Crystal base of the negative half of quantum orthosymplectic superalgebras

Abstract

We construct a crystal base of the negative half of a quantum orthosymplectic superalgebra. It can be viewed as a limit of the crystal bases of q -deformed irreducible oscillator representations. We also give a combinatorial description of the embedding from the crystal of a q -oscillator representation to that of the negative half subalgebra given in terms of a PBW type basis. It is given as a composition of embeddings into the crystals of intermediate parabolic Verma modules, where the most nontrivial one is from an oscillator module to a maximally parabolic Verma module with respect to a quantum subsuperalgebra for

. A new crystal theoretic realization of Burge correspondence of orthosymplectic type plays an important role in the description of this embedding.