topological vector spaces.

Posted by: Andrew Stacey on October 9, 2007 2:44 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Andrew explained it all very nicely, but let me repeat what he said in more lowbrow terms.

There’s a covariant functor

$\mathbb{C}[-] : Set \to Vect$

sending each set $X$ to $\mathbb{C}[X]$, which is the complex vector space with $X$ as basis.

There’s a contravariant functor

$\mathbb{C}^{-} : Set \to Vect$

sending each set $X$ to $\mathbb{C}^X$, which is the complex vector space of functions from $X$ to $\mathbb{C}$.

Since one of these functors is covariant while the other is contravariant, it doesn’t make sense to ask if there’s a natural transformation from one to the other.

Moreover, if $X$ is infinite, $\mathbb{C}[X]$ and $\mathbb{C}^X$ are not isomorphic.

We can cure the latter problem by restricting attention to finite sets. We get a covariant functor

$\mathbb{C}[-] : FinSet \to Vect$

and a contravariant functor

$\mathbb{C}^{-} : FinSet \to Vect$

These assign isomorphic vector spaces to any finite set $X$. But alas, it still doesn’t make sense to ask if they’re naturally isomorphic.

To cure this problem, we can restrict to the groupoid of finite sets, with bijections as morphisms. Any contravariant functor from a groupoid to a category can be turned into a covariant one, by cleverly sticking an ‘inverse’ in the right place.

Using this trick, we get two covariant functors from $FinSet$ to $Vect$. The first assigns to each finite set the vector space having that set as basis, and does the obvious thing on morphisms. The second assigns to each finite set the vector space of functions on that set, and does the obvious thing on morphisms… but with an inverse cleverly stuck into the formula!

These functors are then naturally isomorphic.

This is the sense in which we don’t need to worry about the difference between $\mathbb{C}[X]$ and $\mathbb{C}^X$ when $X$ is a finite set.

Of course, the fact that it took me this many paragraphs to explain how we don’t need to worry about the difference, means that in fact we really do need to worry about the difference — until all this stuff becomes second nature.

Posted by: John Baez on October 9, 2007 7:33 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

But Andrew wasn’t restricting to the underlying groupoid of finite sets. He was instead defining a covariant functor on finite sets, $X \mapsto \mathbb{C}^X$, where the effect on a function $g: X \to Y$ is to send it to the linear map

$(f: X \to \mathbb{C}) \mapsto (g_{*}(f): y \mapsto \sum_{x: g(x) = y} f(x)),$

which is an idea reminiscent of Kan extension. In fact, it really is the taking of an adjoint (in the usual linear algebra sense): if we denote the pulling back or restriction along $g$ by $g^{*}: \mathbb{C}^Y \to \mathbb{C}^X$, then we have

$\langle g_{*}(f), h \rangle = \langle f, g^{*}(h) \rangle$

where the inner product is defined by taking the basis $X$ to be self-dual (which was also implicit in Andrew’s comment, when he referred to the delta function).

Posted by: Todd Trimble on October 9, 2007 8:45 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

“Since one of these functors is covariant while the other is contravariant, it doesn’t make sense to ask if there’s a natural transformation from one to the other.”

Uh, good point. I guess my low-level thinking is not as categorically enlightened as I had hoped. You learn something new every day, and this one just started!

Posted by: James on October 9, 2007 11:00 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

“Since one of these functors is covariant while the other is contravariant, it doesn’t make sense to ask if there’s a natural transformation from one to the other.”

Actually, it does make sense if you twist your head round a bit.

Suppose we have functors $F, G: \mathbf{A}^{op} \times \mathbf{A} \to \mathbf{B}.$ One can talk about dinatural transformations from $F$ to $G$. Such a thing is a family $(\alpha_A: F(A, A) \to G(A, A))_{A \in \mathbf{A}}$ of maps in $\mathbf{B}$, such that for each map in $\mathbf{A}$, a certain hexagon commutes. See e.g. Categories for the Working Mathematician.

In particular, suppose we have functors $P: \mathbf{A} \to \mathbf{B}, \quad Q: \mathbf{A}^{op} \to \mathbf{B}.$ By composing $P$ and $Q$ with the two product-projections of $\mathbf{A}^{op} \times \mathbf{A}$, we obtain functors $F$ and $G$ of the form above. The phrase ‘natural transformation from $P$ to $Q$’ can then be interpreted as ‘dinatural transformation from $F$ to $G$’. Such a thing is a family $(\alpha_A: P(A) \to Q(A))_{A \in \mathbf{A}}$ of maps, such that for each map $f: A \to A'$ in $\mathbf{A}$, a certain square commutes… Hmm, not ready to draw commutative diagrams yet, but it’s a square in which one side is equal to the composite of the other three. In one-dimensional notation, it says $\alpha_A = Q(f) \circ \alpha_{A'} \circ P(f).$

Now that the question makes sense, what’s the answer? It’s still no! Though if my back-of-the-envelope calculations are correct (and this particular envelope was already heavily scribbled on), it’s ‘yes’ if you restrict to the category of sets and injections.

