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 $X^{(1)}$?

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 $X^{(1)}$?