## Abstract

Let [Figure presented], be a reductive Lie algebra over an algebraically closed field F with charF=p>0. Suppose G satisfies Jantzen's standard assumptions. Then the structure of Z, the center of the enveloping algebra [Figure presented], is described by (the extended) Veldkamp's theorem. We examine here the deviation of Z from this theorem, in case [Figure presented] [Figure presented] or [Figure presented] and p|n. It is shown that Veldkamp's description is valid for [Figure presented]. This implies that Friedlander–Parshall–Donkin decomposition theorem for [Figure presented] holds in case p is good for a semi-simple simply connected G (excluding, if p=2, A_{1}-factors of G). In case [Figure presented] or [Figure presented] we prove a fiber product theorem for a polynomial extension of Z. However Veldkamp's description mostly fails for [Figure presented] and [Figure presented]. In particular Z is not Cohen–Macaulay if n>4, in both cases. Contrary to a result of Kac–Weisfeiler, we show for an odd prime that [Figure presented] and [Figure presented] do not generate [Figure presented]. We also show for [Figure presented] that the codimension of the non-Azumaya locus of Z is at least 2 (if n≥3), and exceeds 2 if n>4. This refutes a conjecture of Brown–Goodearl. We then show that Z is factorial (excluding [Figure presented]), thus confirming a conjecture of Premet–Tange. We also verify Humphreys conjecture on the parametrization of blocks, in case p is good for G.

Original language | English |
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Pages (from-to) | 217-290 |

Number of pages | 74 |

Journal | Journal of Algebra |

Volume | 504 |

DOIs | |

State | Published - 15 Jun 2018 |

### Bibliographical note

Publisher Copyright:© 2018 Elsevier Inc.

## Keywords

- Enveloping algebra
- Prime characteristic
- pgl
- psl
- sl

## ASJC Scopus subject areas

- Algebra and Number Theory

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