Posted by: Tom Leinster on October 10, 2007 12:34 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

I’ve seen $\mathbb{C}\langle X \rangle$ used to denote the vector space with basis $X$, although similar pointy-bracket notation is used also for the field of quotients of a polynomial ring.

I personally like either $\mathbb{C} \cdot X$ or $X \cdot \mathbb{C}$ to denote the $X$-fold sum (coproduct) of copies of $\mathbb{C}$. This notation is current in other contexts, like enriched category theory, as in the tensor $v \cdot a$ of an object of a $V$-enriched category by an object of $V$, where $(-) \cdot a$ is left adjoint to the representable $hom(a, -): A \to V$ (in the enriched sense).

Posted by: Todd Trimble on October 9, 2007 1:17 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

How silly of me. $\mathbb{C}(X)$ is notation used for the field of quotients. But I think $\mathbb{C}\langle X \rangle$ is sometimes used for the algebra of non-commuting polynomials.

Posted by: Todd Trimble on October 9, 2007 1:23 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

I often use $\mathbb{C}[X]$ for the complex vector space with $X$ as basis. Why? Because when $X = G$ is a group, this is called the ‘group algebra’ of $G$, and denoted $\mathbb{C}[G]$.

However, in this particular lecture, I wanted to use the notation $n$ for the $n$-element set. Nobody uses $\mathbb{C}[n]$ for the complex vector space with this set as basis — everyone uses $\mathbb{C}^n$. So, since we’re mainly talking about finite sets anyway, I decided to use $\mathbb{C}^X$ as my notation for the vector space with $X$ as basis.

Posted by: John Baez on October 9, 2007 3:01 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

“You can ask Jim about the difference between ‘naturally’ and ‘canonically’.)”

This is an interesting subject. Could you elaborate on this? Thanks

Posted by: Goncalo Marques on October 9, 2007 7:02 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Could you elaborate on this?

Briefly: two functors $F,G : C \to D$ are ‘naturally isomorphic’ if they do isomorphic things to any object, and this isomorphism can be chosen in a way that’s compatible with all morphisms in $C$. They’re ‘canonically isomorphic’ if they do they do isomorphic things to any object, and this isomorphism can be chosen in a way that’s compatible with all isomorphisms in $C$. The second one is weaker.

But let me be a bit more precise.

I’ll assume you know the usual concept of naturally isomorphic functors.

Jim says two functors

$F, G : C \to D$

are canonically isomorphic if they become naturally isomorphic when restricted to the groupoid whose objects are those of $C$, but whose morphisms are just the isomorphisms of $C$.

There are lots of situations where this comes up. In particular, even if $F$ is covariant and $G$ is contravariant, we can talk about $F$ and $G$ being canonically isomorphic, since we can always turn a contravariant functor from a groupoid into a covariant one.

I used this trick in a previous comment to note that $\mathbb{C}^X$ and $\mathbb{C}[X]$ are canonically isomorphic when $X$ is a finite set — even though they’re not naturally isomorphic.

(But, I didn’t come out and use the phrase ‘canonically isomorphic’.)

Posted by: John Baez on October 9, 2007 7:43 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Jim says two functors

$F,G : C \to D$

are canonically isomorphic if they become natural isomorphic when restricted to the [core of $C$]

Cool. Didn’t know that. The claim is that we can give “canonical” a technically precise sense?

Surely you have a list of (further) examples with which to convince oneself that this is indeed the right way to formalize “canonical isomorphism”?

I mean, suppose I doubted it: convince me!

Posted by: Urs Schreiber on October 9, 2007 7:53 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Sorry, I don’t feel in the mood to go through the various senses in which people use the word ‘canonically’ and see how many fit this definition. A bunch do; a bunch don’t. The main thing is to have some term for the concept of ‘naturally — with respect to isomorphisms’.

Posted by: John Baez on October 9, 2007 8:09 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

The main thing is to have some term for the concept of ‘naturally — with respect to isomorphisms’.

Okay. But I must say I am lacking a feeling for why we would want to call that particular notion “canonical isomorphism”.

What bothers me a bit is that this would imply that two functors can be canonically isomorphic without being isomorphic.

That seems to go against the grain of the usual use of the word “canonical”. No? Usually we’d want to see two things that are isomorphic, and then say: “Ah, but they are not just isomorphic, but even canonically isomorphic: there is a god-given choice of isomorphism’”.

On the other hand, for the definition you mentioned, any two functors which are isomorphic are also canonically isomorphic.

Maybe the notion “isomorphic after pulled back to the core of the domain” could be called

essentially isomorphic

instead, or something like that?

