On \(C\)-small conjugacy classes in a reductive group. (Q649090)

Let \(G\) be an almost simple reductive group with Weyl group \(W\). Let \(B\) be a Borel subgroup of \(G\). Assume the characteristic is zero or a good prime for \(G\). Let \(w\) be an elliptic element of \(W\) which has minimal length in its conjugacy class. It is shown that in almost all cases there exists a semisimple class \(\gamma\) in \(G\) whose intersection with \(BwB\) has dimension \(\dim(B)\). The exception is a specific class in type \(E_8\). The Springer representation of \(\gamma\) is connected with the Springer representation associated with the \textit{unipotent} class that also may be associated with \(w\).