The t-analogue of string functions for the affine Kac-Moody algebras[HBNI Th54]

Abstract

The author studies Lusztig's t-analogue of weight multiplicities associated to the irreducible integrable highest weight modules of affine Kac-Moody algebras. First, for the level one representation of twisted affine Kac-Moody algebras, obtained an explicit closed form expression for the corresponding t-string function using constant term identities of Macdonald and Cherednik. The closed form involves the generalised exponents of the graded pieces of the twisted affine algebra, considered as modules for the underlying finite dimensional simple Lie algebra. This extends previous work on level 1 t-string functions for the untwisted simply-laced affine Kac-Moody algebras. Next, for the Lie algebra A(1) 1 , the author gives a basis for the weight spaces of its basic representation, which is compatible with the affine Brylinski-Kostant filtration defined by Slofstra. Using this basis, given an alternative derivation of the expression for the t-string function of the basic representation. Finally, obtained explicit formula for the t-string function of irreducible integrable highest weight A(1) 1 -modules of all levels. This is generalisation of a theorem of Kac and Peterson.