Posted by: Urs Schreiber on October 10, 2007 9:57 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Here’s a comment by Chris Rogers on Lecture 1:

Hi Dr. Baez,

I have a quick question: In the quantum case we are talking about Vect, which intuitively seems to have a lot of structure built into it that we care about i.e. structure that is important to quantum mechanics. Aren’t we “cheating” classical mechanics by just treating it as Set, when classical mechanics really lives in the category of symplectic manifolds, which has a lot more relevant structure going on than Set? And if we are talking about group theory in this context, aren’t we eventually going to want to say something about the relationship between canonical transformations (not just functions between sets) and representations of unitary operators? I guess I’m a little confused.

Thanks,
Chris

Aren’t we “cheating” classical mechanics by just treating it as Set, when classical mechanics really lives in the category of symplectic manifolds, which has a lot more relevant structure going on than Set?

Right, definitely. When I wrote “classical” on the board, what I really meant is not so much “classical mechanics” as “classical logic” — i.e., the way we treat Set as the foundations of mathematics. I actually hinted at this, but I didn’t want to make a big deal about it. There’s really too much to say about this…

The category of symplectic manifolds, or even better (maybe) Poisson manifolds, is actually much more like the category of vector spaces than people tend to realize. They’re both non-cartesian - see

for details on what ‘cartesian’ means. The category Set is cartesian. For more on cartesian versus noncartesian categories, try:

In any event, what matters most in this seminar is how group actions on sets are related to group representations on vector spaces, and the extent to which we can find a ‘purely combinatorial’ description of portions of quantum mechanics. We’re not really going to talk much about classical mechanics.

Best,
jb

Posted by: John Baez on October 8, 2007 8:09 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

chris rogers asked: Aren’t we cheating classical mechanics by just treating it as Set, when classical mechanics really lives in the category of symplectic manifolds, which has a lot more relevant structure going on than Set?

john baez mostly agreed. i’ll give a contrary viewpoint here even if the situation isn’t completely clear. we’re _not_ really just treating classical mechanics as living in the category of sets; it’s more like we’re treating it as living in the category of free modules over the rig r of truth-values, or perhaps over some extension rig of r such as the rig of “costs”. this is very parallel to the way we’re treating quantum mechanics as living in the category of free modules over the rig r’ of rational numbers, or perhaps over some extension rig of r’ such as the rig of complex numbers.

a homomorphism between free modules in this context is a _relation_ rather than a
function. an equivariant such relation is a union of “double cosets”, or rather of the orbits-in-cartesian-products to which they correspond. this is how double cosets got to be so important as to deserve a much more suggestive name than “double cosets”.

the category of sets and relations (or its close cousin the category of sets and cost-matrixes, arguably a better formalization of classical mechanics than the original dehydrated elephant symplectic geometry) is very similar to the category of vector spaces and linear operators in many ways but is the archetype of “classical” in the same way that the category of vector spaces and linear operators over the rig of rational (or real or complex) numbers is the archetype of “quantum”. there’s more parallelism between “classical” and “quantum” than a lot of people realize.

it may come up in the seminar though how the meaning of “quantum” is somewhat up for grabs, as is the meaning of “geometric” (in “geometric representation theory”). john seemed to suggest in the first seminar lecture that “geometric” here means something like “classical” or at least “classically inspired even if still quantum”, which i thought was an interesting idea.

Posted by: james dolan on October 9, 2007 10:22 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

the category of sets and relations…is the archetype of “classical”

I’ll take that as support for my position.

Posted by: David Corfield on October 9, 2007 10:47 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Here’s an article on Vermeer’s camera.

Posted by: Mike Stay on October 8, 2007 9:23 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Excellent!

For those poor folks who can’t watch the video of my lecture yet and are wondering what the heck Mike is talking about:

I began my discussion of projective geometry with a little spiel about its roots in Renaissance painting… and the class got to talking about Vermeer’s use of the camera obscura to help with perspective. Thinking about these things is a good way to get an intuition for the projective plane.

Projective geometry will be very important in this seminar! For more, try my page on octonionic projective geometry (just ignore the stuff about octonions), and also week106 and week145.

Posted by: John Baez on October 8, 2007 9:39 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

duu ~ nope ~ sorry ~ can’t get lectures here ~ vid or otherwise :( just the pdf’s would be nice ~ for starters at least ~ vids for download would also be best ~ in the interim ~ just drop them on utube ~ that works for some :)

Posted by: k on October 9, 2007 1:36 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

You mentioned Renaissance projective technique. Are you aware of the Hockney dispute that many of the masters used cameras? That camp claims that cameras were long a secret within the guild and patchwork perspective errors show the use of narrow view lenses.

Posted by: RodMcGuire on October 9, 2007 7:17 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Yes, I’m aware of the Hockney debate — thanks for the link!

What most people don’t know is that the masters used, not just projective geometry, but also groupoidification. Cheaters!

Posted by: John Baez on October 9, 2007 8:04 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

The lecture streams beautifully from Sheffield! And that’s just on bog-standard home broadband. I like the software - click and play, simple and effective. Also, it seems one can instantaneously jump to various times in the video.

Whose satchel was that obscuring the right hand board?

I think we are all sorely missing Derek Wise’s notes!

