Balanced parabolic quotients and branching rules for Demazure crystals

Abstract

We study a subset of a parabolic quotient in a simply laced Weyl group W--stable under an automorphism $$\sigma $$ź--which we call the balanced parabolic quotient. This subset describes the interaction between the branching rule for a Levi subalgebra, Demazure modules, and $$\sigma $$ź-invariant weight spaces in $$\sigma $$ź-stable simple modules for the corresponding Lie algebra. The Hasse diagram of the balanced parabolic quotient in types A and D under the Bruhat order is a directed forest with a remarkable self-similarity property, and its order is related to the Fibonacci sequence. We characterize an element of a balanced quotient on the level of the root system of W and find that the subalgebras of the Borel associated with these elements decompose into the direct sum of two subalgebras: one contained in the Borel for a Levi subalgebra and another consisting of $$\sigma $$ź-invariants.