  Wright-Fisher diffusions describe the evolution of the type composition of an
infinite haploid population with two types, type $0$ and type $1$, subject to
neutral reproductions, selection, and sometimes mutations. In the present paper
we study a Wright-Fisher diffusion in a L\'evy environment that gives a
selective advantage to both types. Classical methods using the Ancestral
Selection Graph (ASG) fail in the study of this model because of the complexity
of the combinatorics of the ASG, resulting from the two-sided selection. We
propose a new method that consists in encoding the relevant information of the
ASG into a function. We show that the expectations of the coefficients of this
function form a (non-stochastic) semi-group and deduce that they satisfy a
linear system of differential equations. As a result we obtain a series
representation for the fixation probability $h(x)$ (where $x$ is the initial
proportion of individuals of type $0$ in the population) as a infinite sum of
polynomials whose coefficients satisfy explicit linear relations. Our approach
then allows to derive Taylor expansions at any order for $h(x)$ near $x=0$ and
to obtain an explicit recursive formula for the coefficients.
