Decomposition of Unipotents for E6 and E7: 25 Years After

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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).”