# Borel-Weil-Bott, Langlands and Hitchin

Let $G$ be a compact semi-simple Lie group and $G_\mathbb{C}$ be its complexification. We denote by $B$ a 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 induced representation of $G$ on $H^0(G_\mathbb{C}/B,\mathcal{L}_\lambda)$ is indeed the highest weight representation $L_\lambda$ of $G$.

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, 1608.01754, 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.

• A couple of small comments: Bott's further work is not directly relevant here, while a search for older questions using just 'Borel-Weil' turns up quite a few (which may or may not be helpful to you). Also, "the" Borel subgroup $B$ should be "a" Borel subgroup; since all of these are conjugate, a fixed choice is OK. But to avoid complications in the definition of the line bundle, it's usually simplest to let $B$ correspond to the negative rather than positive roots, relative to a fixed maximal torus contained in $B$. – Jim Humphreys Oct 16 '16 at 17:36
• Langlands' re-interpretation of work of Harish-Chandra explains how discrete series representations of (not necessarily compact) real or complex reductive groups are in correspondence with certain discrete combinatorial data (which could even be certain weights of a maximal torus, my memory of the details is not good). A reference is Knapp's book on semisimple Lie groups. – znt Oct 16 '16 at 21:35