intersections of $SO_2^n, SL_2^n$ with $SO_{2n}, Sp_{2n}$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T05:53:38Z http://mathoverflow.net/feeds/question/87052 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87052/intersections-of-so-2n-sl-2n-with-so-2n-sp-2n intersections of $SO_2^n, SL_2^n$ with $SO_{2n}, Sp_{2n}$ J. Martel 2012-01-30T18:10:09Z 2012-02-02T12:34:34Z <p>Let $\mathbb{R}^{2n}$ be endowed with the canonical symplectic structure $(\omega, J)$, where $\omega$ the usual nondegenerate symplectic form and $J$ the usual almost complex structure (ie. $\omega(\cdot, J\cdot)$ gives the usual dot product). </p> <p>Suppose we have $n$ 2-planes $P_i$ which give an $\omega$-orthogonal decomposition $\mathbb{R}^{2n}=\oplus_{i=1 \ldots n} P_i$ (ie. $\omega(P_i, P_j)=0$ for $i \neq j$). Then both $SO(P_1) \times \ldots \times SO(P_n)$ and $SL(P_1) \times \ldots \times SL(P_n)$ acts as symmetries summandwise.</p> <p>We also have two symmetry groups $SO(2n), Sp_{2n}$ acting on $\mathbb{R}^{2n}$ defined relative to the fixed structure $(\omega, J)$. The question is then: determine the intersections $SO(2n) \cap. SO(P_1) \times \ldots \times SO(P_n)$ and $Sp_{2n} \cap. SL(P_1) \times \ldots \times SL(P_n)$. </p> http://mathoverflow.net/questions/87052/intersections-of-so-2n-sl-2n-with-so-2n-sp-2n/87059#87059 Answer by algori for intersections of $SO_2^n, SL_2^n$ with $SO_{2n}, Sp_{2n}$ algori 2012-01-30T19:56:09Z 2012-02-02T12:34:34Z <p>[upd: this answers the old version of the question, which has since been changed.]</p> <p>The intersection is $SO(2)^n=SO(2)\times SO(2)\times\cdots\times SO(2)$, as perhaps expected. The inclusion $SO(2)^n\subset (SL_2)^n\cap SO(2n)$ is clear. The other inclusion follows from this observation: If a linear map that preserves a vector subspace is to be orthogonal, it must preserve the metric restricted to the subspace.</p>