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

摘要

The paper is devoted to the generalization of Lusztig’s q-analog of weight multiplicities to the Lie superalgebras mathfrakgl(n,m)mathfrak{gl}(n,m) and mathfrakspo(2n,M).mathfrak{spo(}2n,M). We define such q-analogs K _λ,μ (q) for the typical modules and for the irreducible covariant tensor mathfrakgl(n,m)mathfrak{gl}(n,m) -modules of highest weight λ. For mathfrakgl(n,m),mathfrak{gl}(n,m), the defined polynomials have nonnegative integer coefficients if the weight μ is dominant. For mathfrakspo(2n,M)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 mathfrakgl(n,m)mathfrak{gl}(n,m) -module of highest weight λ and a dominant weight μ is the generating series of a simple statistic on the set of semistandard hook-tableaux of shape λ and weight μ. This statistic can be regarded as a super analog of the charge statistic defined by Lascoux and Schützenberger.