  Wright-Fisher diffusions describe the evolution of the type composition of an
infinite haploid population with two types (say type $0$ and type $1$) subject
to neutral reproductions, and possibly selection and mutations. In the present
paper we study a Wright-Fisher diffusion in a L\'evy environment that gives a
selective advantage to sometimes one type, sometimes the other. Classical
methods using the Ancestral Selection Graph (ASG) fail in the study of this
model because of the complexity, resulting from the two-sided selection, of the
structure of the information contained in the ASG. We propose a new method that
consists in encoding the relevant combinatorics of the ASG into a function. We
show that the expectations of the coefficients of this function form a
(non-stochastic) semigroup 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 an infinite sum of polynomials whose
coefficients satisfy explicit linear relations. Our approach then allows to
derive Taylor expansions at every order for $h(x)$ near $x=0$ and to obtain an
explicit recursion formula for the coefficients.
