  We denote by A the ring of Laurent polynomials in the indeterminate v and by
K its field of fractions. In this paper, we are interested in representation
theory of the "generic" q-Schur algebra S_q(n,r) over A. We will associate to
every non-degenerate symmetrising trace form \tau on KS_q(n,r) a subalgebra
J_{\tau} of KS_q(n,r) which is isomorphic to the "asymptotic" algebra \J(n,r)_A
defined by J. Du. As a consequence, we give a new criterion for James'
conjecture.

Representation Theory