Let $G$ be a connected linear algebraic group over an algebraically
closed field $k$ of characteristic $p$. An element $x\in G$ is
called *regular* if its centralizer has minimal dimension among
all the elements of $G$. Suppose now that $G$ is connected simple,
let $u\in G$ be a regular unipotent element, and let $U$ be the
unipotent radical of the Borel containing $u$. Then Springer [Some
arithmetical results on semi-simple Lie algebras] has shown that
the centralizer $C\_{U}(u)$ is connected, provided $p$ is a good
prime for $G$.


1. Is it true that if $p$ is a good prime for $G$, then $C\_{U}(x)$
is connected for all $x\in U$?
2. Let $V$ by an arbitrary connected unipotent group over $k$. Is
it possible to describe some 'small' subset $S$ of $V$, such that
the connectedness of $C\_{V}(x)$ for all $x\in S$ implies the connectedness
of $C\_{V}(x)$ for all $x\in V$? In particular, can $S$ be taken
to be the set of regular elements in $V$?



Connectedness of centralizers and regular elements in unipotent groups
Suppose that $G$ is semisimple and that the characteristic is *very good* for $G$. This means that the characteristic is good, and doesn't divide $n$ if $A\_{n-1}$ is an irreducible
component of the root system of $G$. [I'll ignore the issue of whether or not "good but not very good" primes present any real issue -- anyhow if you take GL(n) instead of SL(n) any problem should go away.]


Under this assumption, there is a $G$-equivariant isomorphism between the nilpotent variety and the unipotent variety of $G$ (a "Springer isomorphism"). This isomorphism gives by restriction an isomorphism $\operatorname{Lie}(U) \to U$ which is $B$-equivariant.


Now observe that $C\_{\operatorname{Lie}(U)}(x)$ is equal to the fixed points of the action of $Ad(x)$ on $\operatorname{Lie}(U)$, and is connected (since a linear space) 


The [choice of a] Springer isomorphism identifies $C\_{\operatorname{Lie}(U)}(x)$ with
the fixed points of $\operatorname{Int}(x)$ on $U$, namely with $C\_U(x)$. So indeed $C\_U(x)$
is connected.


