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The paper is devoted to the generalization of Lusztig's q-analog of weightmultiplicities to the Lie superalgebras gl(n,m) and spo(2n,M). We define suchq-analogs K_{lambda,mu}(q) for the typical modules and for the irreduciblecovariant tensor gl(n,m)-modules of highest weight lambda. For gl(n,m), thedefined polynomials have nonnegative integer coefficients if the weight mu isdominant. For spo(2n,M), we show that the positivity property holds when mu isdominant and sufficiently far from a specific wall of the fundamental chamber.We also establish that the q-analog associated to an irreducible covarianttensor gl(n,m)-module of highest weight lambda and a dominant weight mu is thegenerating series of a simple statistic on the set of semistandardhook-tableaux of shape lambda and weight mu. This statistic can be regarded asa super analog of the charge statistic defined by Lascoux and Schutzenberger.

On q-analogs of weight multiplicities for the Lie superalgebras gl(n,m) and spo(2n,M)