Tensor product over Lie group isomorphic to that over its Lie algebra

Question

Hi: I have a question about the tensor product of representation of Lie groups as follows:

Let $G$ be a connected, complex Lie group with Lie algebra $\mathfrak{g}$. Let $V$ and $W$ are representations with respect to $G$, and so these induce representations (with respect to $\mathfrak{g}$) on $V$ and $W$. That is, we obtain module structures: $V$ and $W$ are $\mathbb{C}[G]$-modules and $U(\mathfrak{g})$-modules as well (where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$)). Is it true that $V\otimes_{\mathbb{C}[G]}W \cong V\otimes_{U(\mathfrak{g})}W$ in the category of vector spaces?

Thanks very much!

Answer 1

As Tobias already indicated in his comment, usually you consider the tensor product of two $G$- resp. ${\mathfrak g}$-modules over ${\mathbb C}$, but in case you mean $X\otimes_G Y$ resp. $X\otimes_{\mathfrak g} Y$ as $(X\otimes_{\mathbb C} Y)^G$ resp. $(X\otimes_{\mathbb C} Y)^{\mathfrak g}$ (that's the same as providing $X$ with the right ${\mathscr U}({\mathfrak g})$-structure given by $x.v := -v.x$ for $x\in X$ and $v\in{\mathfrak g}$ and then taking the usual tensor product $X\otimes_{{\mathscr U}({\mathfrak g})} Y$) the question splits in two parts:

• First, check that the functor $(G\text{-mod},\otimes_{\mathbb C})\to({\mathfrak g}\text{-mod},\otimes_{\mathbb C})$ is monoidal, that is, that deriving the diagonal $G$-action $g.(v\otimes w) := (g.v)\otimes (g.w)$ on the tensor product $V\otimes_{\mathbb C}W$ of two $G$-modules results in the usual ${\mathfrak g}$-action $X.(v\otimes w) := (X.v)\otimes w + v\otimes (X.w)$ on $V\otimes_{\mathbb C} W$.

  You can do this by decomposing $G\to\text{GL}(V\otimes_{\mathbb C} W)$ as $$G\xrightarrow{\Delta} G\times G\xrightarrow{\rho_V\times\rho_W}\text{GL}(V)\times\text{GL}(W)\xrightarrow{\alpha}\text{GL}(V\otimes_{\mathbb C} W).$$ and noting that the derivative of $\Delta$ at $e$ is again $\Delta: {\mathfrak g}\to{\mathfrak g}\oplus{\mathbb g}$, while $\text{d}(\rho_V\times\rho_W)_e = \text{d}(\rho_V)_e\times\text{d}(\rho_W)_e$, and finally $\alpha$ is the restriction of the bilinear map $\text{End}(V)\times\text{End}(W)\xrightarrow{\alpha}\text{End}(V\otimes_{\mathbb C}W)$ the derivative of which is (as for any bilinear map) given by $\text{d}(\alpha)_{(f,g)}(\sigma,\tau) = \alpha(f,\tau)+\alpha(\sigma,g) = f\otimes\tau + \sigma\otimes g$.

• Secondly, check that taking $G$-invariants $V\mapsto V^G:=\{v\in V\ |\ g.v=v\text{ for all } g\in G\}$ is the same as taking ${\mathfrak g}$-invariants $V\mapsto V^{\mathfrak g} := \{v\in V\ |\ X.v=0\text{ for all } X\in{\mathfrak g}\}$ if $G$ is connected - see $Hom_G(\pi,\sigma)$ = $Hom_{\mathfrak{g}}(d\pi,d\sigma)$?.