Is there an example of a representation $\rho: G \rightarrow GL(V)$ for some finite group $G$ where say $W \subset V$ is a $G$-invariant subspace for $\rho$ but the orthogonal complement (in the standard sense) $W^{\perp}$ is not G-invariant? I understand one could "unitarize" the representation using Weyl's averaging trick (and getting a new inner product) but my question is to find an example where one cannot do away with this averaging (and use the usual inner product)

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    $\begingroup$ What is the "usual inner product"? $\endgroup$ – Alex B. Jul 3 '20 at 12:31
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    $\begingroup$ Also, what if this usual inner product is not G-invariant? $\endgroup$ – David A. Craven Jul 3 '20 at 12:44
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    $\begingroup$ That’s not “Weyl’s unitary trick”. Weyl’s trick is to replace a noncompact group by a compact one to which Hurwitz’s averaging method then applies. $\endgroup$ – Francois Ziegler Jul 3 '20 at 13:10
  • $\begingroup$ @FrancoisZiegler Thanks for pointing out. I was using the term based on some lecture notes I stumbled upon. For instance see Theorem 3.10 here:math.berkeley.edu/~teleman/math/RepThry.pdf $\endgroup$ – Fdost Jul 13 '20 at 5:16

Try $G=\{1,-1\}$ and $\rho\colon G \to GL_2\mathbb{R}$ where $\rho(-1)$ is the matrix

$$\left(\begin{matrix}-1&2\\ 0&1\end{matrix}\right)$$

Take $W$ to be the span of

$$\left(\begin{matrix}1\\ 0\end{matrix}\right)$$

and use the standard inner product on $\mathbb{R}^2$.


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