Group representation irreducibility

We introduced previously the generalized characteristic polynomial defined by PC(λ)=detC(λ), where C(λ)=C+diag(λ1,…,λn) for C∈Mat(n,ℂ) and λ=(λk)nk=1∈ℂn and gave the explicit formula for PC(λ). In this article we define an analogue of the resolvent C(λ)−1, calculate it and the expression (C(λ)−1a,a) for a∈ℂn explicitly. The obtained formulas and their variants were applied to the proof of the irreducibility of unitary representations of some infinite-dimensional groups.