Differential operators and crystals of extremal weight modules

We give a combinatorial realization of extremal weight crystals over the quantum group of type A+∞ and their Littlewood–Richardson rule. Based on this description, we show that the Grothendieck ring generated by the isomorphism classes of extremal weight A+∞-crystals is isomorphic to the Weyl algebra of infinite rank, and hence each isomorphism class is realized as a differential operator or non-commutative Schur function acting on the algebra of symmetric functions. We also find a duality between extremal weight A+∞-crystals and generalized Verma A∞-crystals appearing in the crystal of the Fock space with infinite level, which recovers the generalized Cauchy identity for Schur operators in a bijective and crystal theoretic way.