Decomposition of the Uniform Projection of the Weil Character

In this paper we obtain a decomposition formula of the uniform projection of the Weil character of a finite reductive dual pair consisting of a symplectic group and an even orthogonal group. This is the first and major step to give an explicit description of the Howe correspondence on unipotent characters confirming a conjecture by Aubert-Michel-Rouquier.
Comment: This work previously appeared under arXiv:1901.00623v1 (and v2), which has been split into two parts. The second part is now available as arXiv:1901.00623v3