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We develop a theory of singular support for various infinite dimensionalstacks and establish several functoriality properties. Then we apply thistheory to compute the singular support of the Grothendieck-Springer affineSpringer sheaf and rely it to the local constancy conjecture ofGoresky-Kottwitz-McPherson on the homology of affine Springer fibers along theroot valuation stratification.

Support singulier et homologie des fibres de Springer affines