On Brauer's second main theorem and characters in blocks with nilpotent coefficient extensions (Q1403890)

Let \(\mathcal O\) be a complete discrete valuation ring of characteristic \(0\) and residue field \(k={\mathcal O}/J({\mathcal O})\) of characteristic \(p>0\). A block \(b{\mathcal O}G\) with defect pointed group \(P_\gamma\) is called nilpotent, if for any local pointed group \(Q_\delta\subset P_\gamma\) and any \(x\in G\) such that \((Q_\delta)^x\subset P_\gamma\), there are \(z\in G\) and \(v\in P\) such that \(x=zv\). Moreover, \(b\) is said to have nilpotent coefficient extensions if there is an extension \({\mathcal O}'\) of \(\mathcal O\) and a nilpotent block \(b'\) of \({\mathcal O}'G\) such that \(bb'=b'\). The main result of the paper is a version of Brauer's Second Main Theorem over arbitrary ground fields. This is then applied to obtain a formula on characters in blocks with nilpotent coefficient extensions.