# stability of parabolic bundles beyond the Weyl alcove and Hecke transforms

When considering (quasi)parabolic vector bundles over a smooth complete curve $X$ with marked points $p_i$, it goes back to the foundational work of Seshadri and collaborators that one needs to specify certain parabolic weights, associated to each of the marked points, to get a decent moduli space.

The weights live in the Weyl alcove of GL(n). The parabolic weights plus the degree of the bundle form the parabolic degree, which is the key ingredient in the parabolic version of slope-stability: a parabolic bundle $E$ is (semi-)stable if for every subbundle $F$ of $E$ (which inherits a canonical parabolic structure itself), one has $$\frac{pdeg(F)}{rk(F)}\leq \frac{pdeg(E)}{rk(E)}.$$ Seshadri and Mehta showed that there exists a moduli space of semi-stable parabolic bundles.

A modern point of view on this (expounded by Constantin Teleman) is to take the stack of all quasi-parabolic bundles (without any weights assigned) - say with fixed determinant to make life easier. The Picard group of this stack consists of one copy of $\mathbb{Z}$ and, for each marked point, a copy of the Picard for the corresponding flag-variety. With a choice of (rational) parabolic weights one can now associate a line bundle over this stack, and the sub-stack of semi-stable parabolic bundles for this choice of weights corresponds to the complement of the base-locus for large powers of this line bundle.

The parabolic weights are also crucial in the correspondence with representations of $\pi_1{\large(}X \setminus \{p_i\}{\large)}$, where they determine the conjugacy class of the monodromy around the marked points.

My question concerns the restriction on these parabolic weights to live in the Weyl alcove. As long as the weights only move around in the Weyl alcove, the variation of the moduli space is well-studied (by Seshadri, Thaddeus, Boden-Hu), but the literature doesn't seem to mention what happens when leaving the Weyl alcove.

It appears to be folklore-knowledge however that one can shift the parabolic weights by applying a Hecke transform to the bundle, using its parabolic structure. These Hecke transforms are also (quasi)parabolic bundles with the same rank and flag types but a different degree, they should be (semi-)stable for a shifted choice of parabolic weights if and only if the original bundle was stable for the original weights (hence reducing the problem to one with weights in the alcove).

My question is: is this written down anywhere, and if not, how does it go exactly? In particular, how does it look for parabolic $SL(n)$ bundles - can one obtain a version of this story only involving bundles with trivial determinant?

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