Multiplicative functions on free groups and irreducible representations (Q1901073)

Let \(\Gamma\) be a free group on infinitely many generators. Fix a basis \(A\) for \(\Gamma\). For any group element \(x\), denote by \(|x|\) its length with respect to this basis. Let \(e\) denote the group identity. A multiplicative function \(\phi\) on \(\Gamma\) is a function satisfying the conditions \(\phi(e) = 1\) and \(\phi(xy) = \phi(x) \phi(y)\) whenever \(|xy|= |x|+ |y|\). It is known that a multiplicative function is positive definite if and only if \(\phi(x) = \phi(x^{-1})\) and \(|\phi(x)|\leq 1\) for all \(x\). Denote by \(\pi_\phi\) the representation of \(\Gamma\) associated with \(\phi\). We prove that \(\pi_\phi\) is irreducible if and only if \(\sum_{a \in A} {|\phi(a)|^2 \over 1 + |\phi(a) |^2} = + \infty\).