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
- G = GL(n), the involution is inverse transpose, and K = O(n).
- G = K x K, the involution switches the two factors, and K is the diagonal.
Then many amazing theorems hold:
- G is the complexification of a real Lie group GR (essentially unique) such that K is the complexification of GR’s maximal compact subgroup.
- K acts on the flag manifold G/B with finitely many orbits.
These form a poset under inclusion of closures.
- GR does too, and its poset is the reverse of K’s (Matsuki duality).
- 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
- To classify these involutions, or even pairs of commuting involutions (called the classification of real symmetric spaces),
following the books of [Helgason].
- To study the posets K\G/B combinatorially, following the papers of [Richardson-Springer], [Incitti], and [Hulman].
- The Morse theory proof of Matsuki duality, using the Yang-Mills functional.
- 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.
- The definition of, motivation for, and algorithmic computation of Kazhdan-Lusztig polynomials for K-orbits on G/B.
- (My own work) Computing positively the decomposition of G-irreps under K.