Local structure of representation varieties (Q1115492)

The isomorphism classes [\(\rho\) ] of simple n-dimensional representation \(\rho\) of the finitely generated group \(\Gamma\) form an algebraic variety. In the paper under review the tangents to formal curves through a point in the variety are studied. Let k be an algebraically closed field of characteristic zero (usually \({\mathbb{C}})\), G an affine algebraic group scheme over k. The functor from commutative k-algebras to sets sending the algebra A to Hom(\(\Gamma\),G(A)) is represented by an affine scheme \({\mathcal R}(\Gamma,G)\) of finite type over k and whose k-points comprise an affine algebraic set denoted R(\(\Gamma\),G). In the case where \(G=GL_ n\) we shall use the notation \({\mathcal R}_ n(\Gamma)\) and \(R_ n(\Gamma)\). The group scheme \(GL_ n\) acts on \({\mathcal R}_ n(\Gamma)\) by conjugation and let \({\mathcal S}{\mathcal S}_ n(\Gamma)\) be its categorical quotient. Inside \({\mathcal R}_ n(\Gamma)\) is the \(GL_ n\)-stable open subscheme \({\mathcal R}_ n(\Gamma)^ s\) of simple representations (\(\rho\) in \(Hom(\Gamma,GL_ n(A))\) is simple if \(\rho\) (\(\Gamma)\) spans \(M_ n(A)\) over A). \(GL_ n\) acts with closed orbits on \({\mathcal R}_ n(\Gamma)^ s\) and its image \({\mathcal S}_ n(\Gamma)\) in \({\mathcal S}{\mathcal S}_ n(\Gamma)\) is a geometric quotient of \({\mathcal R}_ n(\Gamma)^ s\) locally trivial for the étale topology. The k-points of these schemes are denoted \(SS_ n(\Gamma)\) and \(S_ n(\Gamma).\) Let \({\mathcal R}\) denote an affine k-scheme of finite type with coordinate ring \({\mathcal A}\) and variety \(R={\mathcal R}(k)\) of k-points. \(A={\mathcal A}_{red}\) is the coordinating ring R. Let r be a k-point in R and \({\mathfrak m}\subseteq A\) the corresponding maximal ideal. In the following definitions k[\(\epsilon\) ] denotes the dual numbers over k: The tangent space \(T_ r({\mathcal R})\) is the fibre of \({\mathcal R}(k[\epsilon])\to {\mathcal R}(k)=R\) over k. The tangent space \(T_ r(R)\) is the fibre of \(Alg_ k(A,[\epsilon])\to Alg_ k(A,k)=R\) over r. The tangent cone \(TC_ r(R)\) is \(Alg_ k(gr_{{\mathfrak m}}(A)k).\) We also need the following definition: The set \(C_ r({\mathcal R})\) of formal curves in \({\mathcal R}\) at r is the fibre of \({\mathcal R}(k[[t]])\to {\mathcal R}(k)=R\) over r. The image of \(C_ r({\mathcal R})\) in \(T_ r(R)\) under the natural map k[[t]]\(\to k[\epsilon]\) is denoted \(CT_ r(R)\) and called the set of curve tangents to R at r. By definition r is non- singular on R when \(TC_ R(R)=T_ r(R)\). If also \({\mathcal R}\) is reduced at r then \(CT_ r(R)=T_ r({\mathcal R})\) and we say R is scheme non- singular at r. If \({\mathcal R}=({\mathcal R}_ n\Gamma)\), \(r=\rho \in R_ n(\Gamma)\) we have \({\mathcal R}_ n(\Gamma)(k[\epsilon])=Hom(\Gamma,GL_ n(k[\epsilon])\), \(R_ n(\Gamma)(k[[t]])=Hom(\Gamma,GL_ n(k[[t]]).\) There is a pro-affine algebraic group A(\(\Gamma)\) and a group homomorphism \(h:\quad \Gamma \to A(\Gamma)\) with Zariski-dense image such that any homomorphism \(\bar h:\quad \Gamma \to G\) from \(\Gamma\) to a pro- algebraic group G factors as \(h=\bar hj\) where \(\bar h:\quad A(\Gamma)\to G\) is a pro-algebraic group homomorphism. The pro-algebraic group A(\(\Gamma)\) is a semi-direct product of its prounipotent radical U(\(\Gamma)\) and any maximal proreductive subgroup. These latter are all conjugate by elements of U(\(\Gamma)\) and we fix and denote it P(\(\Gamma)\). U(\(\Gamma)\) is acted on by \(\Gamma\) via j. The main results of the paper under review are the following: Theorem 1. Let \(\rho\) : \(\Gamma \to GL_ n(k)\) be a semi-simple representation then \[ T_{\rho}({\mathcal R}_ n(\Gamma))=Hom_{\Gamma}(U(\Gamma),\quad I+\epsilon M_ n(k))\oplus B^ 1(\Gamma,Ad\circ \rho). \] Theorem 2. Let \(\rho\) : \(\Gamma \to GL_ n(k)\) be a semi-simple representation, let \(V=\{\phi \in Hom_{\Gamma}(U(\Gamma),\quad I+\epsilon M_ n(k))| \quad \exists \phi_ t\in Hom_{\Gamma}(U(\Gamma),I+rM_ n(k[[t]])\quad with\quad \phi_ t=\phi (mod t^ 2).\) Then \[ CT_{\rho}(R_ n(\Gamma))=V\times B^ 1(\Gamma,Ad\circ \rho). \] If \(\rho\) is simple, then \(CT_{[\rho]}(S_ u(\Gamma))\subset V.\) Corollary. Let \(\rho\) : \(\Gamma \to GL_ n(k)\) be a semi-simple representation, and let V be as in theorem 2. Then, if \(V=Hom_{\Gamma}(U(\Gamma),I+\epsilon M_ n(k))\), \(\rho\) is non- singular on \(R_ n(\Gamma)\) and \[ T_{\rho}(R_ n(\Gamma))=T_{\rho}(R_ n(\Gamma))=Z^ 1(\Gamma,Ad\circ \rho). \] If in addition \(\rho\) is simple, then \(\rho\) is non-singular on \(S_ n(\Gamma)\) and \(T_{\rho}(S_ n(\Gamma))=H^ 1(\Gamma,Ad\circ \rho).\) Theorem 3. Assume \(L=Lie(U(\Gamma))\) is graded by its lower central series. Let \(\rho:\quad \Gamma \to GL_ n(k)\) be semi-simple. Then \(CT_{\rho}(R_ n(\Gamma))=TC_{\rho}(R_ n(\Gamma))\). If \(\rho\) is simple, then \(CT_{[\rho]}(S_ n(\Gamma))=TC_{[\rho]}(S_ n(\Gamma)).\) Theorem 4. Let \(\Gamma\) be nilpotent of rank at least two, and let \(\rho \in R_ n(\Gamma)\) be semi-simple. If some component of \(\rho\) has multiplicity two or more, then \(CT_{\rho}(R_ n(\Gamma))\) is not a linear subspace of \(T_{\rho}(R_ n(\Gamma))\). In particular, \(\rho\) is a singular point of \(R_ n(\Gamma).\) At the end of the paper a description of the tangent cones to representation varieties is given.