A note on exponents vs root heights for complex simple Lie algebras

We give an elementary combinatorial proof of a special case of a result due to Bazlov and Ion concerning the Fourier coefficients of the Cherednik kernel. This can be used to give yet another proof of the classical fact that for a complex simple Lie algebra, the partition formed by its exponents is dual to that formed by the numbers of positive roots at each height.Comment: 5 page

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

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 $V\otimes V^*$. 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.Comment: 15 page

A note on exponents vs root heights for complex simple Lie algebras

We give an elementary combinatorial proof of a special case of a result due to Bazlov and Ion concerning the Fourier coefficients of the Cherednik kernel. This can be used to give yet another proof of the classical fact that for a complex simple Lie algebra, the partition formed by its exponents is dual to that formed by the numbers of positive roots at each height