Brauer character inner product

For class functions on the p-regular elements of a finite group,Equivalently, summing over p-regular conjugacy-class representatives $x$ gives weights $1/|C_G(x)|$.

If $S_i$ are the simple modules of a split group algebra and $P_i$ are their projective covers, then

For p-regular elements $g,h$,where $S$ ranges over the simple modules and $P_S$ is the projective cover of $S$.