Let $G$ be a compact semi-simple Lie group and $G_\mathbb{C}$ be its complexification. We denote $B$ by the Borel subgroup of $G_\mathbb{C}$.

Given a dominant weight $\lambda$, one can construct a line bundle by means of Borel construction
$$
\mathcal{L}_\lambda:=G_\mathbb{C}\times_{\chi_\lambda} \mathbb{C}\rightarrow G_\mathbb{C}/B
$$
where $\chi_\lambda:B \to \mathbb{C}^\times$ is the extension of the character to $B$. Then, the Borel-Weil-Bott theorem states that the natural action of $G$ on $H^0(G_\mathbb{C}/B,\mathcal{L}_\lambda)$ is indeed the highest weight representation $L_\lambda$.

I heard that generalizations of the Borel-Weil-Bott theory for non-compact groups are related to Langlands duality. How are they related? Moreover, since Kapustin-Witten has formulated geometric Langlands duality as Fourier-Mukai transformations of Hitchin moduli spaces (S-duality of 4d N=4 TQFT), I wonder if one can connect the Borel-Weil-Bott theory for non-compact groups to Hitchin moduli spaces.

The motivation of this question comes from the (equivariant) Verlinde formula for Hitchin moduli spaces (See [1501.01310](https://arxiv.org/abs/1501.01310), [1608.01754](https://arxiv.org/abs/1608.01754), [1608.01761](https://arxiv.org/abs/1608.01761)). The Borel-Weil-Bott theorem for compact groups can be considered as a toy model of the Verlinde formula for $Bun_G$. I wonder how one can formulate Borel-Weil-Bott theory corresponding to (equivariant) Verlinde formula for Hitchin moduli spaces.