Support varieties for modules over Chevalley groups and classical Lie algebras. (English) Zbl 1182.20039
Let \(G\) be a connected reductive \(\overline{\mathbb{F}}_p\)-algebraic group defined and split over the finite field \(\mathbb{F}_p\) of \(p\) elements, let \(G_1\) denote the first Frobenius kernel of \(G\), and let \(G(\mathbb{F}_p)\) denote the finite (untwisted) Chevalley group obtained as the \(\mathbb{F}_p\)-rational points of \(G\). Any rational \(G\)-module \(M\) can be considered as a \(G_1\)-module and as a \(G(\mathbb{F}_p)\)-module via restriction.
In 1987, B. J. Parshall [Representations of finite groups, Proc. Conf., Arcata/Calif. 1986, Pt. 1, Proc. Symp. Pure Math. 47, 233-248 (1987; Zbl 0649.20043)] asked whether the support variety \(|G_1|_M\) of \(M\) for \(G_1\) is directly related to the support variety \(|G(\mathbb{F}_p)|_M\) of \(M\) for \(G(\mathbb{F}_p)\)? Moreover, he asked whether the projectivity of \(M\) as a \(G_1\)-module implies the projectivity of \(M\) as a \(G(\mathbb{F}_p)\)-module?
In [Invent. Math. 138, No. 1, 85-101 (1999; Zbl 0937.17006)] the second and the third author of the paper under review have related the dimensions of these support varieties by the inequality \(\dim\,|G(\mathbb{F}_p)|_M\leq\frac 12\dim\,|G_1|_M\) (which immediately gives an affirmative answer to Parshall’s second question). In the paper under review the authors provide an affirmative answer to Parshall’s first question if \(p\) is good, the restricted nullcone is normal, and a certain technical condition is satisfied (which all hold if \(p\) is not smaller than the Coxeter number \(h\) of the corresponding root system).
As an application of their main result the authors extend results of J. L. Alperin and G. Mason [Bull. Lond. Math. Soc. 25, No. 6, 553-557 (1993; Zbl 0820.20017)] to non-simply laced root systems if \(q=p\geq h\) as well as the non-existence of non-projective periodic simple modules unless the rank of \(G\) is one due to I. Janiszczak and J. C. Jantzen [J. Lond. Math. Soc., II. Ser. 41, No. 2, 217-230 (1990; Zbl 0732.20024)] by using the geometry of nilpotent orbits in the Lie algebra of the corresponding algebraic group. Moreover, it is demonstrated how the complexities of \(G\)-modules over \(G(\mathbb{F}_p)\) can be computed provided their support varieties over \(G_1\) are known. In particular, the complexities of all simple \(G(\mathbb{F}_p)\)-modules are determined for any simple group of rank two and the complexity of the induced module \(H^0(\lambda)\) over \(\mathrm{GL}_n(\mathbb{F}_p)\) is computed for \(n\leq 5\) and any restricted \(\lambda\) as well as for \(\lambda=6\), \(7\) if \(\lambda\) is not large.
In 1987, B. J. Parshall [Representations of finite groups, Proc. Conf., Arcata/Calif. 1986, Pt. 1, Proc. Symp. Pure Math. 47, 233-248 (1987; Zbl 0649.20043)] asked whether the support variety \(|G_1|_M\) of \(M\) for \(G_1\) is directly related to the support variety \(|G(\mathbb{F}_p)|_M\) of \(M\) for \(G(\mathbb{F}_p)\)? Moreover, he asked whether the projectivity of \(M\) as a \(G_1\)-module implies the projectivity of \(M\) as a \(G(\mathbb{F}_p)\)-module?
In [Invent. Math. 138, No. 1, 85-101 (1999; Zbl 0937.17006)] the second and the third author of the paper under review have related the dimensions of these support varieties by the inequality \(\dim\,|G(\mathbb{F}_p)|_M\leq\frac 12\dim\,|G_1|_M\) (which immediately gives an affirmative answer to Parshall’s second question). In the paper under review the authors provide an affirmative answer to Parshall’s first question if \(p\) is good, the restricted nullcone is normal, and a certain technical condition is satisfied (which all hold if \(p\) is not smaller than the Coxeter number \(h\) of the corresponding root system).
As an application of their main result the authors extend results of J. L. Alperin and G. Mason [Bull. Lond. Math. Soc. 25, No. 6, 553-557 (1993; Zbl 0820.20017)] to non-simply laced root systems if \(q=p\geq h\) as well as the non-existence of non-projective periodic simple modules unless the rank of \(G\) is one due to I. Janiszczak and J. C. Jantzen [J. Lond. Math. Soc., II. Ser. 41, No. 2, 217-230 (1990; Zbl 0732.20024)] by using the geometry of nilpotent orbits in the Lie algebra of the corresponding algebraic group. Moreover, it is demonstrated how the complexities of \(G\)-modules over \(G(\mathbb{F}_p)\) can be computed provided their support varieties over \(G_1\) are known. In particular, the complexities of all simple \(G(\mathbb{F}_p)\)-modules are determined for any simple group of rank two and the complexity of the induced module \(H^0(\lambda)\) over \(\mathrm{GL}_n(\mathbb{F}_p)\) is computed for \(n\leq 5\) and any restricted \(\lambda\) as well as for \(\lambda=6\), \(7\) if \(\lambda\) is not large.
Reviewer: Jörg Feldvoss (Mobile)
MSC:
| 20G05 | Representation theory for linear algebraic groups |
| 20G40 | Linear algebraic groups over finite fields |
| 20G10 | Cohomology theory for linear algebraic groups |
| 20C33 | Representations of finite groups of Lie type |
| 17B55 | Homological methods in Lie (super)algebras |
| 17B50 | Modular Lie (super)algebras |
Keywords:
reductive algebraic groups; Frobenius kernels; finite Chevalley groups; support varieties; restricted nullcones; normal varieties; nilpotent orbitsReferences:
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