## Varieties of representations of finitely generated groups.(English)Zbl 0598.14042

Mem. Am. Math. Soc. 336, 117 p. (1985).
Let $$\Gamma$$ be a finitely generated group and k an algebraically closed field of characteristic zero. The authors study the finite dimensional theory of $$\Gamma$$ over k.
In section 1 they define the basic objects of the paper. They introduce the functor $${\mathfrak R}_ n(\Gamma)$$ from commutative k-algebras to sets defined by $${\mathfrak R}_ n(\Gamma)(A)=Hom(\Gamma,GL_ n(A))$$, and if $$f: A\to B$$ is a k-algebra homomorphism, $$f_*: {\mathfrak R}_ n(\Gamma)(A)\to {\mathfrak R}_ n(\Gamma)(B)$$ denotes the function sending $$\rho: \Gamma \to GL_ n(A)$$ into the composite $$\Gamma \to GL_ n(A)\to GL_ n(B)$$. Then they show that this functor is representable by an affine algebra, and so is an affine scheme. This scheme has an algebraic action of $$GL_ n$$ and a universal categorical quotient $$\delta \delta_ n(\Gamma)$$. $${\mathfrak R}_ n(\Gamma)$$ contains an open subscheme $${\mathfrak R}^ s_ n(\Gamma)$$ consisting of the simple representations and its image $$\delta_ n(\Gamma)$$ in the categorical quotient $$\delta \delta_ n(\Gamma)$$ is a geometric quotient $${\mathfrak R}^ s_ n(\Gamma)$$ by $$GL_ n$$. The k-points of the schemes above yield (possibly reducible) k-varieties $$R_ n(\Gamma)$$ (parametrizing degree n representations over k), $$R^ s_ n(\Gamma)$$ (parametrizing simple representations), $$S_ n(\Gamma)$$ (parametrizing isomorphism classes of semisimple representations). In section 1 the authors study the orbits of representations from $$R_ n(\Gamma)$$, too. In particular, they show that such orbit 0($$\rho)$$ is closed iff $$\rho$$ is a semisimple representation.
In section 2 the authors study the tangent spaces of the representation varieties $$R_ n(\Gamma)$$. This section contains the following Weil results about these tangent spaces. The tangent space to $$R_ n(\Gamma)$$ at $$\rho$$ can be identified with a subspace of the space $$Z^ 1(\Gamma,Ad\circ \rho)$$ of one-cocycles of $$\Gamma$$ with coefficients in the representation Ad$$\circ \rho$$ (where $$Ad: GL_ n(k)\to Aut(M_ n(k))$$ is the adjoint representation). The tangent space of the scheme $${\mathfrak R}_ n(\Gamma)$$ at $$\rho$$ is actually equal to $$Z^ 1(\Gamma,Ad\circ \rho)$$. The tangent space to the orbit 0($$\rho)$$ at $$\rho$$ is equal to the space $$B^ 1(\Gamma,Ad\circ \rho)$$ of one- coboundaries. In particular, it follows from the above facts that if $$H^ 1(\Gamma,Ad\circ \rho)=0$$, then the orbit 0($$\rho)$$ is open in $$R_ n(\Gamma)$$ and $$\rho$$ is non-singular on $$R_ n(\Gamma)$$. If $$H^ 1(\Gamma,Ad\circ \rho)=0$$ for all $$\rho \in R_ n(\Gamma)$$, then $$R_ n(\Gamma)$$ consists of finitely many orbits and $$SS_ n(\Gamma)$$ is finite. In this case any representation in $$R_ n(\Gamma)$$ is semisimple. As a consequence of the above results the authors obtain the following statement. If every representation of $$\Gamma$$ of degree 2n is semisimple, then up to isomorphism there are finitely many representations of $$\Gamma$$ of degree n.
