##
**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.

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 |