On Lusztig's q-analogues of all weight multiplicities of a representation

2014/06/05 by Dmitri I. Panyushev, Panyushev, Dmitri I.
Mathematics · #17B10 #17B20 #20G10 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B10 #msc:17B20 #msc:20G10

paper · pdf · doi:10.48550/arxiv.1406.1453

15 pages

arxiv created 2014/06/05 · arxiv updated 2014/06/06

Abstract

Let \mathfrak g be a complex semisimple Lie algebra. We obtain new properties of the q-analogue of weight multiplicities in finite-dimensional representations of \mathfrak g. In particular, it is proved that certain weighted sum of q-analogues of all weights of a representation V equals the q-analogue of the zero weight multiplicity in the reducible representation V⊗ V^*. This also provides another formula for the \mathbb Z[q]-valued symmetric bilinear form on the character ring of \mathfrak g that was introduced by R.Gupta (Brylinski) in 1987.

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