Let $G=GL_n(q)$ be the general linear group over $\mathbb{F}_q$ and $T$ be the torus of diagonal matrices. We also pick a Levi subgroup of the form $L=GL_{n_1}(q)\times GL_{n_2}(q) \times \cdots \times GL_{n_k}(q)$. In this context one can define the so-called Deligne-Lusztig functor $R_{L}^G.$ I'm interested in the case $R^G_L(1)$ and its connection with flag varieties.
More precisely,let fix a standard parabolic subgroup $P \supseteq L$: one can check that $R^G_L(1)=\mathbb{C}[G/P]$ and so $$R^G_L(1)(g)=\#\{x \in G/P \ \ \text{s.t} \ \ g\cdot x=x \} .$$
Now $G/P$ corresponds to $\mathbb{F}_q$ rational points of the flag variety: so, $$G/P=\{\mathbb{F}_q^n \supsetneq V_1 \supsetneq V_2 \supsetneq \cdots V_k \supsetneq 0\} $$ with differences of dimension prescribed by $L$ and with $R^G_L(1)$ we are counting fixed points of the $G$-action.
On the other hand there is a (more general) combinatorial formula for $R^G_L$ which specializes nicely in this context. Let $g=su$ be the Jordan decomposition and $M=C_{G}(s)$. We have $$R^G_L(1)(g)=\dfrac{1}{|W_L||M|}\sum_{w \in W_L}\sum_{ h | M\supseteq hT_wh^{-1} }Q^{M}_{hT_wh^{-1}}(u) $$ where $Q^M_{hT_wh^{-1}}$ are the Green functions and $T_w$ is the $w$-twisted form of the torus $T$.
Now, the proof I know of this last formula, which I read in the book by Digne/Michel "Representations of finite groups of Lie type " is very general (works for any $L$ and any reductive group) and algebraic/abstract somehow. I was wondering whether in this particular case there was a more explicit and maybe combinatorial connection between the counting of flags fixed by a certain element, Green functions and so on.
