  In this paper we investigate the image of the center $Z$ of the distribution
algebra $Dist(GL(m|n))$ of the general linear supergroup over a ground field of
positive characteristic under the Harish-Chandra morphism $h:Z \to Dist(T)$
obtained by the restriction of the natural map $Dist(GL(m|n))\to Dist(T)$. We
define supersymmetric elements in $Dist(T)$ and show that each image $h(c)$ for
$c\in Z$ is supersymmetric. The central part of the paper is devoted to a
description of a minimal set of generators of the algebra of supersymmetric
elements over Frobenius kernels $T_r$.
