A formula with nonnegative terms for the degree of the dual variety of a homogeneous space

Let $G$ be a reductive group and $P$ a parabolic subgroup. For every $P$-regular dominant weight $\lambda$ let $X(\lambda )$ denote the variety $G/P$ embedded in the projective space by the embedding corresponding to the ample line bundle $\mathcal L(\lambda )$. Writing $\lambda =\rho _P+\sum _{i=1}^n m'_i\omega _i$, we prove that the degree $d(\lambda )^\vee$ of the dual variety to $X(\lambda )$ is a polynomial with nonnegative coefficients in $m'_1,\dots , m'_n$. In the case of homogeneous spaces $G/B$ we find an expression for the constant term of this polynomial.