Speaker: David Vogan (MIT)

Title: Restricting real group representations to K

Abstract: It's an old idea of Harish-Chandra that representations of a real
reductive G can be studied by understanding their restrictions to a
maximal compact K.  Traditionally this was done using the Cartan-Weyl
highest weight theory for K representations.

I will recall a (closely related but very different) description of K
representations, in which the highest weight is replaced by the
Harish-Chandra parameter of a discrete series on a Levi subgroup of G.
Using this new parameter, one can define a very natural "height" of a
representation of K, taking non-negative integer values. As evidence
that height is natural, I offer

Conjecture. Suppose G is split, and Π₁ and Π₂ are two (minimal)
principal series representations of G. If Π₁ and Π₂ have the same
restriction to the center of G, then the sets of heights of K-types of
Π₁ and of Π₂ are the same.

For example, split G₂ has principal series having two different
restrictions to K. In each, the first heights appearing are

0, 3, 6, 9, 10,12, 15...

F4 has principal series representations having three different
restrictions to K. But in all three, the first heights appearing are

0, 8, 11, 15,16, 21, 24, 26, 27, 29, 30, 32, 33, 36....

The numbers for split E₈ (which also has three types of principal
series) are

0, 29, 46, 57, 58, 68, 75, 84, 87, 91, 92...

It seems more or less impossible to find a similar statement using
highest weights. I will give some evidence for this conjecture.

Of course there should be an elementary construction of these sequences
of heights from the root system of G and the central character; I have
no idea how to make such a construction.

If there is time, I will talk about the possibility of finding parallel
statements for p-adic groups.