ON THE JORDAN BLOCK STRUCTURE OF A PRODUCT OF LONG AND SHORT ROOT ELEMENTS IN IRREDUCIBLE REPRESENTATIONS OF ALGEBRAIC GROUPS OF TYPE B_r

摘要

The behavior of a product of commuting long and short root elements of the group of type B_r in p-restricted irreducible representations is investigated. For such representations with certain local properties of highest weights, it is shown that the images of these elements have Jordan blocks of all a priori possible sizes. For a p-restricted representation with highest weight a_1ω_1+ …+ a_rω_r, this fact is proved when a_j≠ p - 1 for some j ＜ r - 1 and one of the following conditions holds: (1)a_r≠p-1 and r-2∑i=1a_i≥p-1; (2)2a_(r-1)+a_r＜p, r-3∑i=1a_i≠0 for 2a_(r-1)+a_r=p-2 or p-1 and r-3∑i=1 a_i≠0 or (r - 3) (p - 1) for a_r = p - 1. Bibliography: 19 titles.