Seminar Algebra and Geometry: Donna Testerman (EPFL, Lausanne)

Double centralizers of unipotent elements in simple algebraic groups.

In joint work with Ross Lawther, and building on work of Proud and Seitz, we considered properties of C_G(C_G(u)) =Z(C_G(u)), where G is a simple algebraic group defined over an algebraically closed field of good positive characteristic or of characteristic 0, and u is a unipotent element. In particular, we determined the dimension of the double centralizer.In subsequent work, Simion considered the case of bad characteristic, and determined the dimension for G of exceptional type.We discuss the above work, describe the results and in particular, indicate the differences between good and bad characteristic. Furthermore, in the case where the characteristic is good or 0 and u is an even element (that is, the associated cocharacter has even weights on Lie(G)), there is a particularly nice formula for this dimension. We indicate a recent case-free proof of one of the inequalities implied by this formula, obtained by myself in characteristic 0 and in collaboration with McNinch in positive characteristic.

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