# An analogue of Deligne--Lusztig theory for real groups?

I am considering the following analogue of Deligne--Lusztig theory:

Take e.g. $G=GL_n(\mathbb{C})$, and let $F$ be the complex conjugate, then we have $G^F=GL_n(\mathbb{R})$. Consider the Lang map'' $L\colon g\mapsto g^{-1}F(g)$; it is usually no more surjective. Let $B$ be the standard Borel subgroup, $U$ its unipotent radical, and $T$ the standard maximal torus, then there is a natural action of $G(\mathbb{R})\times T(\mathbb{R})$ on the real manifold $L^{-1}(U)$, hence induce an action on a suitable cohomology of it.

In the Deligne--Lusztig theory for finite reductive groups with $F$ being the Frobenius, the $\ell$-adic cohomology of the varieties $L^{-1}(wU)$, $w$ runs over the Weyl group, produce all irreducible representations, parametrised by the characters of the tori. I am wondering what happens for the above real group settings?

It seems this is a very natural analogue, but a Google search does not return any reference. Is there some reason making this construction non-interesing for real reductive groups?

• I find it helpful to think about $\mathbb{P}^1 \setminus \mathbb{P}^1(\mathbb{F}_q)$ as somewhat analogous to the upper half plane (or perhaps more precisely the upper and lower half planes $\mathbb{P}^1 \setminus \mathbb{P}^1(\mathbb{R})$). In case of finite reductive groups one gets interesting representations (Drinfeld's observation), and holomorphic sections for $SL_2(\mathbb{R})$ gives the discrete series. Beyond that I have no idea. I guess these ideas are not pursued as much because BB localisation gives a pretty convincing picture for real groups. – Geordie Williamson Jan 21 '16 at 20:57
• I just realised that I am repeating myself: mathoverflow.net/questions/109461/… – Geordie Williamson Jan 21 '16 at 21:16