On the Jordan Block Structure of a Product of Long and Short Root Elements in Irreducible Representations of Algebraic Groups of Type B r

Abstract

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 a1ω1 +· · ·+arωr, this fact is proved when aj ≠ p − 1 for some j < r − 1 and one of the following conditions holds: (1) $$ {a}_r\ne p-1\kern0.75em and\kern0.5em {\displaystyle \sum_{i=1}^{r-2}{a}_i\ge p-1;} $$ and (2) $$ 2{a}_{r-1}+{a}_r<p,{\displaystyle \sum_{i=1}^{r-2}r-3}{a}_i\ne 0\;for\;2{a}_{r-1}+{a}_r=p-2\; or\;p-1\; and\;{\displaystyle \sum_{i=1}^{r-3}{a}_i\ne 0} $$ or (r−3) (p-1) for a r = p−1.