Connectedness of Springer Fibers

Question

Let $G$ be a connected, simply-connected, complex semisimple Lie group with Lie algebra $\frak{g}$. Let $\mu:T^*\mathcal{B}\rightarrow\mathcal{N}$ be the Springer resolution of $\mathcal{N}$. If $G=\operatorname{SL}_n(\mathbb{C})$, then the Springer fibers $\mu^{-1}(e)$, $e\in\mathcal{N}$, are known to be connected projective varieties. Are the Springer fibers known to be connected varieties for arbitrary $G$?

Answer 1

The answer is yes. I'm not sure what sources you are working with, but much of this theory originates in the work of Spaltenstein (Lecture Notes in Math. 946, Springer, 1982). While the fibers are connected, they are not irreducible as varieties but the irreducible components are shown to be of equal dimension, etc. For most of this treatment, there is no need to work over the complex field or even in characteristic 0, by the way. Expositions were given in Steinberg's Tata Institute lectures (written up by Deodhar) and in my 1995 AMS book Conjugacy Classes in Semisimple Algebraic Groups where the connectedness theorem is proved in section 6.5.

It's worth adding that much remains unknown about the Springer fibers in terms of their precise geometric structure and their cohomology. But Spaltenstein developed a lot of detailed information about dimensions, numbers of irreducible components, and the like.

Answer 2

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.