Rational representations of algebraic groups: Tensor products and filtrations.

*(English)*Zbl 0586.20017
Lecture Notes in Mathematics. 1140. Berlin etc.: Springer-Verlag. VII, 254 p. DM 38.50 (1985).

Let G be a connected, affine algebraic group over an algebraically closed field and B a Borel subgroup of G. For each one dimensional rational B- module L the induced G-module \(Ind^ G_ BL\) is defined. These modules are of fundamental importance in the representation theory of G; in characteristic 0 one obtains every simple rational G-module in this way and in arbitrary characteristic \(Ind^ G_ BL\), when non-zero, has a simple socle and each simple G-module occurs as the socle of some such induced module. The formal character of \(Ind^ G_ BL\) is independent of the characteristic, being given by Weyl’s character formula, but the submodule structure depends very heavily upon the characteristic and very little is known about this structure in characteristic p. For G semisimple, the induced modules also have an interpretation as the global sections of line bundles on the quotient variety G/B and so provide a bridge between the representation theory of G and the geometry of G/B.

A good filtration of a rational G-module V is an ascending chain of submodules \(O=V_ 0,V_ 1,V_ 2,..\). of V such that V is the union of the \(V_ i\) and, for each \(i>0\), \(V_ i/V_{i-1}\) is either 0 or isomorphic to \(Ind^ G_ BL\) for some rational one dimensional B-module L. It is known that for G semisimple and simply connected each rationally injective indecomposable G-module has a good filtration. In the monograph under review the author studies for a connected, affine algebraic group G over an algebraically closed field k, the following hypotheses.

Hypothesis 1. For all rational G-modules V, V’ which have a good filtration the tensor product \(V\otimes V'\) has a good filtration.

Hypothesis 2. For every rational G-module V which has a good filtration and every parabolic subgroup P of G the restriction of V to P has a good filtration.

In chapters 1, 2 and 5 the author considers some general results on group cohomology and the derived functors of induction which are needed for the specific calculations in chapters 4, 6, 7, 8, 9 and 10. The main purpose of chapter 1 is to establish the notation and explain the relationship between various left exact functors. Chapter 2 contains results computing the cohomology of some modules for parabolic subgroups. This chapter also contains a deduction from Kempf’s vanishing theorem of Weyl’s character formula for the character of \(Ind^ G_ BL\) for a reductive group G and one-dimensional B-module L.

In chapter 3 the author makes various reductions to the hypotheses so that they become susceptible to the case by case analysis which follows.

In chapter 4 he proves the hypotheses for the classical groups. The argument here is independent of the characteristic and it has been possible to treat the groups of type B, C and D in a unified manner. The restriction of \(Y(\lambda_ i)\) to a proper parabolic subgroup of maximal dimension has only 4 successive quotients in a good filtration (for i in ”general position”) and the module structure is much the same in all three types B, C, and D.

Some additional homological algebra is needed to deal with the exceptional groups and this is given in chapter 5. The hypotheses are proved for \(G_ 2\) in chapter 6 however it would not be difficult to treat this case without the benefit of chapter 5. In chapter 7, treating \(F_ 4\), he found it necessary to consider separately the cases of odd and even characteristic.

The subject, \(E_ 6\), of chapter 8 is altogether easier but again there is a division in the proof into odd and even characteristics. Chapters 9 and 10 are devoted to the remaining exceptional groups \(E_ 7\) and \(E_ 8\). The procedure here is to analyse first the modules \(Y(\lambda_ i)\), corresponding to the terminal vertices \(\alpha_ i\) of the Dynkin diagram, and then exterior powers of these modules are used to deal with \(Y(\lambda_ r)\) for an arbitrary fundamental dominant weight \(\lambda_ r.\)

Chapter 11 opens with an example of a reductive subgroup H of a reductive group G and a G-module V such that V has a good filtration but the restriction of V to H does not. The remainder of the chapter is devoted to applications of the hypotheses to rational cohomology, homomorphisms between Weyl modules, canonical products on induced modules and filtrations over \({\mathbb{Z}}\) of Weyl modules for Kostant’s \({\mathbb{Z}}\)-form \(U_{{\mathbb{Z}}}\) of the enveloping algebra U(\({\mathfrak g})\) of a complex semisimple Lie algebra \({\mathfrak g}.\)

The final chapter is devoted to a number of observations on issues not directly concerned with the hypotheses but which nevertheless have mainly arisen in the course of the work on the hypotheses. The issues discussed are the injective indecomposable modules for a parabolic subgroup of a reductive group, Kempf’s vanishing theorem for rank 1 groups, Kempf’s vanishing theorem in characteristic zero and the exactness of induction.

