Let $q=p^f$ be an odd prime power and $P$ be a maximal parobolic subgroup of $GO^\varepsilon(n,q)$ stabilising a totally singular $k$-subspace. It is known that $P$ has shape $A{:}(B\times C)$, where $A$ is a special $p$-group of order $q^{k(k-1)/2+k(n-2k)}$ with center of order $q^{k(k-1)/2}$, $B=GL(k,q)$ and $C=GO^\varepsilon(n-2k,q)$. Then what is the conjugation action of $B$ and $C$ on $A$?
On the structure of maximal parobolic subgroups of orthogonal groups over finite fields
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group-theory
finite-groups
1 Answers
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The special $p$-group $A$ has two elementary abelian layers, $M$ and $N$, of orders $q^{k(k-1)/2}$ and $q^{k(n-2k)}$, and these can be thought of as modules of dimensions $k(k-1)/2$ and $k(n-2k)$ for $B$ and $C$ over ${\mathbb F}_q$.
Then $M$ is the exterior square module for $B$ and the trivial module for $C$, whereas $N$ is the tensor product of the natural modules for $B$ and $C$. So $N$ is the direct sum of $(n-2k)$ copies of the natural module for $B$ and of $k$ copies of the natural module for $C$.
The parabolic subgroups of the other classical groups that preserve forms have similar structures.
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0Yes, "exterior square" is just an alternative name for "alternating square". – 2012-12-29