MATH 7310: Flag Manifolds and Symmetric Spaces (Fall 2012)

Instructor: Allen Knutson

Prerequisites: Some Lie-theoretic background, like root systems, Dynkin diagrams, and the classification of irreps in terms of their highest weights will be assumed.

Let G be a complex Lie group with an involution, and K the fixed-point subgroup. Two standard examples are

  1. G = GL(n), the involution is inverse transpose, and K = O(n).
  2. G = K x K, the involution switches the two factors, and K is the diagonal.

Then many amazing theorems hold:

  1. G is the complexification of a real Lie group GR (essentially unique) such that K is the complexification of GR’s maximal compact subgroup.
  2. K acts on the flag manifold G/B with finitely many orbits.

These form a poset under inclusion of closures.

  1. GR does too, and its poset is the reverse of K’s (Matsuki duality).
  2. If G = K x K, this poset is K’s Bruhat order, so one can think of the poset K\G/B as a generalization of Bruhat order.

There are many possible topics to cover, depending on attendance. Some are

  1. To classify these involutions, or even pairs of commuting involutions (called the classification of real symmetric spaces),
    following the books of [Helgason].
  2. To study the posets K\G/B combinatorially, following the papers of [Richardson-Springer], [Incitti], and [Hulman].
  3. The Morse theory proof of Matsuki duality, using the Yang-Mills functional.
  4. Localization of (g,K)-modules, the algebraic version of GR-modules, as K-invariant D-modules on the flag manifold, following [Milicic]’s Penrose volume survey.
  5. The definition of, motivation for, and algorithmic computation of Kazhdan-Lusztig polynomials for K-orbits on G/B.
  6. (My own work) Computing positively the decomposition of G-irreps under K.