Decomposition of Unipotents for E6 and E7: 25 Years After

Abstract

In this paper I sketch two new variations of the method of decomposition of unipotents in the microweight representations (E₆,ϖ₁) and (E₇,ϖ₇). To put them in the context, I first very briefly recall the two previous stages of the method, an A₅-proof for E₆ and an A₇-proof for E₇, first developed some 25 years ago by Alexei Stepanov, Eugene Plotkin, and myself (a definitive exposition was given in my paper “A third look at weight diagrams”) and an A₂-proof for E₆ and E₇ developed by Mikhail Gavrilovich and myself in early 2000. The first new twist outlined in this paper is an observation that the A₂-proof actually effectuates reduction to small parabolics, of corank 3 in E₆ and corank 5 in E₇. This allows one to revamp proofs and to sharpen existing bounds in many applications. The second new variation is a D₅-proof for E₆, based on stabilization of columns with one zero. [I devised also a similar D₆-proof for E₇, based on stabilization of columns with two adjacent zeros, but it is too abstruse to be included in a casual exposition.] Also, I list several further variations. Actual detailed calculations will appear in my paper “A closer look at weight diagrams of types (E₆,ϖ₁) and (E₇,ϖ₇).”