| Description |
All rings are assumed to be commutative with 1.
Let $K$ be a ring, let $S\subseteq A$ be $K$-algebras and $G$ an algebraic group.
This is a well known problem: to describe lattice of subgroups between
$G(S)$ and~$G(A)$.
The talk is about this problem for a Chevalley--Demazure group scheme
$G=\G(\Phi)$
with a root system $\Phi\ne A_1$ over $K=\Z$. For a ring $R$ let $E(R)=
\E(\Phi)$
denote the elementary subgroup of $G(R)$. We consider a slightly bigger lattice,
namely, the lattice of subgroups between $E(S)$ and $G(A)$.
The standard description of this lattice is called standard sandwich calssification(SSC).\begin{defn}
Fix a triple $(\Phi,S,A)$. The SSC holds if given a subgroup
$H$ between $E(S)$ and $G(A)$ there exists a unique subring $R$ between
$S$ and $A$ such that
$$
E(R)\le H\le N_A(R)
$$
\noindent
where $N_A(R)$ denotes the normalizer of $E(R)$ in $G(A)$.
\end{defn}
Recently I have proved that for doubly laced root systems (i.\,e.
$\Phi=B_l,C_l$
the SSC holds for an arbitrary pair of rings provided that 2 is invertible in
$R$.
Together with another my result and a result of Ya. Nuzhin this gives a final
answer
to the question for which field extensions $A/S$ and root systems the SSC holds.
By simple group theoretical arguments the SSC can be extended to subgroups of
$G(A)$ nomalized by $E(S)$.
I shall exhibit known results and my new results mentioned above and
show the main steps of the proof illustrating them with examples of $G=\SL_n$
(if the step gets through for this group scheme) or $G=\Sp_{2n}$.
Also I shall state a conjecture about the final answer and show some immediate
problems in frames of this conjecture.
|