Parabolic induction and Jacquet modules of representations of O(2n,F)

For the sum of the Grothendieck groups of the categories of smooth finite length representations of O(2n, F) (resp., SO(2n, F)), n ≥ 0, (F a p-adic field), the structure of a module and a comodule over the sum of the Grothendieck groups of the categories of smooth finite length representations of GL(n, F), n ≥ 0, is achieved. The multiplication is defined in terms of parabolic induction, and the comultiplication in terms of Jacquet modules. Also, for even orthogonal groups, the combinatorial formula, which connects the module and comodule structures, is obtained