On the behaviour of small quadratic elements in representations of the special linear group with large highest weights

M. V. Velichko   ; I. D. Suprunenko

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 15, Tome 343 (2007), pp. 84-120

Résumé

For almost all $p$-restricted irreducible representations of the groups $A_n(K)$ in characteristic $p>0$ with highest weights large with respect to $p$ the Jordan block structure of images of small quadratic unipotent elements in these representations is determined. It is proved that if $\varphi$ is an irreducible $p$-restricted representation of $A_n(K)$ in characteristic $p>0$ with highest weight $$ m_1\omega_1+\ldots+m_n\omega_n, \quad \sum_{i=1}^n m_i\ge p-1, $$ not too few of the coefficients $m_i$ are less than $p-1$ and $n$ is large enough with respect to the codimension of the fixed subspace of an element $z$ under consideration, then $\varphi(z)$ has blocks of all sizes from 1 to $p$.

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