  Motivated by affine Schubert calculus, we construct a family of dual graded
graphs $(\Gamma_s,\Gamma_w)$ for an arbitrary Kac-Moody algebra $\g(A)$. The
graded graphs have the Weyl group $W$ of $\g(A)$ as vertex set and are labeled
versions of the strong and weak orders of $W$ respectively. Using a
construction of Lusztig for quivers with an admissible automorphism, we define
folded insertion for a Kac-Moody algebra and obtain Sagan-Worley shifted
insertion from Robinson-Schensted insertion as a special case. Drawing on work
of Stembridge, we analyze the induced subgraphs of $(\Gamma_s,\Gamma_w)$ which
are distributive posets.
