Support varieties for Weyl modules over bad primes. (English) Zbl 1123.20038
The study of cohomological support varieties has led to numerous advances in many areas of representation theory. However, there remains a need for explicit computations of support varieties. Let \(G\) be a reductive algebraic group over an algebraically closed field of prime characteristic \(p>0\), and let \(G_1\) denote the first Frobenius kernel of \(G\). The representation theory of \(G_1\) is equivalent to the representation theory of the restricted enveloping algebra of the Lie algebra of \(G\). A fundamental family of modules for \(G\) is the set of Weyl modules (parameterized by the dominant weights). These can be restricted to modules over \(G_1\). Under the assumption that \(p\) is good, the support varieties over \(G_1\) of Weyl modules were computed by D. K. Nakano, B. J. Parshall, and D. C. Vella [J. Reine Angew. Math. 547, 15-49 (2002; Zbl 1009.17013)], confirming a conjecture of J. C. Jantzen [Bull. Lond. Math. Soc. 19, 238-244 (1987; Zbl 0623.17008)].
The main result of this paper is a computation of these support varieties when \(p\) is not a good prime. In the bad prime case, a uniform description of the support varieties cannot be given. However, the authors do show that for all primes a common formula holds for the dimension of the support varieties in terms of a certain subset of roots determined by the highest weight of the Weyl module.
In general, the support variety of a \(G_1\)-module is contained in the restricted nullcone of the Lie algebra of \(G\). Moreover, in the case that the module is a rational \(G\)-module (such as a Weyl module), the support variety is closed under the adjoint action of \(G\) on its Lie algebra. In the good prime case, the support variety of a Weyl module is always the closure of a Richardson orbit. The computations here involve investigations of orbits and previous computations of the restricted nullcone over bad primes by the authors [J. Algebra 292, No. 1, 65-99 (2005; Zbl 1124.17003)].
The paper concludes with some interesting observations: these support varieties are irreducible; for bad primes, the closures of all but possibly three orbits in the restricted nullcone can be realized as the support variety of a module; and all but one Richardson orbit in the restricted nullcone has closure equal to the support variety of a Weyl module.
The main result of this paper is a computation of these support varieties when \(p\) is not a good prime. In the bad prime case, a uniform description of the support varieties cannot be given. However, the authors do show that for all primes a common formula holds for the dimension of the support varieties in terms of a certain subset of roots determined by the highest weight of the Weyl module.
In general, the support variety of a \(G_1\)-module is contained in the restricted nullcone of the Lie algebra of \(G\). Moreover, in the case that the module is a rational \(G\)-module (such as a Weyl module), the support variety is closed under the adjoint action of \(G\) on its Lie algebra. In the good prime case, the support variety of a Weyl module is always the closure of a Richardson orbit. The computations here involve investigations of orbits and previous computations of the restricted nullcone over bad primes by the authors [J. Algebra 292, No. 1, 65-99 (2005; Zbl 1124.17003)].
The paper concludes with some interesting observations: these support varieties are irreducible; for bad primes, the closures of all but possibly three orbits in the restricted nullcone can be realized as the support variety of a module; and all but one Richardson orbit in the restricted nullcone has closure equal to the support variety of a Weyl module.
Reviewer: Christopher P. Bendel (Menomonie)
MSC:
| 20G05 | Representation theory for linear algebraic groups |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 17B45 | Lie algebras of linear algebraic groups |
| 17B50 | Modular Lie (super)algebras |
| 20G10 | Cohomology theory for linear algebraic groups |
Keywords:
cohomology; Weyl modules; support varieties; Frobenius kernels; reductive algebraic groups; restricted enveloping algebras; dominant weights; Richardson orbitsReferences:
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