Posted by: Bruce Bartlett on October 11, 2007 10:34 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Bruce wrote:

The lecture streams beautifully from Sheffield! And that’s just on bog-standard home broadband.

Great! It’s nice to hear someone outside the US can watch this thing. Please try to find out why Eugenia had trouble with the streaming video, also presumably from Sheffield.

Maybe she has ‘sub-bog-standard’ broadband? Also known as ‘skinnyband’?

I like the software - click and play, simple and effective.

Yeah, it works fine for me. Unfortunately, in response to some of the complaints on this blog entry, the UCR multimedia folks have switched to a different streaming format that doesn’t work at all for me, together with a downloadable format that’s taking forever for me to download. We’ll see how it goes…

Also, it seems one can instantaneously jump to various times in the video.

Yeah, that’s a feature I really like.

I think we are all sorely missing Derek Wise’s notes!

Yeah, me too! I keep trying to get U.C. Davis to fire him, so far to no avail. Luckily, you can now download Alex Hoffnung’s notes of Lecture 1 — and pretty soon, some other people’s lecture notes. We’ll see what you like best.

More stuff coming soon.

Posted by: John Baez on October 12, 2007 5:14 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

John wrote:

Unfortunately, in response to some of the complaints on this blog entry, the UCR multimedia folks have switched to a different streaming format that doesn’t work at all for me, together with a downloadable format that’s taking forever for me to download. We’ll see how it goes…

Ouch, that’s no good that it’s messed things up for you! For what it’s worth, I’m getting the streaming about twice as fast now than before (it’s gone from a factor of 4 to a factor of 2 slower than proper time), but I’d still prefer to download if possible. Is there a public link for downloads, or is that still experimental?

Posted by: Greg Egan on October 12, 2007 10:03 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

In fact I was in London when I was having the streaming problems, using bog-standard broadband but wireless. It works fine from my office in Sheffield down a wire.

Posted by: Eugenia Cheng on October 12, 2007 12:10 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

A video of the first lecture is now available for download here. It’s big — 556 megabytes. If you click on this link you should see a blue letter Q on your screen (assuming you have QuickTime loaded). Your webbrowser should say it’s transferring data from mediaserve.ucr.edu… and then you can expect to wait a long time for the file to download.

Go on a hike, have dinner, maybe dessert too, and when you’re done you’ll have your very own movie of me lecturing about geometric representation theory.

In general, the downloadable videos should appear here. My second lecture is already available — more about that soon.

I’m glad Eugenia can now watch the video in streaming form. I’ll be interested to hear Greg’s report if he tries the downloadable version. I hope his computer has enough disk space.

We’re just beginning to work the bugs out of this business. It might have been easier to learn how to take a video, chop it into 10-minute fragments, and upload it to YouTube. But ultimately, my university shouldn’t need to rely on YouTube to run online classes.

Posted by: John Baez on October 12, 2007 8:12 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

The download of the first lecture took 6 hours (Australian bog-standard broadband is 256 kilobaud, i.e. about 25 kilobytes/sec) but it went without a hitch, and the result plays flawlessly.

Thanks very much for adding this option! For those of us with slow connections, we can always download the movie overnight, and it’s much more watchable than a stream when you can’t keep up with the data rate. And while the files are big they’re not unmanageable; you can even fit a whole lecture on a CD, which might be handy.

Posted by: Greg Egan on October 13, 2007 7:53 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Thanks for the report!

I’m glad it works. I’m sorry it takes so long to download the darn thing. The picture resolution is higher than the typical video you’d find on YouTube — but you’ll be grateful for that if you watch lecture 4, where I accidentally started writing too small.

Posted by: John Baez on October 13, 2007 8:18 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

I’m sure you have done it with malice aforethought but why don’t you mention ‘partitions’ of n instead of uncombed…

Posted by: jim stasheff on October 13, 2007 2:58 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

‘Partitions’ means two things already, neither of which is an uncombed Young diagram:

1. A partition of a finite set is a way of writing it as a disjoint union of nonempty subsets.
2. A partition of a natural number is a way of writing it as a sum of nonzero natural numbers, where we don’t care about the order.

An ‘uncombed Young diagram’ is a way of writing a natural number as a sum of nonzero natural numbers where we do care about the order. So,

XX
X
X

(or 2+1+1) is different from

X
XX
X

(or 1+2+1). If we decide we don’t care about the order, we can ‘comb’ our uncombed Young diagram, making sure the rows get shorter as we march down. This gives an ordinary Young diagram — which is just a way of representing a partition in sense 2 above.

So, there are three confusingly similar, deeply related but crucially distinct concepts floating around, two of which are already called ‘partitions’. We decided to call the third one something else.

Posted by: John Baez on October 13, 2007 7:59 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

To make it clear I’m not being pedantic just for the fun of it, let me illustrate how these concepts are really different.

