RESTRICTIONS OF LARGE IRREDUCIBLE REPRESENTATIONS OF THE CLASSICAL GROUPS TO NATURALLY EMBEDDED SMALL SUBGROUPS CANNOT BE SEMISIMPLE

Abstract

It is proved that the restriction of a p-restricted representation of a classical algebraic group G of rank r in characteristic p > 0 to a naturally embedded semisimple subgroup cannot be completely reducible (semisimple) if the subgroup has a simple component of rank m small enough with respect to r and the highest weight is large enough with respect to p. It suffices to assume that r ≥ 2m and that the highest weight is equal to ∑ ^r _i=1 a_i ω _i with ∑ ^r _i=1 a_i ≥ 2p − 1 if p ≠ 2 or G ≠ C_r (K) and ∑ ^r _i=1 a_i ≥ 4 for p = 2 and G = C_r (K).

ACKNOWLEDGMENTS

This research has been supported by the Belarus Basic Research Foundation, project F98-180. The author would like to thank George McNinch for the information on some recent publications on the topic.