A Littlewood-Richardson rule for the MacDonald inner product and bimodules over wreath products

We prove a Littlewood-Richardson type formula for $(s_{\lambda \mathrm{/}\mu },s_{\nu \mathrm{/}\kappa }{)}_{t^k,t}$, the pairing of two skew Schur functions in the MacDonald inner product at $q=t^k$ for positive integers $k$. This pairing counts graded decomposition numbers in the representation theory of wreath products of the algebra $C[x]\mathrm{/}{x}^k$ and symmetric groups.