Glauberman-Isaacs correspondence and \(\pi\)-Brauer characters (Q1340259)

Assume that \(A\) and \(G\) are finite groups of coprime order and that \(A\) acts on \(G\). Denote \(C=C_ G(A)\). If \(\chi\in\text{Irr}_ A(G)\), then the Glauberman-Isaacs correspondence gives a unique element \(\chi^*\in\text{Irr}(C)\). Also, if \(G\) is a \(\pi\)-separable group and \(\chi\) is a class function on \(G\), denote by \(\chi^ 0\) the restriction of \(\chi\) to the set of \(\pi\)-elements of \(G\). The set of Isaacs' \(\pi\)- Brauer characters of \(G\) is \(I_ \pi(G) =\{\chi^ 0\mid\chi\in\text{Irr}(G)\), \(\chi^ 0\) is not of the form \(\chi^ 0 =\xi^ 0 +\eta^ 0\) for \(\xi,\eta\in\text{char}(G)\}\). The map \(^ 0\) defines a bijection between the subset \(B_ \pi(G)\subset\text{Irr}(G)\) and \(I_ \pi(G)\). In this paper the author investigates the \(\pi\)-restriction of \(A\)- invariant characters of \(G\) and of their Glauberman-Isaacs correspondents. Assume in the sequel that \(G\) is \(\pi\)-separable. The first main result says that if \(\chi,\psi\in\text{Irr}_ A(G)\) and \(\chi^ 0 =\psi^ 0\in I_ \pi(G)\) then \((\chi^*)^ 0 = (\psi^*)^ 0\in I_ \pi(C)\). It is known that if \(\phi\) is an \(A\)-invariant \(\pi\)-Brauer character, then \(\phi^* = (\chi^*)^ 0\), where \(\chi\) is the unique element of \(B_ \phi(G)\) such that \(\chi^ 0 =\phi\). The author shows that it is unnecessary to assume that \(\chi\in B_ \pi(G) :\phi^* = (\psi^*)^ 0\) for any \(\psi\in\text{Irr}_ A(G)\) with \(\phi^* =\phi\). These results can be reformulated as follows: if \(\chi\in\text{Irr}_ A(G)\) is such that \(\chi^ 0\) is a \(\pi\)-Brauer character, then \((\chi^ 0)^* = (\chi^*)^ 0\). If in addition \(A\) is solvable, this result is generalized to the case when \(\chi^ 0 =\sum d_ \phi\phi\) is a sum of \(A\)-invariant \(\pi\)- Brauer characters, where \(d_ \phi\geq 0\). In this case, \((\chi^*)^ 0 =\sum b_ \phi\phi^*\), where \(0\leq b_ \phi\leq d_ \phi\) for all \(\phi\in I_ A(G)\). Moreover, if \(A\) is a \(p\)-group, then \(b_ \phi\equiv\pm d_ \phi\) for all \(\phi\in I_ A(G)\).