Decomposition of unipotents for $\mathrm E_6$ and $\mathrm E_7$: 25~years after

N. A. Vavilov

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 27, Tome 430 (2014), pp. 32-52

Résumé

In this paper I sketch two new variations of the method of decomposition of unipotents in the microweight representations $(\mathrm E_6,\varpi_1)$ and $(\mathrm E_7,\varpi_7)$. To put them in context, I first very briefly recall the two previous stages of the method, an $\mathrm A_5$-proof for $\mathrm E_6$ and an $\mathrm A_7$-proof for $\mathrm E_7$, first developed some 25 years ago by Alexei Stepanov, Eugene Plotkin and myself (a definitive exposition was given in my paper “A thirdlook at weight diagrams”), and an $\mathrm A_2$-proof for $\mathrm E_6$ and $\mathrm E_7$ developed by Mikhail Gavrilovich and myself in early 2000. The first new twist outlined in this paper is an observation that the $\mathrm A_2$-proof actually effectuates reduction to small parabolics, of corank 3 in $\mathrm E_6$ and of corank 5 in $\mathrm E_7$. This allows to revamp proofs and sharpen existing bounds in many applications. The second new variation is a $\mathrm D_5$-proof for $\mathrm E_6$, based on stabilisation of columns with one zero. [I devised also a similar $\mathrm D_6$-proof for $\mathrm E_7$, based on stabilisation of columns with two adjacent zeroes, 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 $(\mathrm E_6,\varpi_1)$ and $(\mathrm E_7,\varpi_7)$".