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Let $X_\sigma$ be an affine toric variety for an action of a torus $T$ and let $\mathcal{P}$ be a toric principal $G$-bundle over $X_\sigma$ where $G$ is an affine algebraic group (here base field $k$ is algebraically closed and characteristic $0$, in fact I am content with $k = \mathbb{C}$). By a toric principal $G$-bundle $\mathcal{P}$ we mean $T$ acts on $\mathcal{P}$ lifting its action on $X_\sigma$ and the $T$-action and $G$-action on $\mathcal{P}$ commute.

My question is wether $\mathcal{P}$ is $T$-equivariantly trivial. By this, we mean the following: there exists a homomorphism $\phi: T \to G$ such that $\mathcal{P}$ is $T$-equivariantly isomorphic to $X_\sigma \times G$ where $T$ acts on this product diagonally by acting on $G$ via $\phi$ and left multiplication, that is, $t \cdot (x, g) = (t \cdot x, \phi(x)g)$.

It is well-known and not difficult to show that this is true when $G = GL(n)$ (that is, one has a toric vector bundle). As far as I know by a result of Biswas-Dey-Poddar (2015) which itself relies on an earlier result of Heinzner-Kutzschebauch (1995), this is true for when $X_\sigma$ is complex affine space. I was wondering if this is known for other affine toric varieties.

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  • $\begingroup$ I am confused. If $X=\mathbf{A}^1$, then there is a $\mathbf{Z}$ worth of $\mathbf{G}_m$-equivariant structures on the line bundle $\mathcal{O}_X$. Doesn't this contradict the statement you want for $G=GL(1)$? $\endgroup$ – Piotr Achinger Apr 18 '18 at 11:34
  • $\begingroup$ @PiotrAchinger This agrees with what I am saying: each integer in $n \in \mathbb{Z}$ gives the trivial line bundle $\mathcal{O}_X$ a $\mathbb{G}_m$-action, that is $t \in \mathbb{G}_m$ acts on $(x, z) \in \mathcal{O}_X = X \times \mathbb{A}^1$ by $(tx, t^n z)$. So each choice of $n \in \mathbb{Z}$ gives a toric line bundle structure to $\mathcal{O}_X$ (or a $T$-linearization if you like). $\endgroup$ – Kiu Apr 18 '18 at 19:27
  • $\begingroup$ Ok, I interpreted "$T$-equivariantly trivial" as "trivial as an equivariant bundle". $\endgroup$ – Piotr Achinger Apr 19 '18 at 5:25

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