Flagged Littlewood-Richardson tableaux and branching rule for classical groups

We give a new formula for the branching rule from ${\rm GL}_n$ to ${\rm O}_n$ generalizing the Littlewood's restriction formula. The formula is given in terms of Littlewood-Richardson tableaux with certain flag conditions which vanish in a stable range. As an application, we give a combinatorial formula for the Lusztig $t$-weight multiplicity $K_{\mu 0}(t)$ of type $B_n$ and $D_n$ with highest weight $\mu$ and weight $0$.