Support singulier et homologie des fibres de Springer affines

Abstract: We develop a theory of singular support for various infinite dimensional stacks and establish several functoriality properties. Then we apply this theory to compute the singular support of the Grothendieck-Springer affine Springer sheaf and rely it to the local constancy conjecture of Goresky-Kottwitz-McPherson on the homology of affine Springer fibers along the root valuation stratification.