degeneration of reductive group If $A$ is a mixed characteristic complete DVR (I'm only actually interested in $\mathbf{Z}_p$) and $G/A$ is a closed subgroup scheme of $GL(n)$ whose generic fibre is connected reductive and split, is the identity component of the special fibre also connected reductive and split?
Oh dear I'm getting very confused about this question. Is there a closed subgroup scheme of $GL(2)$ over $A$ whose generic fibre is trivial but whose special fibre is a Borel subgroup of $GL(2)$? Is life really as bad as that?
 A: This may also be SGA3's example, since I am too lazy to go check, but my standard example of a reductive group degenerating to something solvable is to define a multiplication $\ast_t$ on $2 \times 2$ matrices by
$$\begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} \ast_t \begin{pmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{pmatrix} = \begin{pmatrix} x_{11} y_{11} + t x_{12} y_{21} & x_{11} y_{12} + x_{12} y_{22} \\
x_{21} y_{11} + x_{22} y_{21} & t x_{21} y_{12} + x_{22} y_{22} \end{pmatrix}.$$
For $t \neq 0$, this is simply the ordinary $2 \times 2$ matrices under the change of coordinates $\left( \begin{smallmatrix} z_{11} & z_{12} \\ t z_{21} & z_{22} \end{smallmatrix} \right)$, so it is associative, and it gives a group when restricted to the open set $z_{11} z_{22} - t z_{21} z_{12} \neq 0$. I hope this shows you that there is nothing mysterious in the given formulas.
In the limit $t=0$, the above formulas still define an associative multiplication, which forms a group $G_0$ on the open set $z_{11} z_{22} \neq 0$. Explicitly,
$$\begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} \ast_0 \begin{pmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{pmatrix} = \begin{pmatrix} x_{11} y_{11} & x_{11} y_{12} + x_{12} y_{22} \\
x_{21} y_{11} + x_{22} y_{21} &  x_{22} y_{22} \end{pmatrix}.$$
Clearly, there is a map $G_0 \to \mathbb{G}_m^2$ projecting a matrix onto its diagonal entries. The kernel, matrices of the form $\left( \begin{smallmatrix} 1 & x_{12} \\ x_{21} & 1 \end{smallmatrix} \right)$, is clearly a copy of $\mathbb{G}_a^2$. So we have a short exact sequence $1 \to \mathbb{G}_a^2 \to G_0 \to \mathbb{G}_m^2 \to 1$ and $G_0$ is solvable.
As grghxy points out, every flat affine group scheme of finite type over a dvr embeds in some $GL_N$, so you are not adding anything with that condition.
This example can be modified to suit many tastes: Replace $z_{11} z_{22} - t z_{12} z_{21} \neq 0$ by $z_{11} z_{22} - t z_{12} z_{21} = 1$ if you like simple Lie groups better than reductive ones; quotient by $\pm \mathrm{Id}$ if you like your groups in adjoint form; work with $2 \times 2$ matrices and $\ast_t$ commutator if you like Lie algebras. Replace $t$ by $p$ in the above formulas to work over $\mathbb{Z}_p$.
A: SGA3 XIX section 5 has a terrifying example of $PGL(2)$ degenerating into something solvable. Thanks to grghxy.
