Intersection Cohomology Methods in Representation Theory*

In recent years, the theory of group representations has greatly benefited from a new approach provided by the topology of singular spaces, namely intersection cohomology (IC ) theory. Let G be a connected reductive algebraic group over an algebraically closed field (which will be assumed to be C unless otherwise specified) ; although G itself is clearly a non-singular variety, the study of the representations of G (or its Lie algebra, or its forms over various fields) leads to singular varieties connected with G. The main theme of this article is that the invariants of IC theory, applied to the singular varieties arising from G appear again and again in a broad range of problems of representation theory: construction of representations, computation of their character, construction of nice bases for representations. Chronologically, both the earliest [22, 23] and the most recent [45] such problem was that of constructing canonical bases in certain representations of Iwahori-Hecke algebras (resp. in irreducible, finite dimensional representations of G). The relevant singularities were Schubert singularities (resp. quiver singularities) ; their local IC provided the necessary ingredients. Certain misterious and apparently intractable quantities in representation theory, like the multiplicities in the Jordan-Holder series of Verma modules, were recognized to be expressible in terms of certain local IC spaces and this has led to understanding and computing them. IC methods are needed for the classification of complex irreducible representations of a reductive group over a finite field; furthermore, the character values of such representations appear to be intimately related to certain IC complexes, the character sheaves. IC methods have been used to give a new construction of the Springer representations of Weyl groups; an elaboration of this idea has been used to construct representations of affine Iwahori-Hecke algebras and hence of p-adic groups.