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

Question

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?

Answer 1

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.

Answer 2

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 $\mathrm{End}_G(V)=\mathbb{H}.$