Unipotent elements from subsystem subgroups of type \(A_2\)
in representations of the special linear group
A. Osinovskaya
Let \(K\) be an algebraically closed field of odd characteristic \(p\), let \(G\) be a simply connected algebraic group of type \(A_r\) over \(K\), \(r\geq 3\), and let \(\omega_i\), \(1\leq i\leq r\), be the fundamental weights with respect to a fixed maximal torus \(T\subset G\) and a choice of a Borel subgroup \(B\supset T\).
A rational irreducible representation of \(G\) with the highest weight \(\sum^r_{i=1}a_i\omega_i\) is called \(p\)-restricted if and only if all \(a_i < p\). A subsystem subgroup of a semisimple algebraic group is a subgroup generated by root subgroups associated with all roots from a certain subsystem of its root system.
Suppose that \(H\subset G\) is a subsystem subgroup of type \(A_2\), \(x\in H\) is a regular unipotent element, \(\varphi\) is a rational irreducible \(p\)-restricted representation of \(G\), \(\omega=\sum^r_{k=1}a_k\omega_k\) is the highest weight of \(\varphi\), \(\omega^*\) is the highest weight of the representation dual to \(\varphi\), \({\rm J}_{\varphi}(x)\) is the set of Jordan block sizes for \(\varphi(x)\) (without their multiplicities), \(s(\varphi)=\min_{1\leq i\leq r-1}(2a_i+2a_{i+1})\), \(m(\varphi)=\min(\sum^r_{k=1}2a_k+1,p)\), \(\mathbb{N}\) is the set of nonnegative integers.
Theorem 1.
Let \(G\), \(x\), and \(\varphi\) be the same as above and \(s(\varphi) < p\). Then
\[
\{k\in\mathbb{N} \mid 1\leq k\leq m(\varphi), k\equiv m(\varphi)({\rm mod}\ 2) \}\subset{\rm J}_{\varphi}(x),
\]
except in the following cases:
\((i)\)
\(r=3\), \(\omega=a_2\omega_2\) and \(a_2\) is odd;
\((ii)\)
\(r=3\), \(\omega\) or \(\omega^*=a_1\omega_1+a_2\omega_2+a_3\omega_3\), \(2a_1+2a_2 < p\), \(a_2+a_3=p-1\), and \(a_1\) is odd;
\((iii)\)
\(p=3\) and \(s(\varphi)=2\).
In such cases,
\[
\{k\in\mathbb{N} \mid 3\leq k\leq m(\varphi), k\equiv m(\varphi)({\rm mod}\ 2) \}\subset{\rm J}_{\varphi}(x)
\]
and, in the case \((i)\),
\[
1\notin{\rm J}_{\varphi}(x).
\]
This result is a part of the programme of investigating the behavior of unipotent elements in modular representations of classical algebraic groups.
Acknowledgement. The research is supported by the Belarusian Republican Foundation for Fundamental Research, project F12R-050.