There are 5 partitions of a 3-element set:

{ {1}, {2}, {3} }

{ {1, 2}, {3} }

{ {1, 3}, {2} }

{ {1}, {2, 3} }

{ {1, 2, 3} }

There are 3 partitions of the number 3:

1+1+1

2+1

3

And, there are 4 uncombed 3-box Young diagrams:

X
X
X

XX
X

X
XX

XXX

or if you prefer:

1+1+1

2+1

1+2

3

Posted by: John Baez on October 13, 2007 8:07 AM | Permalink | Reply to this

### verma modules and d-modules

i’ve been struggling recently to try to understand some of the ideas in kirwan’s “an introduction to intersection homology theory” (and/or the second edition co-authored by woolf). i’ll try to state here some of the questions that are bugging me although of course the ones that are really bugging me i’m having trouble articulating even to myself.

i’m starting with the last chapter of the book (chapter 8 in the first edition, chapter 12 in the second edition), on “the kazhdan-lusztig conjecture”, because that’s one of the main things that i’m interested in learning about, and because that’s the chapter that i’ve come closest to being able to understand in the past.

(i have the vague impression that the second edition was to some extent designed to try to convert the book into one that can be read in the forwards direction instead of backwards. since i’m still trying to read it backwards myself, i’m not finding the second edition changes very relevant so far.)

the last chapter seems pretty decipherable up to and including the part (page 151 in the first edition, about page 204 in the second edition) where they describe the functor f that extracts a representation of the semi-simple lie algebra g from a “d-module” over the corresponding flag manifold. then (page 152) they start appealing to heavy machinery from earlier in the book (“riemann-hilbert correspondence” and “intersection sheaf complex”) to obtain particular d-modules whose images under f will be verma modules and/or irreducible modules with highest weights in the affine weyl orbit of 0.

since i haven’t yet worked my way backwards to understanding that heavy machinery, though, i’m hoping that the particular d-modules in question can be described in a more direct and explicit lowbrow way, without the heavy machinery. so that’s my question (for now): can you give such an explicit lowbrow (but pleasant) description of these d-modules??

(“lowbrow” and “pleasant” are defined here according to my personal taste. for example “parabolic induction” and “schubert calculus on flag manifolds” are in the lowbrow and pleasant direction while “intersection sheaf complex” and “perverse sheaf” are in the opposite direction, for now.)

i’m starting here with the attitude that a d-module is (insofar as i understand it yet) essentially a sheaf of systems of differential equations, with the sections over a given open set standing in for the “unknown functions” which are to be solved for. so, i’d like to understand what is this alleged system of differential equations that corresponds to (for example) the verma module with highest weight 0.

since the verma module “v_w” with highest weight w is the module possessing a generic maximal weight vector of weight w, i can see how for example a module homomorphism from v_w to the module given by smooth complex-valued functions on some manifold m on which the lie algebra acts as vector fields is essentially a function f satisfying the differential equation stating that “f is a maximal weight vector of weight w”; is that all that’s going on here?? with the flag manifold playing the role of m, or something like that?? unfortunately i don’t seem to be able to get this interpretation to mesh yet with other things that kirwan says.

i could try to say a lot more about my vague intuitions about what’s going on here but i might get bogged down if i did so for now i’ll just post this as is.

Posted by: james dolan on October 15, 2007 10:17 AM | Permalink | Reply to this

### Re: verma modules and d-modules

James, just to avoid any potential confusion, you are talking about generalized Verma modules as opposed to ordinary Verma modules, right?

Hmm, I have never seen the book to which you refer but perhaps this paper or this paper might have some clues for what you are looking for.

Posted by: Charlie Stromeyer Jr on October 15, 2007 2:09 PM | Permalink | Reply to this

### Re: verma modules and d-modules

James, just to avoid any potential confusion, you are talking about generalized Verma modules as opposed to ordinary Verma modules, right?

i’m pretty sure that i’m talking about just plain ordinary verma modules here.

Hmm, I have never seen the book to which you refer but perhaps this paper or this paper might have some clues for what you are looking for.

those might be a bit more sophisticated than what i’m looking for at the moment, but there might be something useful in the references.

Posted by: james dolan on October 15, 2007 2:48 PM | Permalink | Reply to this

### Re: verma modules and d-modules

Hi,

First, there are several good references
for this story (aka Beilinson-Bernstein localization): my favorite is Gaitsgory’s recent notes http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf
- I have several other references listed
on my web page at http://www.math.utexas.edu/users/benzvi/Langlands.html
including a nice survey
by Milicic. Of course there’s the original Beilinson-Bernstein papers and Bernstein’s D-modules notes as well, and the chapter
on D-modules in Gelfand-Manin’s Encyclopedia book has a nice overview.

The relation between Lie algebra reps and D-modules is precisely analogous to the relation between modules over a commutative algebra and [quasicoherent] sheaves on varieties. If a Lie algebra
g acts on a space X then Ug (the enveloping algebra) maps to global differential operators on X, and thus there are natural adjoint functors back and forth from g-reps to D-modules on X: namely global sections and tensor.
(Global sections of a D-module are a module over Ug, while for a Ug module
we can tensor it with diffops D_X over Ug, ie induce). These functors are pretty easy
to write down in concrete situations.
B-B theorem then says that for X=flag variety for g this is an equivalence of categories (if we restrict to Ug-modules with a fixed dominant regular infinitesimal character, eg the [finite!] Weyl group orbit of 0 you’re considering).

