The ‘Mautner phenomenon’ for
unitary representations is a general assertion of the form that if x(t) is a one
parameter subgroup of a Lie group G, π a unitary representation of G on a Hilbert
space ℋ = ℋ(π) and v a vector in ℋ which is fixed by x(t); i.e., π(x(t))v = v, then v
must also be fixed by a generally much larger subgroup H of G. How much larger H
is than the original one parameter group depends in a general way on how
noncommutative the group G is. Our purpose here is to establish a very general
result of this nature which we believe to be the best possible result of this
kind.
