In characteristic 0 or good prime characteristic, there are standard ways to relate the unipotent variety $\mathcal{U}$ of a simple algebraic group $G$ and the nilpotent variety $\mathcal{N}$ of its Lie algebra $\mathfrak{g}$. Recall that $p$ is good just when it fails to divide any coefficient of the highest root (the root system being irreducible), bad otherwise. The only possible bad primes are $2,3,5$. In characteristic 0, algebraic versions of the exponential and logarithm maps provide Ad $G$-equivariant isomorphisms in both directions, whereas in good characteristic $p>0$, the less direct arguments of Springer here yield similar results. [The isogeny type of $G$ adds some complications here.]

There is scattered literature on the varieties $\mathcal{U}$ and $\mathcal{N}$ when $p$ is bad, often treated case-by-case, e.g., four papers by Lusztig posted on arXiv starting here, along with papers by his student T. Xue. A serious challenge when $p$ is bad is to find a uniform explanation for the failure of the numbers of unipotent classes and nilpotent orbits to agree in some cases: the details were worked out by Holt-Spaltenstein and others. In spite of this breakdown in $G$-equivariance, a natural question can be raised:

Are the two varieties $\mathcal{U}$ and $\mathcal{N}$ isomorphic in all characteristics, for example when $G$ is simply connected?

The answer does not seem to be written down explicitly (?), but for example one can see indirectly that both varieties have the same dimension in all characteristics: the number of roots. A paper here by S. Keny, a former student of Steinberg, showed case-by-case that regular nilpotent elements always exist and form a dense orbit in $\mathcal{N}$. By definition, the isotropy group in $G$ of such an element has dimension equal to the rank of $G$.

In characteristic 0 or good prime characteristic, there are standard ways to relate the unipotent variety $\mathcal{U}$ of a simple algebraic group $G$ and the nilpotent variety $\mathcal{N}$ of its Lie algebra $\mathfrak{g}$. Recall that $p$ is good just when it fails to divide any coefficient of the highest root (the root system being irreducible), bad otherwise. The only possible bad primes are $2,3,5$. In characteristic 0, algebraic versions of the exponential and logarithm maps provide Ad $G$-equivariant isomorphisms in both directions, whereas in good characteristic $p>0$, the less direct arguments of Springer in The unipotent variety of a semi-simple group yield similar results. [The isogeny type of $G$ adds some complications here.]

There is scattered literature on the varieties $\mathcal{U}$ and $\mathcal{N}$ when $p$ is bad, often treated case-by-case, e.g., four papers by Lusztig posted on arXiv starting with Unipotent elements in small characteristic, along with papers by his student T. Xue. A serious challenge when $p$ is bad is to find a uniform explanation for the failure of the numbers of unipotent classes and nilpotent orbits to agree in some cases: the details were worked out by Holt–Spaltenstein and others. In spite of this breakdown in $G$-equivariance, a natural question can be raised:

Are the two varieties $\mathcal{U}$ and $\mathcal{N}$ isomorphic in all characteristics, for example when $G$ is simply connected?

The answer does not seem to be written down explicitly (?), but for example one can see indirectly that both varieties have the same dimension in all characteristics: the number of roots.   Existence of regular nilpotent elements in the Lie algebra of a simple algebraic group in bad characteristics by S. Keny, a former student of Steinberg, showed case-by-case that regular nilpotent elements always exist and form a dense orbit in $\mathcal{N}$. By definition, the isotropy group in $G$ of such an element has dimension equal to the rank of $G$.

In characteristic 0 or good prime characteristic, there are standard ways to relate the unipotent variety $\mathcal{U}$ of a simple algebraic group $G$ and the nilpotent variety $\mathcal{N}$ of its Lie algebra $\mathfrak{g}$. Recall that $p$ is good just when it fails to divide any coefficient of the highest root (the root system being irreducible), bad otherwise. The only possible bad primes are $2,3,5$. In characteristic 0, algebraic versions of the exponential and logarithm maps provide Ad $G$-equivariant isomorphisms in both directions, whereas in good characteristic $p>0$, the less direct arguments of Springer here yield similar results. [The isogeny type of $G$ adds some complications here.]

There is scattered literature on the varieties $\mathcal{U}$ and $\mathcal{N}$ when $p$ is bad, often treated case-by-case, e.g., four papers by Lusztig posted on arXiv starting here, along with papers by his student T. Xue. A serious challenge when $p$ is bad is to find a uniform explanation for the failure of the numbers of unipotent classes and nilpotent orbits to agree in some cases: the details were worked out by Holt-Spaltenstein and others. In spite of this breakdown in $G$-equivariance, a natural question can be raised:

Are the two varieties $\mathcal{U}$ and $\mathcal{N}$ isomorphic in all characteristics, for example when $G$ is simply connected?

The answer does not seem to be written down explicitly (?), but for example one can see indirectly that both varieties have the same dimension in all characteristics: the number of roots. A paper here by S. Keny, a former student of Steinberg, showed case-by-case that regular nilpotent elements always exist and form a dense orbit in $\mathcal{N}$. By definition, the isotropy group in $G$ of such an element has dimension equal to the rank of $G$.

In characteristic 0 or good prime characteristic, there are standard ways to relate the unipotent variety $\mathcal{U}$ of a simple algebraic group $G$ and the nilpotent variety $\mathcal{N}$ of its Lie algebra $\mathfrak{g}$. Recall that $p$ is good just when it fails to divide any coefficient of the highest root (the root system being irreducible), bad otherwise. The only possible bad primes are $2,3,5$. In characteristic 0, algebraic versions of the exponential and logarithm maps provide Ad $G$-equivariant isomorphisms in both directions, whereas in good characteristic $p>0$, the less direct arguments of Springer here yield similar results. [The isogeny type of $G$ adds some complications here.]

There is scattered literature on the varieties $\mathcal{U}$ and $\mathcal{N}$ when $p$ is bad, often treated case-by-case, e.g., four papers by Lusztig posted on arXiv starting here, along with papers by his student T. Xue. A serious challenge when $p$ is bad is to find a uniform explanation for the failure of the numbers of unipotent classes and nilpotent orbits to agree in some cases: the details were worked out by Holt-Spaltenstein and others. In spite of this breakdown in $G$-equivariance, a natural question can be raised:

Are the two varieties $\mathcal{U}$ and $\mathcal{N}$ isomorphic in all characteristics, for example when $G$ is simply connected?

The answer does not seem to be written down explicitly (?), but for example one can see indirectly that both varieties have the same dimension in all characteristics: the number of roots. A paper here by S. Keny, a former student of Steinberg, showed case-by-case that regular nilpotent elements always exist and form a dense orbit in $\mathcal{N}$. By definition, the isotropy group in $G$ of such an element has dimension equal to the rank of $G$.