Irreducible representations of the group of infinite upper triangular matrices (Q1088790)

Let B(n, \({\mathbb{R}})\) be the group of real upper triangular \(n\times n\) matrices with \(a_{ii}=1\), \(i=1,...,n\). Let \(B_ 0(\infty, {\mathbb{R}})\) and B(\(\infty, {\mathbb{R}})\) be the inductive and projective limits of \(\{\) B(n, \({\mathbb{R}})\}\), respectively. If \(T_ n(g)\) is an irreducible unitary representation of B(n, \({\mathbb{R}})\), it can be lifted to a representation of each of the groups B(m, \({\mathbb{R}})\), \(m>n\), which is trivial on the supplement of B(n, \({\mathbb{R}})\) in B(m, \({\mathbb{R}})\). Therefore, \(T_ n(g)\) defines representations of \(B_ 0(\infty, {\mathbb{R}})\) and B(\(\infty, {\mathbb{R}})\). It is proved that every irreducible unitary representation of \(B_ 0(\infty, {\mathbb{R}})\) and of B(\(\infty, {\mathbb{R}})\) is of this type.