# How to detect if a subgroup lands inside an orthogonal group?

Equivalently, my question may be phrased as, "Are there defining characteristics of representations of orthogonal (symmetric form-preserving) groups?"

Here I am working with a unitary representation of subgroups of the reductive group $GL(n, \mathbb{C})$, and I am seeking a criterion which I may apply to representations of arbitrary $H \leq GL(n, \mathbb{C})$.

For $(V, \rho)$ a representation of $H$ such that $H \leq O(n, \Bbb C)$, one such criterion that I had thought may be true is that this representation must necessarily be self-dual i.e., isomorphic to the dual representation. Is this in fact the case?

Any references would be appreciated. I am currently reading Dynkin's "Maximal Subgroups of the Classical Groups" in search of a lead. Many thanks for taking the time to read this.

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Yes, of course that is the case. The natural representation of the orthogonal group is self dual,hence so is its restriction to $H$. But not conversely: the symplectic group also has self dual natural representation. –  Aakumadula May 18 at 15:03
There are criteria to tell if the form that the "self-dual" group $H$preserves is orthogonal or symplectic, in terms of the action by a particular element of the centre of $H$. All this is classical, and I am sure is in standard textbooks. –  Aakumadula May 18 at 15:05
@Aakumadula thank you. I will consult Fulton and Harris, I will surely find it in there –  Josh Izzard May 18 at 15:13