The common ancestor type distribution of a $Lambda$-Wright-Fisher process with selection and mutation

Using graphical methods based on a `lookdown and pruned version of the {em ancestral selection graph}, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population. This extends results from Lenz, Kluth, Baake, and Wakolbinger (Theor. Pop. Biol., 103 (2015), 27-37) to the case of heavy-tailed offspring, directed by a reproduction measure $Lambda$. The representation is in terms of the equilibrium tail probabilities of the line-counting process $L$ of the graph. We identify a strong pathwise Siegmund dual of $L$, and characterise the equilibrium tail probabilities of $L$ in terms of hitting probabilities of the dual process.