The finite dimensional irreducible representations of a semisimple algebraic group $G$ are parametrized by their highest weights (relative to some choice of positive roots in the root system $R$ for $G$). Denote by $L(\lambda)$ the irreducible representation with dominants highest weight $\lambda$ and let $[L{\lambda}]$ denote its formal character. In this talk we want to discuss the problem on how to determine these characters.
One way to construct $L(\lambda)$ is to realize it as the unique simple $G$-module inside the global sections $H^0(\lambda)$ of the line bundle on the flag manifold for $G$ induced by $\lambda$. The character $[H^0(\lambda)]$ is well known to be given by Weyl's character formula and in characteristic zero this solves our problem because in this case we have $L(\lambda)=H^0(\lambda)$. In positive characteristics we write $$[L(\lambda)]=\Sigma_\mu a_{\lambda,\mu}[H^0(\mu)]$$ where $a_{\lambda,\mu}$ are integers. Our problem is then to compute these integers.
In 1979 G. Lusztig formulated a conjecture (in the following referred to as the modular conjecture) which gave an algorithm for finding the numbers $a_{\lambda,\mu})$ (as valued at 1 of the socalled Kazhdan-Lusztig polynomials attached to the affine Weyl group associated with $G$). Later he proposed analogous conjectures in the following two cases
Very recently M. Kashiwara and and T. Tanisaki have announced a proof of step one. Step two has been taken by D. Kazhdan and G. Lusztig whereas J. C. Jantzen, W. Soergel and the speaker have made progress on the third step. For instance we have shown that the conjecture in case (b) holds for all roots of unity of order at least equal to the Coxeter number iff the modular conjecture holds in all large characteristics.
In this talk we shall give an account of this program with special emphasis on the relations between the modular and the quantum cases.