Abstract
In this paper, we study Jacobi forms of half-integral index for any even integral positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A 1=〈2〉). We construct Jacobi forms of singular (respectively, critical) weight in all dimensions n≥8 (respectively, n≥9). We give the Jacobi lifting for Jacobi forms of half-integral indices and we obtain an additive lifting construction of new reflective modular forms which are natural generalizations to O(2,n) (n=4, 5 and 6) of the Igusa modular form Δ 5.
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