Let $\pi:\tilde{\mathfrak{g}}\rightarrow \mathfrak{g}$, be the Grothendieck-Springer resolution, where $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$.

We know that $\pi$ is small thus $\pi_{*}\mathbb{Q}_{\ell}[\mathrm{dim}\,\mathfrak{g}]$ is an irreducible perverse sheaf on $\mathfrak{g}$.

Do we know on which stratas this sheaf is locally constant?

Let $\pi:\tilde{\mathfrak{g}}\rightarrow \mathfrak{g}$, be the Grothendieck-Springer resolution, where $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$.

We know that $\pi$ is small thus $\pi_{*}\mathbb{Q}_{\ell}[\mathrm{dim}\,\mathfrak{g}]$ is an irreducible perverse sheaf on $\mathfrak{g}$.

Is this sheaf locally constant on the stratas where the dimension of Springer fibers is constant?

Let $\pi:\tilde{\mathfrak{g}}\rightarrow \mathfrak{g}$, be the Springer resolution, where $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$.

We know that $\pi$ is small thus $\pi_{*}\mathbb{Q}_{\ell}$ is an irreducible perverse sheaf on $\mathfrak{g}$.

Do we know on which stratas this sheaf is locally constant?

Let $\pi:\tilde{\mathfrak{g}}\rightarrow \mathfrak{g}$, be the Grothendieck-Springer resolution, where $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$.

We know that $\pi$ is small thus $\pi_{*}\mathbb{Q}_{\ell}[\mathrm{dim}\,\mathfrak{g}]$ is an irreducible perverse sheaf on $\mathfrak{g}$.

Do we know on which stratas this sheaf is locally constant?