In his notes on K-theory, A. Okounkov states the following exercise:
The group $\operatorname{GL}(2)$ acts naturally on $\mathbb{P}^1$ and on line bundles $\mathcal{O}(k)$ over it. Push forward these line bundles under $\pi\colon\mathbb{P}^1 \to \text{ pt}$ using an explicit $T$-invariant ($T$ is supposed to be the maximal torus in $\operatorname{GL}(2)$) Čech covering of $\mathbb{P}^1$.
The pushforward is defined as:
$$\pi_*\mathcal{O}(k) = \sum_i (-1)^i [R^i\pi_*\mathcal{O}(k)].$$
If $\mathbb{P}^1$ is given by the coordinates $x_0,x_1$, the Čech covering $$U_1 = \{y_1 = x_1/x_0 \mid x_0\neq 0\}, U_2 = \{z_0 = x_0/x_1 \mid x_1\neq 0\}$$ is $T$-invariant.
By Theorem II.8.5 in Hartshorne,
$$R^i\pi_*\mathcal{O}(k) = H^i(\mathbb{P}^1,\mathcal{O}(k))^\sim.$$
How do I conclude and where does the action come into play?