On the flag variety in the Kazhdan-Lusztig context we are considering D-modules — aka [quasicoherent] sheaves with a flat connection — which on each Schubert variety are just flat vector bundles (hence trivial of some rank). e.g. for sl_2 we’re looking at P^1 and we consider D-modules which are constant on A^1 and
at the point infinity.

The main (only?) thing one needs right
now from the theory of D-modules is the
functor j_* of pushforward (extension).
This is easiest perhaps to see for the inclusion of a closed subvariety: D-module is a sheaf with an action of D, differential operators. If you’re a D-module on a closed subvariety Z of X you already know how to act by all functions on X, by restriction, but not by all vector fields on X — so you induce, you let the normal vector fields to Z act freely. So in the transverse direction to Z you look like the symmetric algebra on the normal bundle – or more poetically like delta functions. Thus the extension can be thought of as distributions on X supported on Z and valued in your original module.

One can show that the category of D-modules we’re considering has a single simple object for each stratum, and that’s called the IC
sheaf. for P^1 that would be the constant vector bundle on P^1 for the open cell and the delta-functions at the point infinity for the closed one. These correspond to two simple representations of sl_2: the trivial one-dim rep, and the Verma module with highest weight -2 – note 0,-2 is the Weyl group orbit of 0, shifted to be centered at -1 (=-rho).

In general as you notice the pushforward
of a D-module from a closed subvariety
is an induced module, like the Verma — in fact the Verma module with highest weight lambda (antidominant) is just the
delta-functions on the one-point orbit on the flag variety (considered as a module over appropriate twisted differential operators).

On the other hand we have another natural D-module which is all functions on the big cell (ie functions on A^1 for sl_2) — this is a contragredient Verma module (coinduced representation). Note that the
constant functions sit inside of this – this is one of the characterizing features of the IC sheaf, it’s a sub of the *-extension for the corresponding orbit (and a quotient of the !-extension, which in this case would be the Verma.)

BTW as for Riemann-Hilbert correspondence
it’s just the functor that assigns to a flat connection its deRham complex, as a complex of sheaves (ie without taking global sections) — which is a complex of sheaves whose cohomology is locally constant. Similarly a (left) D-module is a (more complicated) sheaf with a flat connection and we may assign to it a deRham complex defined the exact same way. If the module is “small” (holonomic) – like all the ones we’re considering on the flag variety – then the cohomologies
of the deRham complex are locally constant along the strata of some stratification – aka a constructible complex.
So then one can try to describe
the category of D-modules purely topologically – leading to the proof
of the Kazhdan-Lusztig conjectures.

oops. got carried away there. oh well.

Posted by: David Ben-Zvi on October 15, 2007 5:21 PM | Permalink | Reply to this

### Re: verma modules and d-modules

ok, thanks; that was extremely helpful. i still have to mull it over a bit more (and probably talk it over some more with john or todd) before i can ask any more sensible questions, though.

Posted by: james dolan on October 16, 2007 10:18 AM | Permalink | Reply to this

### Re: verma modules and d-modules

Yes, thank you David for teaching us something new. In case anyone is curious, both David and James ARE talking about ordinary Verma modules as in this wikipedia entry . There are also generalized Verma modules as in this wikipedia entry . To figure out the case with D-modules and generalized Verma modules one would probably have to look at the first paper I mentioned, but I won’t be able to see this paper until Saturday.

Posted by: Charlie Stromeyer Jr on October 16, 2007 1:49 PM | Permalink | Reply to this

### Re: verma modules and d-modules

i’m still struggling with this, but i’d like to try to describe some of my thoughts about it so far.

first, it sounds like you’re confirming my guess that the functor from the representation category to the d-module category amounts to interpreting a presentation of a representation as a system of differential equations by means of the realization of the lie algebra as first-order differential operators on the flag manifold.

so i think then that a “germ of a solution” of this system of differential equations at a point f of the flag manifold should be essentially a morphism from the original representation to the representation given by germs of functions at f. and you (and kirwan and woolf) seem to be saying that when the original representation is a verma module induced from the stabilizer borel subalgebra of a flag f1, then the nature of the stalk of germs of solutions of the system at a flag f2 depends (as f1 and f2 vary) only on the double coset characterizing the geometric relationship between f1 and f2. (moreover you seem to be giving specific information about _how_ the nature of the stalk depends on the double coset, but i’ll worry about that later.)

and it’s tempting to think that i can understand the role played by the double coset here as being very similar to the role played by double cosets in an apparently analogous situation in the category of representations of a discrete group g. namely, if h and j are subgroups of g, and r1 and r2 are representations of h and j respectively, then hom_g(ind(r1),co-ind(r2)) is a cartesian product of separate sectors corresponding to the double cosets between h and j in g (where “ind” here indicates “induced representation”), with the sector corresponding to the double coset x being isomorphic to hom_k(r1,r2) where k is “the intersection of h and j when in position corresponding to x”.

