The unitary representation theory of GL(n) of an infinite discrete field
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If K is an infinite field and if G=GL(n, K) with the discrete topology, then all principal-series representations of G are irreducible, and any two such with the same central character are weakly equivalent to one another and to the -regular representation. In addition, every irreducible unitary representation of G which is not one-dimensional weakly contains a representation of the principal series. We deduce that every maximal ideal of C*(G) is either of codimension 1 or else a kernel of a principal-series representation. In particular, except in the exceptional case where K is an infinite algebraic extension of a finite field, the reduced C*-algebra of PGL(n, K) is simple, as was also shown by de la Harpe in many cases. 1989 The Weizmann Science Press of Israel.
