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Journal of the London Mathematical Society 1987 s2-36(1):68-76; doi:10.1112/jlms/s2-36.1.68
© 1987 by London Mathematical Society
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© Oxford University Press

Characters and the q-Analog of Weight Multiplicity

R. K. Gupta

Brown University, Department of Mathematics , Providence, Rhode Island 02912, USA

Let g be a complex semisimple Lie algebra and {Lambda} its character ring of finite-dimensional representations. We prove the basic orthogonality relation in {Lambda} extended by a formal indeterminate q. Our point of view is new even in the case when g = sln as we work with some ‘infinite’ characters in {Lambda}[[q]] as well as those from {Lambda}[q]. Also we use the natural inner product for which the irreducible representations V{lambda} form an orthonormal basis, rather than the usual one in which the Hall-Littlewood symmetric functions are orthogonal.

We are then able to give new, elementary proofs of two facts dealing with the q-analog of weight multiplicity; both were proven by Kato using deep machinery from the theory of spherical analysis on p-adic groups. The first fact, a conjecture of Lusztig, computes the inverse of the infinite upper triangular q-weight multiplicity matrix. We also give a new identity relating (the q-analogs of) general weight multiplicity and zero weight multiplicity.

Our work here has two applications to the study of generalized exponents for sln and to Kostka-Foulkes polynomials, which will be developed in later papers.


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