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?