Zuckerman on KLV (preview)

Even though Andrew Wiles was speaking at the Branford College master’s tea today, I didn’t go. Zuckerman was giving the first of two or three lectures on this whole KLV thing. And I actually took notes!

Unfortunately the scanner in the computer lab was being evil, so I can’t post them quite yet. I’ll definitely have them by Monday, though, and after that I’ll try to explain what I wrote. They should be already more than readable to mathematicians, though.

What I do have is pictures of a conceptual diagram we constructed on the blackboard in his office the other day. I managed to get it mostly into three parts: 1 2 3 (~700KB each). I apologize for the quality of the middle one — I couldn’t use a flash without washing out the board entirely. It should still be readable. The third cuts off some lists of names associated with the topics they’re next to. The second list is “Jantzen, Vogan, Speh”, while in the first list “H.C.” is Harish-Chandra and “Z.” is Zuckerman.

As for what all this means, that’s what these lectures are to explain more thoroughly. Here we see the entire subject circling “characters”, which are certain functions on the groups we’re interested in. Properly defining them was the subject of today’s lecture. In the lower left is a list of examples of the sorts of groups we’re interested in — is the now-(in)famous one. To the right of the diagram is the statement that two special classes of representations, the “standard” and “irreducible” ones, are related in a certain way. On the right is the recipe for computing the irreducible representations into which the Atlas project’s computation fits.