Frobenius pushforward of an equivariant tautological bundle on the flag variety Let $m,n$ be nonnegative integers. Let $k$ be a field of positive characteristic. Let the vector space $V$ over the field $k$ have basis $e_1,...,e_{m+n}, f_1,..., f_n$. Let $k^*$ act on $V$ by $t \cdot e_i = t^{-2i} e_i$, $t \cdot f_i = t^{-2i} f_i$. This action of $k^*$ on $V$ induces an action on the complete flag variety $X=Fl(V)$. Let $Fr: X \to X^{(1)}$ be the Frobenius morphism. Let ${\mathcal O}(\lambda)$ be the standard line bundle on $X$ corresponding to a weight $\lambda$ of $\mathfrak{sl}_{m+2n}$. Non-equivaraintly, it is known that $Fr_*(O(\lambda)) \cong O(\lambda)^{\oplus p^{\rm dim(X)}}$. What is $Fr_*(O(\lambda))$ in the equivariant setting, where the actions of $k^*$ on $X$ and $X^{(1)}$ are induced by the above action of $k^*$ on $V$? In particular, can $Fr_*(O(\lambda))$ be expressed in terms of the line bundles ${\mathcal O}(\mu)$ on $X^{(1)}$?
 A: EDIT. Corrected the statement ($\sigma$ should be $p-1$ times what I wrote) and answered the question in the comment.
In general, the push-forward of a line bundle on the flag variety $G/B$ will not be the direct sum of line bundles. (This holds only on toric varieties.)
However, Haboush [1] (and independently Andersen) has proved that this is true for some special values of $\lambda$. More specifically, he proved that for $\lambda = \sigma + p\mu$ where $\sigma$ is $(p-1)/2$ times the sum of the positive roots (which is $(p-1)$ times the sum of the fundamental weights), one has an equivariant isomorphism
$$ F_* \mathcal{O}(\lambda) \simeq V\otimes \mathcal{O}(\mu) $$
where $V$ is the Steinberg module, irreducible of highest weight $\sigma$. In particular, $F_*\mathcal{O}(\sigma)$ is a trivial vector bundle $V\otimes \mathcal{O}_{G/B}$ corresponding to the Steinberg representation.
Notice that the statement follows from the case $\mu=0$ by the projection formula and the fact that $F^* L = L^p$ for a line bundle $L$:
$$ F_* \mathcal{O}(\sigma + p\mu) = F_* (\mathcal{O}(\sigma)\otimes F^* \mathcal{O}(\mu)) = (F_* \mathcal{O}(\sigma))\otimes \mathcal{O}(\mu) = V\otimes \mathcal{O}(\mu). $$
Now, to answer the question in the comment: in the reference, they are interested in $F_* \mathcal{O}(-\rho)$ where $\rho$ is the sum of the fundamental weights. Since $\sigma = (p-1)\rho = -\rho + p\rho$, we obtain with $\lambda = -\rho = \mu$:
$$ F_* \mathcal{O}(-\rho) \simeq V\otimes \mathcal{O}(-\rho). $$
This isomorphism (saying that $\mathcal{O}(-\rho)$ is an "eigenvector" for $F_*$ with "eigenvalue" $V$) is employed in Samokhin's paper [2], which you may find useful.
[1]  W. J. Haboush, A short proof of the Kempf vanishing theorem, Inventiones mathematicae volume 56, pages 109–112 (1980)
[2] https://arxiv.org/abs/1611.10320
