Endomorphismalgebras and Representation Theory

Endomorphism algebras gure prominently in group representation theory. F or example, if G is a nite group of Lie type, the representation theory of the endomorphism algebra End G Cj G B |sometimes known as the Hecke algebra over C of G|plays a central role in unraveling the complex unipotent c haracters on G 22, 77. Another example arises in the modular representation theory of the nite general linear group G = GL n q o ver an algebraically closed eld k of characteristic p not dividing q. In this so-called non-describing characteristic representation theory, the Hecke algebras H over k associated with symmetric groups provide a link between the representation theory of G and that of quantum groups. In fact, if T denotes the direct sum of the various transitive" q-permutation modules for H, then the endomorphism algebras End H T are Morita equivalent t o q-Schur algebras over k. I n w ork by Dipper and James see 100, 111, 88 the decomposition numbers for kG are proved to be completely determined by the decomposition numbers for certain of these quantized Schur algebras". In turn, the representation theory of these latter algebras relates closely to that of the quantum linear group GL n;q k o ver k. Several years ago, the authors began a general homological investigation of endomorphism algebras 44. Motivation came from several sources, including the theory of Schur algebras, but more particularly work of Dlab, Heath, and Marko 122 on quasi-hereditary endomorphism algebras as well as Soergel's work 322. The latter realizes the principal block O triv for the category O of a complex semisimple Lie algebra g as the module category for