What is this pretty picture?
The
diagram parametrizes the irreducible representations
of SU(2,2) lying in the same block as a
fixed finite dimensional
representation F of the group.
In turn, the nodes in this picture parametrize the K-orbits
in the flag variety B.
The Kazhdan-Lusztig-Vogan conjectures
(now theorems) tell us how
to use sophisticated combinatorics attached to this picture to
compute a couple of interesting things:
- The characters of the irreducible representations
- The composition series of generalized principal series
- The Lowey filtrations of induced representions