  We know from Letellier that if for some triple of partitions the
corresponding Kronecker coefficient is non-zero then the corresponding
multiplicities for unipotent characters of GL(n,q) is also non-zero. A
conjecture of Saxl says that the tensor square of an irreducible character of
the symmetric group corresponding to a staircase partition contains all the
irreducible characters. Therefore Saxl conjecture implies its analogue for
unipotent characters. In this paper we prove the analogue of Saxl conjecture
for unipotent characters and we describe conjecturally the set of all
partitions for which the tensor square of the associated unipotent character
contains all the unipotent characters.
