On the B-canonical splittings of flag varieties

Understand

Let G be a semisimple algebraic group over an algebraically closed field of positive characteristic.

• In this note, we show that an irreducible closed subvariety of the flag variety of G is compatibly split by the unique canonical Frobenius splitting if and only if it is a Richardson variety, i.e.
• an intersection of a Schubert and an opposite Schubert variety.