# intersections of $SO_2^n, SL_2^n$ with $SO_{2n}, Sp_{2n}$

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).

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.

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 $Sp_{2n} \cap. SL(P_1) \times \ldots \times SL(P_n)$.

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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.
Actually I don't see the first inclusion as clear at all: this because $SO_2^n$ (as embedded in $SL_2^n \subset Sp_{2n}$) is not contained in $SO_{2n}$. Let us remind ourselves that $Sp_{2n} \cap SO_{2n}=U_n$--the unitary group. –  J. Martel Jan 30 '12 at 20:25
forgive that last comment. Under all the usual embeddings we certainly do have $SO_2^n$ contained in $SO_{2n}$. What I obected to was the idea that a linear mapping which restricts to an isometry on certain transverse 2-planes must be an isometry of the ambient space. I should clarify the situation i had in mind: –  J. Martel Jan 30 '12 at 20:43
Let us have a decomposition of a symplectic vector space $(V,\omega)$ into $\omega-orthogonal$ 2-planes $\oplus_{i=1 \ldots n} P_i$. Then we have $n$ copies of $SO_2$---one for each 2-plane $P_i$. What I wanted to consider was the quotient of these 2-plane isometries $SO_2^n$ modulo the isometries of the ambient space. To be totally explicit I was asking about the intersection of $\Pi_i SO(P_i)$ with $SO(V)$. Actually it's the quotient $\Pi_i SO(P_i) / SO(V)$ i'm interested in. –  J. Martel Jan 30 '12 at 20:51
jmart -- I was presuming that the metric that $SO(2n)$ preserves is the direct sum of the standard metrics on each on the 2-planes in the decomposition $\mathbb{R}^{2n}=\mathbb{R}^2\oplus\cdots \mathbb{R}^2$. If this is the case, then $SO(2n)$ definitely contains $SO(2)^n$. If not, then could you please specify what the metric is and also what skew-linear form you use to define $Sp_n$. –  algori Jan 30 '12 at 20:51
jmart -- another confusing point: do you take an embedding of a cartesian power of $SL_2$, or the diagonal embedding of $SL_2$ itself? –  algori Jan 30 '12 at 20:53