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Lusztig's q-analogs of weight multiplicities for semisimple Lie algebras are polynomials (in q) with positive coefficients which evaluate to the weight multiplicities at q = 1; they are related to a filtration of the weight spaces, and arise in several other contexts too. I will present a natural generalization of Lusztig's q-analog to certain irreducible representations of the general linear and orthosymplectic superalgebras. It turns out that the positivity property of Lusztig's q-analog extends to all the considered representations of the general linear superalgebra. In the case of the orthosymplectic one, the positivity holds when the weight space is, in a certain sense, "generic". A combinatorial formula for the q-weight multiplicities of the general linear superalgebra is also presented. This talk is based on joint work with C. Lecouvey. It will be mostly self-contained, and does not require any background on Lie superalgebras.
Refreshments at 3:45 in the Math Lounge.