optimistically speaking, then, perhaps the only significant extra wrinkle in the case at hand is that since a verma module is an induced representation of an “infinitesimal group” (aka lie algebra) rather than of an ordinary group, its isomorphism class depends not just on the conjugacy class of subalgebra that it was induced from, but on the specific choice of subalgebra within that conjugacy class. that is, perhaps a morphism from an induced representation to a co-induced representation is still associated with a double coset, but the double coset in question is now built into the relationship between the inducing subalgebra h and the co-inducing subalgebra j.

(pessimistically speaking, though, there are probably all sorts of other complications to worry about as well. for example i seem to be suggesting that the representation of a lie algebra given by the germs of functions (or more generally of sections of an equivariant vector bundle) at a point of a homogeneous space is analogous to a co-induced representation, whereas there’s something fishy about that.)

anyway, i would like a theorem something like this: given a complex lie group g, and given lie subgroups h and j, and representations r1 and r2 of the lie algebras h# and j# respectively (where “x#” here indicates “lie algebra of the lie group x”), the hom-space hom_g#(ind(r1),co-ind(r2)) is isomorphic to hom_k#(r1,r2) where k# is the intersection of h# and j#; and/or perhaps some trickier variant on this idea, with the role of “co-ind(r2)” played by something closer to the stalk of germs at the point with stabilizer subalgebra j# of sections of an appropriate equivariant vector bundle.

then on top of that i’d like to understand the details of how to apply such a theorem in the kazhdan-lusztig context, which i really don’t understand yet.

(one of my goals here is to understand in what sense or senses “geometric representation theory” really is “geometric”. i think that i have a good sense of how induced representations of groups (lie or otherwise) are “geometric”, but it seems trickier to understand the “geometric” interpretation of induced representations of lie _algebras_.)

Posted by: james dolan on October 25, 2007 6:36 AM | Permalink | Reply to this

### Re: verma modules and d-modules

David Ben-Zvi should be better able to address the points you make than I can, but I can say something about the very last issue you raise:

You will want to look at Cartan geometries. More specifically, this paper and this paper examine a type of Cartan geometry called parabolic geometry.

I know only a little about parabolic geometry because it happens to include projective geometry and von Neumann generalized projective geometry via his continuous geometries which is what my interest is.

Posted by: Charlie Stromeyer Jr on October 25, 2007 2:26 PM | Permalink | Reply to this

### Re: verma modules and d-modules

Charlie wrote:

You will want to look at Cartan geometries.

Heh. Jim knows a lot about Cartan geometry. My student Derek Wise did a Ph.D. thesis on Cartan geometry, part of which appears here. I gave a little intro to Cartan geometry and Derek’s work in week243.

I hadn’t heard the term “parabolic geometry” for a Cartan geometry modeled on $G/P$ with $G$ semisimple and $P \subseteq G$ parabolic, but that certainly does cover a bunch of famous examples.

Posted by: John Baez on October 26, 2007 11:25 AM | Permalink | Reply to this

### Re: verma modules and d-modules

It’s good that Jim knows a lot about Cartan geometry. I forgot to tell Jim that he won’t understand the intros of the two papers I referred to because the intros are not detailed enough, however, once you start reading the papers right after the intros then the papers begin to become understandable because they start to explicitly define Cartan geometries via algebras and groups.

Derek Wise’s work looks quite interesting because he applies his model to the correct de Sitter spacetime. I say “correct” only because the last time I checked (which I admit was over 4 years ago) the latest astronomical data showed that we live in de Sitter spacetime.

Posted by: Charlie Stromeyer Jr on October 26, 2007 1:13 PM | Permalink | Reply to this

### Re: verma modules and d-modules

Brief comment: (need to run to airport to your part of the world, speaking at Caltech in a couple of days in case anyone in the area is interested) Induced representations of Lie algebras can be described just as those of Lie groupsby passing to formal groups - Lie algebras (in char zero!) are the same as formal groups, and the same is true for their representations. And if you’d like a formal group is a group object in an appropriate category so reps of Lie algebras are really encompassed by reps of groups..

Anyway given Lie algebras g,h the induced representation of the trivial rep of h to g is given by distributions on exp g/ exp h. A similar picture will hold for inducing any other rep, looking at another D-module of distributions valued in the corresponding vector bundle.

The picture for Verma modules is the same, except we were identifying exp g/exp b with the formal neighborhood of the basepoint in G/B, so thinking of delta functions on G/B supported (set theoretically) at [B]. Of course replacing B with P you can get generalized Vermas… But it’s interesting that ALL reps of g can be written as a “direct integral” of Verma modules in an appropriate sense – this is one way to look at Beilinson-Bernstein (which I think is the way it’s explained in Gaitsgory’s notes I referred to).

