Eta Products and Lattice Points in Simplices.

In Sect. 2.5 we obtained a bijection between the holomorphic eta products of level N and the lattice points in a closed simplicial cone ]> in ℝ^τ(N). Since η(mz)^2k is a cuspidal eta product of level N and weight k for every m|N and every (integral or half-integral) k>0, the half lines from the origin through the standard unit vectors belong to the interior of ]> . Therefore, the first octant {X=(x_m)_mϵℝ^τ(N)ǀX≠0, x_m≥0 for all m|N} belongs to the interior of ]> . [ABSTRACT FROM AUTHOR]