Existence of non-zero $G$-invariant global section of a vector bundle

Question

Let $f:Y\to X$ be a Galois, etale map between smooth quasi-projective varieties with (finite) Galois group $G$. Let $\mathcal{E}$ be a vector bundle on $X$ and let $\mathcal{F}=f^*\mathcal{E}$ on $Y$. Then one can make the identification $H^0(X,\mathcal{E})=H^0(Y,\mathcal{F})^G$. My question is the following: If we know that $H^0(Y,\mathcal{F})\neq0$, then when can we say that $H^0(X,\mathcal{F})^G=H^0(X,\mathcal{E})\neq0$? I know that this is not true in general: one counterexample I was told is: X an elliptic curve with $\mathcal{E}$ a torsion line bundle on $X$, and $Y$ a cover of $X$ that trivializes $\mathcal{E}$. Then $f^*\mathcal{E}$ has global sections but $\mathcal{E}$ doesn't. However, are there some extra assumptions that would guarantee descent of non-zero global sections? I specifically care about the case $\mathcal{E}=\textrm{Sym}^i\Omega^1_X$ but a more general treatment would be great. Thanks in advance!

Answer 1

I will assume that $X$ is a scheme over a field of characteristic prime to the order of $G$.

Let $R_0,R_1,\dots,R_k$ be all irreducible representations of $G$, and assume that $R_0$ is the trivial representation. Clearly, the sheaf $f_\ast\mathcal{O}_{Y}$ has the structure of a representation of $G$ (which is the regular representation pointwise), hence there is a direct sum decomposition $$ f_\ast\mathcal{O}_{Y} \cong \bigoplus_{i=0}^k R_i \otimes \mathcal{V}_i, $$ where $\mathcal{V}_i$ is a vector bundle on $X$ of rank equal to $\dim(R_i)$ (with $\mathcal{V}_0 = \mathcal{O}_X$). Consequently, $$ f_\ast\mathcal{F} \cong f_\ast f^\ast\mathcal{E} \cong \mathcal{E} \otimes f_\ast\mathcal{O}_{Y} \cong \bigoplus_{i=0}^k R_i \otimes (\mathcal{E} \otimes \mathcal{V}_i), $$ and therefore $$ H^0(Y,\mathcal{F}) = \bigoplus_{i=0}^k R_i \otimes H^0(X, \mathcal{E} \otimes \mathcal{V}_i). $$ This shows that a global section of $\mathcal{F}$ can come from a global section of either of the vector bundles $\mathcal{E} \otimes \mathcal{V}_i$, so if you want to ensure that it comes from the summand with $i = 0$, it is enough to check that the other summands have no global sections.