On the graded center of the stable category of a finite pp-group

We show that for any finite pp-group PP of rank at least 22 and any algebraically closed field kk of characteristic pp the graded center Z∗(mod¯(kP)) of the stable module category of finite-dimensional kPkP-modules has infinite dimension in each odd degree, and if p=2p=2, also in each even degree. In particular, this provides examples of symmetric algebras AA for which Z0(mod¯(A)) is not finite-dimensional, answering a question raised in Linckelmann (in press) [1].