Scale functions on linear groups over local skew fields
Certain questions in the theory of probability measures led G. Willis to the definition of the ``scale function'' on a totally disconnected locally compact group. This function with values in the natural numbers gives some insight into the structure of the group. In the present paper the scale function is computed on diagonal matrices in the general or special linear groups or in the group of triangular matrices over a local skew field. It is computed completely on \(\text{GL}(2,K)\) and \(\text{SL}(2,K)\) when \(K\) is a commutative local field. All computed values of the scale function are integral powers of \(q\), the number of elements of the residue field, so that it is possible to recover \(q\) from (partial) knowledge of the scale function.
scientific article; zbMATH DE number 1214131