A good filtration of a rational G-module V is an ascending chain of submodules \(O=V_ 0,V_ 1,V_ 2,..\). of V such that V is the union of the \(V_ i\) and, for each \(i>0\), \(V_ i/V_{i-1}\) is either 0 or isomorphic to \(Ind^ G_ BL\) for some rational one dimensional B-module L. It is known that for G semisimple and simply connected each rationally injective indecomposable G-module has a good filtration. In the monograph under review the author studies for a connected, affine algebraic group G over an algebraically closed field k, the following hypotheses.

Hypothesis 1. For all rational G-modules V, V’ which have a good filtration the tensor product \(V\otimes V'\) has a good filtration.

Hypothesis 2. For every rational G-module V which has a good filtration and every parabolic subgroup P of G the restriction of V to P has a good filtration.

In chapters 1, 2 and 5 the author considers some general results on group cohomology and the derived functors of induction which are needed for the specific calculations in chapters 4, 6, 7, 8, 9 and 10. The main purpose of chapter 1 is to establish the notation and explain the relationship between various left exact functors. Chapter 2 contains results computing the cohomology of some modules for parabolic subgroups. This chapter also contains a deduction from Kempf’s vanishing theorem of Weyl’s character formula for the character of \(Ind^ G_ BL\) for a reductive group G and one-dimensional B-module L.

In chapter 3 the author makes various reductions to the hypotheses so that they become susceptible to the case by case analysis which follows.

In chapter 4 he proves the hypotheses for the classical groups. The argument here is independent of the characteristic and it has been possible to treat the groups of type B, C and D in a unified manner. The restriction of \(Y(\lambda_ i)\) to a proper parabolic subgroup of maximal dimension has only 4 successive quotients in a good filtration (for i in ”general position”) and the module structure is much the same in all three types B, C, and D.

Some additional homological algebra is needed to deal with the exceptional groups and this is given in chapter 5. The hypotheses are proved for \(G_ 2\) in chapter 6 however it would not be difficult to treat this case without the benefit of chapter 5. In chapter 7, treating \(F_ 4\), he found it necessary to consider separately the cases of odd and even characteristic.

The subject, \(E_ 6\), of chapter 8 is altogether easier but again there is a division in the proof into odd and even characteristics. Chapters 9 and 10 are devoted to the remaining exceptional groups \(E_ 7\) and \(E_ 8\). The procedure here is to analyse first the modules \(Y(\lambda_ i)\), corresponding to the terminal vertices \(\alpha_ i\) of the Dynkin diagram, and then exterior powers of these modules are used to deal with \(Y(\lambda_ r)\) for an arbitrary fundamental dominant weight \(\lambda_ r.\)

Chapter 11 opens with an example of a reductive subgroup H of a reductive group G and a G-module V such that V has a good filtration but the restriction of V to H does not. The remainder of the chapter is devoted to applications of the hypotheses to rational cohomology, homomorphisms between Weyl modules, canonical products on induced modules and filtrations over \({\mathbb{Z}}\) of Weyl modules for Kostant’s \({\mathbb{Z}}\)-form \(U_{{\mathbb{Z}}}\) of the enveloping algebra U(\({\mathfrak g})\) of a complex semisimple Lie algebra \({\mathfrak g}.\)

The final chapter is devoted to a number of observations on issues not directly concerned with the hypotheses but which nevertheless have mainly arisen in the course of the work on the hypotheses. The issues discussed are the injective indecomposable modules for a parabolic subgroup of a reductive group, Kempf’s vanishing theorem for rank 1 groups, Kempf’s vanishing theorem in characteristic zero and the exactness of induction.

Reviewer: N.I.Osetinski

##### MSC:

20G05 | Representation theory for linear algebraic groups |

14L17 | Affine algebraic groups, hyperalgebra constructions |

20G10 | Cohomology theory for linear algebraic groups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

14M17 | Homogeneous spaces and generalizations |