Quadratic unipotent blocks in general linear, unitary and symplectic groups

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An irreducible ordinary character of a finite reductive group is called quadratic unipotent if it corresponds under Jordan decomposition to a semisimple element s in a dual group such that s²=1. We prove that there is a bijection between, on the one hand the set of quadratic unipotent characters of GL(n,q) or U(n,q) for all n ≥ 0 and on the other hand, the set of quadratic unipotent characters of Sp(2n,q) for all n ≥ 0. We then extend this correspondence to ℓ-blocks for certain ℓ not dividing q.