Scale functions on \(p\)-adic Lie groups (Q1271529)

To each locally compact, totally disconnected group \(G\), a scale function \(s\) from \(G\) to the semigroup of positive integers has been associated by \textit{G. Willis} [Math. Ann. 300, 341-363 (1994; Zbl 0811.22004)]. This scale function measures the distortion of suitable compact subgroups under conjugation. The present paper shows how to compute the scale function for a \(p\)-adic Lie group, using the (adjoint representation on the) Lie algebra. Explicit formulae are given for the case where \(G\) is one of the groups \(GL(n,K)\), \(SL(n,K)\), \(PGL(n,K)\), or \(PSL(n,K)\), where \(K\) is a local field of characteristic \(0\). The case of positive characteristic has been treated by the author in a previous paper [J. Algebra 205, 525-541 (1998)].