Representations of algebraic groups.

*(English)*Zbl 0654.20039
Pure and Applied Mathematics, Vol. 131. Boston etc.: Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers). XIII, 443 p.; $ 59.50 (1987).

This book, which seems to me very readable, is meant to give its reader an introduction to the representation theory of reductive algebraic groups G over an algebraically closed field k. In the case \(char(k)=0\) the subject is well understood: each G-module is semisimple, the simple G-modules are classified by their highest weights, and one has a character formula for these simple modules - in fact, Weyl’s formula holds. The situation in prime characteristic is much worse: except for the case of a torus, there are non-simple G-modules, except for low rank cases, it is not known a character formula for simple modules, and Weyl’s formula will certainly not carry over (though, according to Chevalley, one property survives: the simple modules are still classified by their highest weights, and the possible highest weights are the “dominant” weights of the group of characters X(T) of a maximal torus T of G). Nowadays this subject is the field of activity of many specialists who made significant progress and developed a beautiful theory. The reader will find this theory in the book under review. Modern developments of the theory are based on many different techniques which have been introduced into the theory, especially during the last fifteen years. This is why the book is divided to two parts: Part I contains a general introduction to the representation theory of algebraic group schemes, whereas Part II then deals with the representations of reductive groups.

The contents of Part I is as follows. Chapters I1 and I2 contain an introduction to schemes and to affine group schemes and their representations. The “functorial” point of view for schemes is adopted. The reader is supposed to have a reasonably good knowledge of varieties and algebraic groups. In Chapter I3, induction functors are defined in the context of group schemes, their elementary properties are proved, and they are used in order to construct injective modules and injective resolutions. These in turn are applied in Chapter I4 to the construction of derived functors, especially to that of the Hochschild cohomology groups and of the derived functors of induction. The values of the derived functors of inductions are interpreted in Chapter I5 as cohomology groups of certain associated bundles on the quotient G/H. Before doing that, one has to understand the construction of the quotient G/H. The situation gets simpler if H is normal in G. This is discussed in Chapter I6. The algebra of distributions on a group scheme G (called also hyperalgebra of G) is described in Chapter I7. The representation theory of a finite group scheme G is discussed in Chapter I8. A special case of this subject, when G arises as Frobenius kernels of algebraic groups over an algebraically closed field k of characteristic \(p\neq 0\), is discussed in Chapter I9. In the final chapter of Part I (Chapter I10) the general properties of the reduction mod p procedure are proved.

