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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 $A$?

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up vote 2 down vote accepted

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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Thanks! And what is the "exterior square module"? Is it the alternating square of the natural module of $B$? – Binzhou Xia Dec 29 '12 at 13:50
Yes, "exterior square" is just an alternative name for "alternating square". – Derek Holt Dec 29 '12 at 19:46

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