Let $G$ be a complex algebraic group with simple Lie algebra $\mathfrak{g} = \operatorname{Lie} G$. Let $\mathcal{B}$ be the flag variety consisting of all the Borel subalgebras of $\mathfrak{g}$.
Let $\mathcal{N} \subseteq \mathfrak{g}$ be the nilpotent cone, choose a nilpotent element $e \in \mathcal{N}$ and extend it to an $\mathfrak{sl}_2$-triple $\{ e, h, f \} \subseteq \mathfrak{g}$. The Slodowy variety is $$ \Lambda_e = \left\{ (x, \mathfrak{b}) \in \mathcal{N} \times \mathcal{B} \mid x \in \mathfrak{b} \text{ and } [x, f] = h \right\}. $$ Letting $W$ be the Weyl group of $G$, for a simple reflection (with respect to a chosen Borel subalgebra $\mathfrak{b}_0$) $s_\alpha \in W$ let $$ Z_{\alpha, e} = \{ (x, \mathfrak{b}_1, \mathfrak{b}_2) \in \mathcal{N} \times Y_\alpha \mid x \in \mathfrak{b}_1 \cap \mathfrak{b}_2 \text{ and } [x, f] = h \}. $$ where $Y_\alpha \subseteq \mathcal{B} \times \mathcal{B}$ is closure of the orbit of $(\mathfrak{b}_0, s_\alpha \mathfrak{b}_0)$ under the diagonal action of $G$. $Y_\alpha$ is known to be smooth, in fact it is a $\mathbb{CP}^1$-bundle over $\mathcal{B}$.
I have a few questions:
- Is $Z_{\alpha, e}$ a smooth variety?
- What is the tangent space of $Z_{\alpha, e}$ at a point? Here, I'm looking for an answer which is analogous to the identification $T_\mathfrak{b} \mathcal{B} \cong \mathfrak{g}/\mathfrak{b}$ where $\mathfrak{b} \in \mathcal{B}$.
- Similarly, what are the tangent spaces to $\Lambda_e$?