The contents of Part II is as follows. The purpose of Chapter II1 is to introduce split reductive group schemes and their most important subgroups, to fix a lot of notations and to mention without proof the main properties of these objects. The algebra of distributions on such a group scheme is described, and the relationship between the representation theories of the group and its algebra of distributions is discussed. Further G is assumed to be reductive, \(T\subset B\) are a fixed maximal torus and Borel subgroup, the ordering of X(T) is chosen in such a way that the weights of T in Lie(B) are negative, L(\(\lambda)\) denotes the simple module with the highest weight \(\lambda\). The main content of Chapter II2 is the discussion of the G-module \(H^ 0(\lambda)\), \(\lambda\in X(T)\), induced by a one-dimensional representation of B defined by \(\lambda\) : it is nonzero iff \(\lambda\) is dominant and L(\(\lambda)\) is the unique simple submodule of \(H^ 0(\lambda)\). Chapter II3 contains a rather simple proof of Steinberg’s tensor product theorem (i.e. the existence of a decomposition of L(\(\lambda)\) into a tensor product of the form \(L(\lambda_ 0)\otimes L(\lambda_ 1)^{(1)}\otimes...\otimes L(\lambda_ r)^{(r)}\) induced by a suitable p-adic expansion of \(\lambda)\), discovered by Cline, Parshall, and Scott. This is based on consideration of some irreducible representations of the Frobenius kernels. Chapter II4 contains the proof (due to Haboush and Andersen) of Kempf’s vanishing theorem, which shows that one special case of the Borel-Bott-Weil theorem holds for any field k (in full generality this theorem does not carry over to prime characteristic): if \(\lambda\) is dominant, then \(H^ i(\lambda)=0\) for all \(i>0\) (here \(H^ i(\mu)\) denotes the i-th cohomology group of a linear bundle of G/B associated to \(\mu)\). This theorem is crucial for the representation theory. In Chapter II5, it is given Demazure’s proof of the Borel-Bott-Weil theorem in the case \(char(k)=0:\) it describes explicitly all \(H^ i(\mu)\) with \(i\in {\mathbb{N}}\), \(\mu\in X(T)\), i.e. for each \(\mu\) there is at most one i with \(H^ i(\mu)\neq 0\), and this \(H^ i(\mu)\) can then be identified with a specific L(\(\lambda)\). Furthermore, it is proved (following Donkin) that Weyl’s character formula yields the alternating sum (over i) of the characters of all \(H^ i(\lambda)\). Chapter II6 is devoted to the linkage principle: if L(\(\mu)\) and L(\(\lambda)\) are both composition factors of a given indecomposable G- module, then \(\mu \in W_ p\cdot \lambda\) (here \(W_ p\) is the group generated by the Weyl group W and all translations by \(p\alpha\) with \(\alpha\) a root; the dot is to indicate a shift in the operation by \(\rho\), the half sum of the positive roots (i.e., \(w.\lambda =w(\lambda +\rho)-\rho))\). The general proof appeared only in 1980 (Andersen). Chapter II7 is devoted to the translation principle, a special case of which was conjectured by Verma: if two dominant weights \(\lambda\), \(\mu\) belong to the same “facet” with respect to the affine reflection group \(W_ p\), then the multiplicity of any L(w.\(\lambda)\) with \(w\in W_ p\) as a composition factor of \(H^ 0(\lambda)\) should be equal to that of L(w.\(\mu)\) in \(H^ 0(\mu)\). This was proved by the author. Kempf’s vanishing theorem implies that one can construct for any k the \(H^ 0(\lambda)\) with \(\lambda\) dominant by starting with the similar object over \({\mathbb{C}}\), taking a suitable lattice stable under a \({\mathbb{Z}}\)-form of G, and then tensoring with k. This approach allows the construction of certain filtrations of \(H^ 0(\lambda)\). One can compute the sum of the characters of the terms in the filtration for large p and use this information to get information about composition factors. All this is considered in Chapter II8. Chapter II9 contains information on representation theory of the group scheme \(G_ rT\) and the applications to some weak version of the Borel-Bott-Weil theorem and the construction of the composition factors of \(H^ 0(\lambda +p\nu)\) from those of \(H^ 0(\lambda)\) if \(\lambda\) and \(\lambda +p\nu\) for some \(\nu\in X(T)\) are the weights that are “small” with respect to \(p^ 2\) and are “sufficiently dominant”. Chapter II10 is devoted to the Steinberg modules \(St_ r\), which are both simple and injective as \(G_ r\)- modules, and some applications. It is proved (following Humphreys) that G is geometrically reductive. One may wonder whether any injective \(G_ r\)-module can be extended to a G-module; for large p this was proved by Ballard, and this is the subject of Chapter II11. The Hochschild cohomology groups \(H^ n(G,M)\) are discussed in Chapter II12, where the theorem due to Friedlander and Parshall is proved: for large p the cohomology ring \(H^.(G_ 1,k)\) is isomorphic to the ring of regular functions on the nilpotent cone in Lie(G). Chapters II13 and II14 are devoted to Schubert schemes and line bundles on these schemes: one can find there a proof of Kempf’s vanishing theorem for Schubert varieties, as well as a proof of the normality of such varieties and a character formula for the space of global sections.

The contents of Part I is as follows. Chapters I1 and I2 contain an introduction to schemes and to affine group schemes and their representations. The “functorial” point of view for schemes is adopted. The reader is supposed to have a reasonably good knowledge of varieties and algebraic groups. In Chapter I3, induction functors are defined in the context of group schemes, their elementary properties are proved, and they are used in order to construct injective modules and injective resolutions. These in turn are applied in Chapter I4 to the construction of derived functors, especially to that of the Hochschild cohomology groups and of the derived functors of induction. The values of the derived functors of inductions are interpreted in Chapter I5 as cohomology groups of certain associated bundles on the quotient G/H. Before doing that, one has to understand the construction of the quotient G/H. The situation gets simpler if H is normal in G. This is discussed in Chapter I6. The algebra of distributions on a group scheme G (called also hyperalgebra of G) is described in Chapter I7. The representation theory of a finite group scheme G is discussed in Chapter I8. A special case of this subject, when G arises as Frobenius kernels of algebraic groups over an algebraically closed field k of characteristic \(p\neq 0\), is discussed in Chapter I9. In the final chapter of Part I (Chapter I10) the general properties of the reduction mod p procedure are proved.