In section 3 the authors construct the embedding of $$R_ n(\Gamma)$$ into an affine space. The fact that this embedding displays $$R_ n(\Gamma)$$ as the fibre of a morphism implies some limits to its dimension. Then the authors define def($$\Gamma)$$, the deficiency of $$\Gamma$$, and prove the following important proposition: If def$$(\Gamma)=rk(\Gamma^{ab})$$, where $$\Gamma^{ab}=\Gamma /(\Gamma,\Gamma)$$, then the trivial representation $$\rho_ 0$$ of $$\Gamma$$ in $$GL_ n(k)$$ is scheme non- singular and the dimension of the unique irreducible component of $$R_ n(\Gamma)$$ through $$\rho_ 0$$ is $$rk(\Gamma^{ab})n^ 2$$. If def$$(\Gamma)=rk(\Gamma^{ab})=1$$, then this unique irreducible component consists of all representations factoring through $$\Gamma^{ab}$$ modulo torsion.
Next the authors show how the Fox calculus helps to calculate the group of one-cocycles. At the end of the section they prove that every simple representation in $${\mathfrak R}_ n(SL_ 2)({\mathbb{Z}}))$$ is scheme non- singular, so the variety $$S_ n(SL_ 2({\mathbb{Z}}))$$ is non-singular for every n.
Section 4. Hochschild and Mostow associate to any group $$\Gamma$$ a pro- affine algebraic group A($$\Gamma)$$ over k which has the following property: representations of $$\Gamma$$ of degree n are in one-to-one correspondence with rational representations of A($$\Gamma)$$. The authors use this property for calculation of the groups $$Z^ 1(\Gamma,\rho)$$ and $$H^ 1(\Gamma,\rho)$$ (they show that there exist isomorphisms $$Z^ 1(A(\Gamma),{\bar \rho})\to Z^ 1(\Gamma,\rho)$$ and $$H^ 1(A(\Gamma),\rho)\to H^ 1(\Gamma,\rho)$$ where $$\rho$$ and $${\bar \rho}$$ respect one another under the above one-to-one correspondence.
In section 5 the authors introduce the notion of twisting: let $$\rho \in R_ n(\Gamma)$$ and $$\chi \in X(\Gamma)=Hom(\Gamma,k^*)$$. The twist of $$\rho$$ by $$\chi$$ is defined to be the representation $$\chi$$ $$\rho$$ given by $$(\chi \rho)(\gamma)=\chi (\gamma)\rho (\gamma)$$. The twist operation $$X(\Gamma)\times R_ n(\Gamma)$$ by $$(\chi,\rho)\to \chi\rho$$ is an algebraic action of the algebraic group $$X(\Gamma)$$ on the variety $$R_ n(\Gamma)$$ stabilizing $$R^ S_ n(\Gamma)$$. The authors study orbits of simple and semisimple representations in $$R_ n(\Gamma)$$ under action of $$X(\Gamma)$$ and their images in $$S_ n(\Gamma)$$ and $$SS_ n(\Gamma).$$
In section 6 the authors apply the results of the previous sections to describe $$SS_ n(\Gamma)$$ and $$S_ n(\Gamma)$$ in the case $$\Gamma$$ is nilpotent. For example they prove the following theorem: Let $$\Gamma$$ be nilpotent. Then the distinct twist isoclasses $$c_{\tau}(\rho)$$ for $$\rho$$ pure semisimple of multiplicity one are a finite partition of $$SS_ n$$ into open-closed subsets (a representation $$\rho \in R_ n(\Gamma)$$ is said to be pure semisimple of multiplicity one if $$\rho =\rho_ 1\oplus...\oplus \rho_ s$$ where $$\rho_ i$$ is simple of dimension $$n_ i$$ and $$\rho_ i$$ is not isomorphic $$\rho_ j$$ for $$i\neq j).$$
Section 7 contains historical remarks.
Reviewer: V.Jančevskiĭ

### MSC:

 14L35 Classical groups (algebro-geometric aspects) 20G05 Representation theory for linear algebraic groups 20C15 Ordinary representations and characters 20J05 Homological methods in group theory
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