Quadratic unipotent blocks of finite general linear, unitary and symplectic groups
Seminar Room 1, Newton Institute
Let G be a finite reductive group. The (complex) irreducible characters of G were classified by Lusztig. They fall into Lusztig series E(G, (s)) where (s) is a semisimple class in a dual group G*. If a character is in E(G, (1)) it is a unipotent character. Let l be a prime different from the characteristic of the field of definition of G. A block of l-modular characters containing unipotent characters is a unipotent block.
Unipotent blocks of finite reductive groups have been intensely investigated by many authors over many years. In particular, correspondences such as perfect isometries (in the sense of Michel Broue) or derived equivalences have been established between unipotent blocks of two general linear groups, a finite reductive group and a "local" subgroup, and so on.
In this talk we will focus on recent work which shows that if we enlarge the set of unipotent blocks to the set of "quadratic unipotent blocks" for some choices of l some surprising results "across types" are obtained. In particular, correspondences are obtained between blocks of a unitary group and a symplectic group, and between blocks of a general linear group and a symplectic group. It would appear that quadratic unipotent blocks are a natural generalization of unipotent blocks in classical groups.