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

Abstract

The paper is devoted to the generalization of Lusztig's q-analog of weight multiplicities to the Lie superalgebras $\mathfrak{gl}(n,m)$ and $\mathfrak{spo(}2n,M).$ We define such q-analogs K ?,μ (q) for the typical modules and for the irreducible covariant tensor $\mathfrak{gl}(n,m)$ -modules of highest weight ?. For $\mathfrak{gl}(n,m),$ the defined polynomials have nonnegative integer coefficients if the weight μ is dominant. For $\mathfrak{spo(}2n,M)$ , we show that the positivity property holds when μ is dominant and sufficiently far from a specific wall of the fundamental chamber. We also establish that the q-analog associated to an irreducible covariant tensor $\mathfrak{gl}(n,m)$ -module of highest weight ? and a dominant weight μ is the generating series of a simple statistic on the set of semistandard hook-tableaux of shape ? and weight μ. This statistic can be regarded as a super analog of the charge statistic defined by Lascoux and Schutzenberger.