# Example of a representation of a finite group where Weyl's unitary trick is necessary?

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)

• What is the "usual inner product"? – Alex B. Jul 3 '20 at 12:31
• Also, what if this usual inner product is not G-invariant? – David A. Craven Jul 3 '20 at 12:44
• 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. – Francois Ziegler Jul 3 '20 at 13:10
• @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 – Fdost Jul 13 '20 at 5:16

## 1 Answer

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 $$\mathbb{R}^2$$.