On the imaginary simple roots of the Borcherds algebra g II 9,1

In a recent paper [O. Barwald, R.W. Gebert, M. Gunaydin and H. Nicolai, preprint KCL-MTH-97-22, IASSNS-HEP-97/20, PSU-TH-178, AEI-029, hep-th/9703084, to appear in Commun. Math. Phys.] it was conjectured that the imaginary simple roots of the Borcherds algebra ɡII9,1 at level 1 are its only ones. We here propose an independent test of this conjecture, establishing its validity for all roots of norm ⩾ − 8. However, the conjecture fails for roots of norm −10 and beyond, as we show by computing the simple multiplicities down to norm −24, which turn out to be remarkably small in comparison with the corresponding E10 multiplicities. Our derivation is based on a modified denominator formula combining the denominator formulas for E10 and ɡII9,1, and provides an efficient method for determining the imaginary simple roots. In addition, we compute the E10 multiplicities of all roots up to height 231, including levels up to l = 6 and norms −42.