# Complexification of $G-$representation where $\mathrm{End}_G(V)\cong\mathbb{C}.$

Let $$G$$ be a compact Lie group and $$V$$ a finite dimensional real $$G$$ representation. Suppose that $$\mathrm{End}_G(V)\cong \mathbb{C}.$$ I want to show that the complexification $$V\otimes_{\mathbb{R}}\mathbb{C}\cong W_1\oplus W_2$$ for some non isomorphic, complex irreducible representations $$W_1,W_2.$$

My progress so far: Since $$G$$ is a compact lie group and $$V$$ is a finite dimensional $$G$$-representation, we may write $$V=W_1^{\oplus n_1}\oplus\dots\oplus W_k^{\oplus n_k}.$$ $$\mathbb{C}\cong \mathrm{End}_G(V)\cong\mathrm{Mat}_{n_1}(\mathbb{C})\oplus \dots\mathrm{Mat}_{n_k}(\mathbb{C}),$$ which is only possible if $$k=1$$ and $$n_1=1.$$ Thus, we have $$V\otimes_{\mathbb{R}}\mathbb{C}\cong W_1\otimes_{\mathbb{R}}\mathbb{C}.$$ I am a bit stuck as to where to go from here. Am I on the right lines? Also what happens if $$\mathrm{End}_G(V)\cong \mathbb{H}$$ instead?

The key idea is that the center of a compact Lie group is trivial, meaning that the only possible irreducible real representations of G are trivial.

Let's analyze each case:

EndG(V)≅C: Based on the Schur orthogonality relations, the only possible irreducible representation of G that appears in the decomposition of V is the trivial representation. Therefore, V must be of the form:

V = W⊕nk,

where W is the trivial representation and nk is the multiplicity of the trivial representation in V.

Since V⊗RC≅W1⊗RC≅1⊗1, we must have W≅1.

EndG(V)≅H: Here, a non-trivial representation of G appears in the decomposition of V. This implies that k>1, and V cannot be isomorphic to W⊗RC. Therefore, if G is a compact Lie group and V is a finite-dimensional real G-representation such that EndG(V)≅C, then its complexification V⊗RC≅W1⊗W2≅1⊗1 for some non-isomorphic, complex irreducible representations W1,W2. However, if EndG(V)≅H, then V⊗RC is not isomorphic to the tensor product of two non-isomorphic, complex irreducible representations.

• Thanks very much! Nov 14, 2023 at 17:12
• Please use MathJax. Nov 14, 2023 at 18:18
• Also, I do not see how you conclude $W \simeq 1$ from what you have; $W$ was defined as the trivial representation just before that anyway. You do not show what OP asks about non-isomorphic $W_1$, $W_2$. Nov 14, 2023 at 18:23
• Also, your very first sentence seems wrong. Nov 14, 2023 at 18:24

So I believe that I have an answer. We know that $$\mathrm{End}_G(V\otimes_{\mathbb{R}}\mathbb{C})\cong \mathrm{End}_GV\otimes_{\mathbb{R}}\mathbb{C}\cong\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}\cong \mathbb{C}\oplus \mathbb{C}.$$ Since $$G$$ is a compact Lie group and $$V$$ is a finite dimensional representation of $$G,$$ so is $$V\otimes_{\mathbb{R}}\mathbb{C},$$ which implies that $$V\otimes_{\mathbb{R}}\mathbb{C}$$ is completely reducible. Hence $$V\otimes_{\mathbb{R}}\mathbb{C}=\bigoplus_{j=1}^kW_i^{\oplus n_j},$$ where the $$n_j$$ are non-isomorphic to one another and are irreducible representations. This implies $$\mathrm{End}_G(V\otimes_{\mathbb{R}}\mathbb{C})\cong \mathrm{Mat}_{n_1}(\mathbb{C})\oplus\dots \mathrm{Mat}_{n_k}(\mathbb{C}).$$ Comparing this to the isomorphism above yields that $$n_1,n_2=1$$ and $$n_i=0$$ for $$i>2.$$ A similar argument, noting that the complexification of $$\mathbb{H}\cong \mathrm{Mat}_2(\mathbb{C}),$$ can be used to figure out the case where $$\mathrm{End}_G(V)=\mathbb{H}.$$