Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 2 / b / Solution

There is a natural isomorphismobtained by evaluating a homomorphism against a covector. Since $P$ is a direct summand of a free module and tensoring a free $kG$-module with $U^{\ast }$ using the diagonal action again gives a free module, $P\otimes U^{\ast }$ is a projective module.

Lift this projective module to an $\mathcal{O}G$-lattice. Projectivity makes the dimension of the homomorphism space equal to the multiplicity of the trivial representation after extending scalars to $K$. The lifted ordinary character vanishes on p-singular elements, while on p-regular elements it is the productOrdinary character orthogonality and the substitution $g\mapsto g^{-1}$ therefore giveEquivalently, this is the Brauer character inner product between the projective character of $P$ and the Brauer character of $U$.