School of Mathematics Seminars and Lectures
Subgroups of a Chevalley group, containing the elementary subgroup over a subring
by Prof. Alexei STEPANOV (St. Petersburg State University, Russia)
Monday, October 31, 2011
from
to
(Asia/Kolkata)
at Colaba Campus ( AG-77 )
at Colaba Campus ( AG-77 )
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. |