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added comment on field of definition
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George Melvin
George Melvin
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Yes: this is discussed in Chriss & Ginzburg 'Representation Theory and Complex Geometry', p.161 Remark 3.3.26. In short, the nilpotent cone is normal and we can apply Zariski's Main Theorem to deduce connectedness of Springer fibres. (This works for reductive $G$, and does not depend on working over $\mathbb{C}$; I'm not sure in what generality the normality condition holds though). This was originally proved by Spaltenstein via a different method, however - EDIT: see Jim Humphreys's answer for this.

Yes: this is discussed in Chriss & Ginzburg 'Representation Theory and Complex Geometry', p.161 Remark 3.3.26. In short, the nilpotent cone is normal and we can apply Zariski's Main Theorem to deduce connectedness of Springer fibres. (This works for reductive $G$). This was originally proved by Spaltenstein via a different method, however - EDIT: see Jim Humphreys's answer for this.

Yes: this is discussed in Chriss & Ginzburg 'Representation Theory and Complex Geometry', p.161 Remark 3.3.26. In short, the nilpotent cone is normal and we can apply Zariski's Main Theorem to deduce connectedness of Springer fibres. (This works for reductive $G$, and does not depend on working over $\mathbb{C}$; I'm not sure in what generality the normality condition holds though). This was originally proved by Spaltenstein via a different method, however - EDIT: see Jim Humphreys's answer for this.

Source Link
George Melvin
George Melvin
  • 1.2k
  • 8
  • 15

Yes: this is discussed in Chriss & Ginzburg 'Representation Theory and Complex Geometry', p.161 Remark 3.3.26. In short, the nilpotent cone is normal and we can apply Zariski's Main Theorem to deduce connectedness of Springer fibres. (This works for reductive $G$). This was originally proved by Spaltenstein via a different method, however - EDIT: see Jim Humphreys's answer for this.