DECOMPOSITION OF UNIPOTENTS FOR E_6 AND E_7: 25 YEARS AFTER

摘要

In this paper I sketch two new variations of the method of decomposition of unipotents in the microweight representations (E_6, ω_1) and (E_7, ω_7). To put them in the context, I first very briefly recall the two previous stages of the method, an A_5-proof for E_6 and an A_7-proof for E_7, 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 A_2-proof for E_6 and E_7 developed by Mikhail Gavrilovich and myself in early 2000. The first new twist outlined in this paper is an observation that the A_2-proof actually effectuates reduction to small parabolics, of corank 3 in E_6 and corank 5 in E_7. This allows one to revamp proofs and to sharpen existing bounds in many applications. The second new variation is a D_5-proof for E_6, based on stabilization of columns with one zero. [I devised also a similar D_6-proof for E_7, 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 (E_5,ω_1) and (E_7,ω_7)." Bibliography: 45 titles.