  We concentrate on inverse scattering transformation for the Sasa-Satsuma
equation with $3\times 3$ matrix spectral and nonzero boundary condition in
this article. To circumvent multi valuedness of eigenvalues, we introduce a
suitable two-sheet Riemann surface to map the original spectral parameter $k$
into a single-valued parameter $z$. The analyticity of the Jost eigenfunctions
and scattering coefficients of Lax pair for the SS equation are analyzed in
details. According to the analyticity of eigenfunctions and scattering
coefficients, the $z$-complex plane is divided into four analytic regions $D_j,
\ j=1, 2, 3, 4$. Since the second column of Jost eigenfunctions is analytic in
$D_{j}, \ j=1, 2, 3, 4$, but in upper-half or lower-half plane, we introduce
certain auxiliary eigenfunctions which are necessary for deriving the analytic
eigenfunctions in $D_{j}$. We find that for the eigenfunctions, scattering
coefficients and the auxiliary eigenfunctions all possess three kinds of
symmetries, which characterize the distribution of discrete spectrum. The
asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering
coefficients are also systematically derived. Then a matrix Riemann-Hilbert
problem with four kind jump conditions associated with the problem of nonzero
asymptotic boundary conditions is established, from which $N$-soliton solutions
is obtained via the corresponding reconstruction formulae. The reflectionless
soliton solutions are explicitly given. As application of the $N$-soliton
formula, we present three kinds of single-soliton solutions according to the
distribution of discrete spectrum.