The contents of Part II is as follows. The purpose of Chapter II1 is to introduce split reductive group schemes and their most important subgroups, to fix a lot of notations and to mention without proof the main properties of these objects. The algebra of distributions on such a group scheme is described, and the relationship between the representation theories of the group and its algebra of distributions is discussed. Further G is assumed to be reductive, \(T\subset B\) are a fixed maximal torus and Borel subgroup, the ordering of X(T) is chosen in such a way that the weights of T in Lie(B) are negative, L(\(\lambda)\) denotes the simple module with the highest weight \(\lambda\). The main content of Chapter II2 is the discussion of the G-module \(H^ 0(\lambda)\), \(\lambda\in X(T)\), induced by a one-dimensional representation of B defined by \(\lambda\) : it is nonzero iff \(\lambda\) is dominant and L(\(\lambda)\) is the unique simple submodule of \(H^ 0(\lambda)\). Chapter II3 contains a rather simple proof of Steinberg’s tensor product theorem (i.e. the existence of a decomposition of L(\(\lambda)\) into a tensor product of the form \(L(\lambda_ 0)\otimes L(\lambda_ 1)^{(1)}\otimes...\otimes L(\lambda_ r)^{(r)}\) induced by a suitable p-adic expansion of \(\lambda)\), discovered by Cline, Parshall, and Scott. This is based on consideration of some irreducible representations of the Frobenius kernels. Chapter II4 contains the proof (due to Haboush and Andersen) of Kempf’s vanishing theorem, which shows that one special case of the Borel-Bott-Weil theorem holds for any field k (in full generality this theorem does not carry over to prime characteristic): if \(\lambda\) is dominant, then \(H^ i(\lambda)=0\) for all \(i>0\) (here \(H^ i(\mu)\) denotes the i-th cohomology group of a linear bundle of G/B associated to \(\mu)\). This theorem is crucial for the representation theory. In Chapter II5, it is given Demazure’s proof of the Borel-Bott-Weil theorem in the case \(char(k)=0:\) it describes explicitly all \(H^ i(\mu)\) with \(i\in {\mathbb{N}}\), \(\mu\in X(T)\), i.e. for each \(\mu\) there is at most one i with \(H^ i(\mu)\neq 0\), and this \(H^ i(\mu)\) can then be identified with a specific L(\(\lambda)\). Furthermore, it is proved (following Donkin) that Weyl’s character formula yields the alternating sum (over i) of the characters of all \(H^ i(\lambda)\). Chapter II6 is devoted to the linkage principle: if L(\(\mu)\) and L(\(\lambda)\) are both composition factors of a given indecomposable G- module, then \(\mu \in W_ p\cdot \lambda\) (here \(W_ p\) is the group generated by the Weyl group W and all translations by \(p\alpha\) with \(\alpha\) a root; the dot is to indicate a shift in the operation by \(\rho\), the half sum of the positive roots (i.e., \(w.\lambda =w(\lambda +\rho)-\rho))\). The general proof appeared only in 1980 (Andersen). Chapter II7 is devoted to the translation principle, a special case of which was conjectured by Verma: if two dominant weights \(\lambda\), \(\mu\) belong to the same “facet” with respect to the affine reflection group \(W_ p\), then the multiplicity of any L(w.\(\lambda)\) with \(w\in W_ p\) as a composition factor of \(H^ 0(\lambda)\) should be equal to that of L(w.\(\mu)\) in \(H^ 0(\mu)\). This was proved by the author. Kempf’s vanishing theorem implies that one can construct for any k the \(H^ 0(\lambda)\) with \(\lambda\) dominant by starting with the similar object over \({\mathbb{C}}\), taking a suitable lattice stable under a \({\mathbb{Z}}\)-form of G, and then tensoring with k. This approach allows the construction of certain filtrations of \(H^ 0(\lambda)\). One can compute the sum of the characters of the terms in the filtration for large p and use this information to get information about composition factors. All this is considered in Chapter II8. Chapter II9 contains information on representation theory of the group scheme \(G_ rT\) and the applications to some weak version of the Borel-Bott-Weil theorem and the construction of the composition factors of \(H^ 0(\lambda +p\nu)\) from those of \(H^ 0(\lambda)\) if \(\lambda\) and \(\lambda +p\nu\) for some \(\nu\in X(T)\) are the weights that are “small” with respect to \(p^ 2\) and are “sufficiently dominant”. Chapter II10 is devoted to the Steinberg modules \(St_ r\), which are both simple and injective as \(G_ r\)- modules, and some applications. It is proved (following Humphreys) that G is geometrically reductive. One may wonder whether any injective \(G_ r\)-module can be extended to a G-module; for large p this was proved by Ballard, and this is the subject of Chapter II11. The Hochschild cohomology groups \(H^ n(G,M)\) are discussed in Chapter II12, where the theorem due to Friedlander and Parshall is proved: for large p the cohomology ring \(H^.(G_ 1,k)\) is isomorphic to the ring of regular functions on the nilpotent cone in Lie(G). Chapters II13 and II14 are devoted to Schubert schemes and line bundles on these schemes: one can find there a proof of Kempf’s vanishing theorem for Schubert varieties, as well as a proof of the normality of such varieties and a character formula for the space of global sections.

Reviewer: V.L.Popov

##### MSC:

20G05 | Representation theory for linear algebraic groups |

20G10 | Cohomology theory for linear algebraic groups |

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

14L15 | Group schemes |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

14L17 | Affine algebraic groups, hyperalgebra constructions |

20G15 | Linear algebraic groups over arbitrary fields |