Frobenius Structures on Tame Hecke Eigensheaves

We construct nonzero perverse Weil Hecke eigensheaves for SL${}_n$ with Borel level at one marked point, after a finite extension of the field of constants. The parameter is a geometrically dense arithmetic PGL${}_n$-local system with tame regular-unipotent monodromy on a once-punctured curve of genus at least two over a finite field of characteristic p > n. The full multi-leg eigenstructure is Frobenius compatible with the prescribed arithmetic eigenvalue, and the geometric sheaf has parabolic nilpotent singular support.