On a result of Moeglin and Waldspurger in residual characteristic 2

Let \(F\) be a \(p\) -adic field, \(\mathbf G\) a connected reductive group over \(F\) , and \(\pi \) an irreducible admissible representation of \(\mathbf G(F)\) . A result of Moeglin and Waldspurger states that, if the residual characteristic of \(F\) is different from \(2\) , then the ‘leading’ coefficients in the character expansion of \(\pi \) at the identity element of \(\mathbf G(F)\) give the dimensions of certain spaces of degenerate Whittaker forms. In this paper, we extend their result to residual characteristic 2. The outline of the proof is the same as in the original paper of Moeglin and Waldspurger, but certain constructions are modified to accommodate the case of even residual characteristic.