Combinatorics of Generalized Exponents

We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type $A_{n-1}$, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type $C_{n}$, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type $A_{2n-1}$, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig $t$-analogs associated to zero-weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula and discuss some implications to relating two type $C$ branching rules. Our methods are expected to extend to the orthogonal types.