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Motivated by affine Schubert calculus, we construct a family of dual gradedgraphs $(\Gamma_s,\Gamma_w)$ for an arbitrary Kac-Moody algebra $\g(A)$. Thegraded graphs have the Weyl group $W$ of $\g(A)$ as vertex set and are labeledversions of the strong and weak orders of $W$ respectively. Using aconstruction of Lusztig for quivers with an admissible automorphism, we definefolded insertion for a Kac-Moody algebra and obtain Sagan-Worley shiftedinsertion from Robinson-Schensted insertion as a special case. Drawing on workof Stembridge, we analyze the induced subgraphs of $(\Gamma_s,\Gamma_w)$ whichare distributive posets.

Dual graded graphs for Kac–Moody algebras