Deformations and liftings of representations (Q2704357)

The paper under review is the first of a series investigating deformations and liftings of representations of finitely generated groups, much of it being devoted to exposition and background.NEWLINENEWLINENEWLINELet \(k\) be an algebraically closed field of characteristic zero, and \(\Gamma\) a finitely generated group. A representation \(\rho\colon\Gamma\to\text{GL}_n(k)\) is called rigid if representations close to \(\rho\) in the variety \(R_n(\Gamma)\) of all representations of degree \(n\) are isomorphic to \(\rho\). Otherwise \(\rho\) is deformable. One of the basic results says that a simple representation \(\rho\) is deformable if and only if there is a nontrivial formal representation \(\rho_t\colon\Gamma\to\text{GL}_n(k[[t]])\) such that \(\rho_0=\rho\), and this is equivalent to the existence of a consistent family of nontrivial lifts \(\rho_a\colon\Gamma\to\text{GL}_n(k[t]/t^{a+1})\). The authors prove the existence of deformations of simple representations in the case when \(\Gamma\) has a finitely generated nilpotent subgroup \(N\) such that \(G/N\) is linearly reductive.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00031].