I’ll try to look at your other questions from Pasadena (certainly what you say in the first couple of paragraphs is right, haven’t thought about the third yet).

Posted by: David Ben-Zvi on October 25, 2007 6:54 PM | Permalink | Reply to this

### Re: verma modules and d-modules

Brief comment: (need to run to airport to your part of the world, speaking at Caltech in a couple of days in case anyone in the area is interested)

i think that i’ll probably be able to attend, but i have to check my schedule.

Induced representations of Lie algebras can be described just as those of Lie groups by passing to formal groups- Lie algebras (in char zero!) are the same as formal groups, and the same is true for their representations.

yes, this is very much the attitude that i’ve been taking towards the subject, thinking of the enveloping algebra as a co-algebraic group whose “co-spectrum” has just a single global point (or dualizing the picture to get the formal algebraic group). to me this is indeed legitimately geometric but i could also try to appreciate a more classical viewpoint according to which a genuine “space” should have enough global points to support a description of the geometry allegedly taking place on it. when i was wondering outloud just how “geometric” geometric representation theory is i was imagining the situation of such a classically-minded person trying to picture what’s going on.

(i’m not exactly an adept of contemporary algebraic geometry but i have a background in for example lawvere’s school of “synthetic differential geometry” which has a similar or perhaps even more extreme commitment towards viewing infinitesimal spaces as legitimate spaces.)

And if you’d like a formal group is a group object in an appropriate category so reps of Lie algebras are really encompassed by reps of groups..

Anyway given Lie algebras g,h the induced representation of the trivial rep of h to g is given by distributions on exp g/ exp h. A similar picture will hold for inducing any other rep, looking at another D-module of distributions valued in the corresponding vector bundle.

yes, in principle this is the way that i’ve been thinking of these induced representations of lie algebras, except that i’ve probably been forgetting to take full advantage of the language of distributions as a means of describing them.

The picture for Verma modules is the same, except we were identifying exp g/exp b with the formal neighborhood of the basepoint in G/B, so thinking of delta functions on G/B supported (set theoretically) at [B].

ok, i think that this might be helping to explain some of what you said about “delta functions” in your original reply, which i didn’t really follow even after john tried to explain it to me. i still have to mull it over a bit more to see whether i really understand it now though.

on the other hand, the picture that you’re describing here of the infinitesimal homogeneous space exp(g#)/exp(b#) as the “formal neighborhood” of one single basepoint in the macroscopic homogeneous space g/b is exactly the picture that i was alluding to in trying to explain how the role played by double cosets in connection with homomorphisms from induced representations to co-induced representations (or something morally similar to co-induced representations) is morally the same though technically different in the case of lie algebra representations as compared to the case of group representations. in the lie algebra case you really have to pick out a special basepoint, and the induced or co-induced representation remembers where that basepoint is, and the double coset corresponding to the geometric relationship between the pair of such basepoints comes into play even before you decide to focus on a particular sector of the hom-space between the representations.

however, i’m still at the stage of just trying to develop a morally correct picture of what’s going on; i’m not claiming that i’ve got the technical details correct yet. some of what seems to be going on in the kazhdan-lusztig context isn’t smoothly meshing yet with the picture that i’m trying to create in my mind.

Of course replacing B with P you can get generalized Vermas.. But it’s interesting that ALL reps of g can be written as a “direct integral” of Verma modules in an appropriate sense- this is one way to look at Beilinson-Bernstein (which I think is the way it’s explained in Gaitsgory’s notes I referred to).

(i’ve taken only a very brief look at gaitsgory’s notes so far, not enough to tell whether i’ll be able to get anything useful out of them.)

I’ll try to look at your other questions from Pasadena (certainly what you say in the first couple of paragraphs is right, haven’t thought about the third yet).

ok, thanks.

Posted by: james dolan on October 25, 2007 9:09 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

Am I right that if one takes D to be the following n-box Young diagram:
X
X

X
then the resulting representation of n! is a regular representation? If I am, then the statement of the theorem about irreps of n! is a bit confusing - after all, all irreps are already included in the regular represetation.

So, is it that the resulting rep is not a regular rep, or rather that the important part of the theorem is that there is some canonical bijection between irreps and Young diagrams?

Posted by: sirix on February 2, 2008 8:09 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 1)

sirix wrote:

Am I right that if one takes D to be the [vertical strip] Young diagram, then the resulting representation of n! is a regular representation?

Yes!

If I am, then the statement of the theorem about irreps of n! is a bit confusing - after all, all irreps are already included in the regular represetation.

That’s understandable, but if memory serves, I think what John said was that inside each $D$-flag representation there is a particular irrep “screaming to get out”. You have to go a little further into the lecture series to find out what is meant, but there is a beautiful “categorified Gram-Schmidt procedure” which gives a recipe for extracting that screaming irrep out of the $D$-flag representation.

Posted by: Todd Trimble on February 3, 2008 3:27 PM | Permalink | Reply to this
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