We give a combinatorial formula for the weight multiplicities of some infinite-dimensional highest weight ${\mathfrak g}$${\mathfrak l}$ (n)-modules. Our proof, which does not rely on Kazhdan–Lusztig combinatorics, uses a reduction to finite characteristics. The character formula for the corresponding modular representations, which has been computed in a 1997 preprint by the authors, is based on a dual pair which has no obvious counterpart in characteristic zero.