Equivalence between Harish-Chandra bimodules and category O

Question

let $\mathfrak{g}$ be a complex semisimple Lie algebra and fix Borel subalgebra and Cardan subalgebra, then we have the BGG category $\mathcal{O}$. It is well known that for a given regular, dominant weight $\lambda \in \mathfrak{h}^*$ there is an equivalence between the category of Harish-Chandra bimodules $\mathcal{H}^1_{\chi_{\lambda}}$ and $\mathcal{O}$ via $\cdot \otimes M(\lambda)$.

By definition, $\mathcal{H}^1_{\chi_{\lambda}}$ is the full category of $\mathfrak{g}\times \mathfrak{g}$-bimodule $N$ such that the $N^{ad}$ is locally finite and semisimple over $\mathfrak{g}$, where $N^{ad}$ is the representation given by $x\cdot n = xn-nx$, for all $x\in \mathfrak{g}$ and $n\in N$, and in addition, $nz = \chi_{\lambda(z)}n$, for all central element $z$ in $U\mathfrak{g}$ and $n \in N$.

My question: Why $N\otimes_{\mathfrak{g}}M(\lambda) \in \mathcal O$, for all Harish-Chandra bimodule $N \in \mathcal{H}^1_{\chi_{\lambda}}$? Thanks very much in advance!

Answer 1

To lie in category $\mathcal{O}$, you need to be $U(\mathfrak{b})$-locally finite, and $U(\mathfrak{h})$ semi-simple. Any element $ v\in N\otimes_{U(\mathfrak{g})}M(\lambda)$ lies in the image of $N_0\otimes_{\mathbb C}M_0$, where $N_0$ is a finite dimensional adjoint invariant subspace of $N$, and $M_0$ is a finite dimensional $U(\mathfrak{b})$-invariant subspace of $M(\lambda)$, since $v$ is a finite sum of tensors $n_i\otimes m_i$, and the $n_i$’s generate $N_0$ under the adjoint action, and $m_i$’s generate $M_0$ under $U(\mathfrak{b})$.

The tensor product $N_0\otimes_{\mathbb C}M_0$ carries the usual tensor product action of $U(\mathfrak{b})$, and the map to $N\otimes_{U(\mathfrak{g})}M(\lambda)$ is equivariant. The image of $N_0\otimes_{\mathbb C}M_0$ is finite-dimensional, $U(\mathfrak{b})$-invariant, and the action of $U(\mathfrak{h})$ is semi-simple, since all these things are true of the two factors. Since every $v$ lies in such a subspace, you are in category $\mathcal{O}$.

Note, the central character condition isn’t needed for this result, but tensor product is unchanged by passing to the largest quotient of the HC bimodule satisfying this condition (since the center acts on $M(\lambda)$ by the same scalar).