Algebraic homogeneous spaces and invariant theory.

*(English)*Zbl 0886.14020
Lecture Notes in Mathematics. 1673. Berlin: Springer. vi, 148 p. (1997).

Let \(k\) denote an algebraically closed field. Let \(G\) be a linear algebraic group acting rationally on \(A,\) a commutative \(k\)-algebra with multiplicative identity. For \(H\) a closed subgroup of \(G\) put \(A^H = \{ a \in A : h \cdot a = a \text{ for all } h \in H \}.\) The \(k\)-algebra \(A^H\) is the main object of the book under review, especially when \(H\) is not reductive. One way of the study of \(A^H\) is by the adjunction argument that states \(A^H = (k[G/H] \otimes_k A)^G,\) where \(G\) acts on \(k[G/H]\) by left translations. The idea of studying the invariants of \(H\) on \(A\) in terms of the invariants of the large group \(G\) on a larger set \((k[G/H] \otimes_k A)\) - discovered in the nineteenth century - is the main technique in order to obtain most of the author’s results about finite generation of rings of invariants of non-reductive groups. (Note that if \(G\) is reductive and if \(k[G/H]\) and \(A\) are finitely generated over \(k,\) then the adjunction argument provides that \(A^H\) is also finitely generated over \(k.\))

For the author’s generalizations to non-reductive groups he provides a more detailed study of \(k[G/H].\) In particular it turns out that one may assume that \(H\) is observable in \(G,\) i.e. \(G/H\) is an open subset of some affine variety. In case \(H\) is observable in \(G\) it is shown that \(k[G/H]\) is finitely generated over \(k\) if and only if \(G/H\) can be embedded as an open subset of an affine variety \(X\) in such a way that \(\text{ dim } (X - G/H) \leq \text{ dim } X - 2.\) As an important instance this occurs when \(H\) is a maximal unipotent subgroup of \(G.\) Let \(V\) denote a finite-dimensional \(H\)-module. Then \(G\) acts on \((k[G] \otimes_k V)^H,\) the induced module of \(V\) from \(H\) to \(G.\) The induced modules are used in order to study observable subgroups, for the proof of the adjunction formula, and to find examples where the algebra of invariants is not finitely generated. In the case \(G\) contains a maximal unipotent subgroup \(G\) is called horospherical. These groups play an important rôle in the study of affine embeddings of \(G/H.\)

Very often the author provides actual constructions of invariants. In a number of instances – including \(k[G/H],\) binary forms, determinantal varieties, and the coadjoint representation – the algebra of invariants is constructed.

Another theme of these notes is the complexity \(c(X)\) of an action of \(G\) on \(X\). This is defined to be the codimension on \(X\) of the \(B\)-orbit, \(B\) a Borel subgroup of \(G,\) having highest dimension. If \(c(X) = 0\) the action of \(G\) on \(k[X]\) is called multiplicity free. Such actions lie behind many of the concrete examples considered in the paper. Another part of these studies are investigations about spherical subgroups \(H\) of \(G,\) these are those with \(c(G/H) = 0.\) In the case where \(G\) is reductive and \(c(X) \leq 1\) is is shown - under some weak conditions on \(k[X]\) - that \(k[X]\) is finitely generated over \(k\). Nagata’s example to Hilbert’s 14th problem [see M.Nagata, Am. J. Math. 81, 766-772 (1959; Zbl 0192.13801)], gives an instance of a variety \(X\) where \(c(X) = 2\) and \(k[X]\) is not finitely generated. The author succeeded in making his exposition throughout as complete as possible. That means in particular that he tried to give almost all proofs. He does not restrict to the case of characteristic zero, where finite-dimensional representations of reductive groups are known to be completely reducible.

This exposition contains a lot of material. A lot of examples are considered. Among them P. Roberts’ example [J. Algebra 132, No. 2, 461-473 (1990; Zbl 0716.13013)], straightening laws, bideterminants, geometric examples. Each of the four chapters is completed by a short introduction and some of them by a bibliographical note. Most of the 23 paragraphs finish with a couple of exercises, good to deepen the techniques of it. It makes a good propaganda for further studies in the relationship between algebraic homogeneous spaces and invariant theoretical questions.

For the author’s generalizations to non-reductive groups he provides a more detailed study of \(k[G/H].\) In particular it turns out that one may assume that \(H\) is observable in \(G,\) i.e. \(G/H\) is an open subset of some affine variety. In case \(H\) is observable in \(G\) it is shown that \(k[G/H]\) is finitely generated over \(k\) if and only if \(G/H\) can be embedded as an open subset of an affine variety \(X\) in such a way that \(\text{ dim } (X - G/H) \leq \text{ dim } X - 2.\) As an important instance this occurs when \(H\) is a maximal unipotent subgroup of \(G.\) Let \(V\) denote a finite-dimensional \(H\)-module. Then \(G\) acts on \((k[G] \otimes_k V)^H,\) the induced module of \(V\) from \(H\) to \(G.\) The induced modules are used in order to study observable subgroups, for the proof of the adjunction formula, and to find examples where the algebra of invariants is not finitely generated. In the case \(G\) contains a maximal unipotent subgroup \(G\) is called horospherical. These groups play an important rôle in the study of affine embeddings of \(G/H.\)

Very often the author provides actual constructions of invariants. In a number of instances – including \(k[G/H],\) binary forms, determinantal varieties, and the coadjoint representation – the algebra of invariants is constructed.

Another theme of these notes is the complexity \(c(X)\) of an action of \(G\) on \(X\). This is defined to be the codimension on \(X\) of the \(B\)-orbit, \(B\) a Borel subgroup of \(G,\) having highest dimension. If \(c(X) = 0\) the action of \(G\) on \(k[X]\) is called multiplicity free. Such actions lie behind many of the concrete examples considered in the paper. Another part of these studies are investigations about spherical subgroups \(H\) of \(G,\) these are those with \(c(G/H) = 0.\) In the case where \(G\) is reductive and \(c(X) \leq 1\) is is shown - under some weak conditions on \(k[X]\) - that \(k[X]\) is finitely generated over \(k\). Nagata’s example to Hilbert’s 14th problem [see M.Nagata, Am. J. Math. 81, 766-772 (1959; Zbl 0192.13801)], gives an instance of a variety \(X\) where \(c(X) = 2\) and \(k[X]\) is not finitely generated. The author succeeded in making his exposition throughout as complete as possible. That means in particular that he tried to give almost all proofs. He does not restrict to the case of characteristic zero, where finite-dimensional representations of reductive groups are known to be completely reducible.

This exposition contains a lot of material. A lot of examples are considered. Among them P. Roberts’ example [J. Algebra 132, No. 2, 461-473 (1990; Zbl 0716.13013)], straightening laws, bideterminants, geometric examples. Each of the four chapters is completed by a short introduction and some of them by a bibliographical note. Most of the 23 paragraphs finish with a couple of exercises, good to deepen the techniques of it. It makes a good propaganda for further studies in the relationship between algebraic homogeneous spaces and invariant theoretical questions.

Reviewer: P.Schenzel (Halle)

##### MSC:

14M17 | Homogeneous spaces and generalizations |

13A50 | Actions of groups on commutative rings; invariant theory |

15A72 | Vector and tensor algebra, theory of invariants |

20G15 | Linear algebraic groups over arbitrary fields |

14L24 | Geometric invariant theory |

14L30 | Group actions on varieties or schemes (quotients) |