A generic algebra associated to certain Hecke algebras

We initiate the systematic study of endomorphism algebras of permutation modules and show they are obtainable by a descent from a certain generic algebra, infinite-dimensional in general, coming from the universal enveloping algebra of (or ). The endomorphism algebras and the generic algebras are cellular (in the latter case, of profinite type in the sense of R.M. Green). We give several equivalent descriptions of these algebras, find a number of explicit bases for them, and describe indexing sets for their irreducible representations. Moreover, we show that the generic algebra embeds densely in an endomorphism algebra of a certain infinite-dimensional induced module.