On the Jordan block structure of a~product of long and short root elements in irreducible representations of algebraic groups of type~$B_r$

T. S. Busel

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 27, Tome 430 (2014), pp. 18-31

Résumé

The behaviour 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 $a_1\omega_1+\dots+a_r\omega_r$ this fact is proved when $a_j\neq p-1$ for some $j$ and one of the following holds: 1) $a_r\neq p-1$ and $\sum_{i=1}^{r-2}a_i\geq p-1$; 2) $2a_{r-1}+a_r$, $\sum_{i=1}^{r-3}a_i\neq0$ for $2a_{r-1}+a_r=p-2$ or $p-1$ and $\sum_{i=1}^{r-3}a_i\neq0$ or $(r-3)(p-1)$ for $a_r=p-1$.