The Green functions were first introduced by Green to compute the character table of \(\textrm{GL}_n(\mathbb {F}_q)\) in 1955. They were later generalized by Deligne and Lusztig for an arbitrary finite group of Lie type \(G(\mathbb {F}_q)\) using \(\ell \) -adic cohomological methods (1976). They proved that these Green functions satisfy an orthogonality relation (we call the first orthogonality relation). Ten years later Kawanaka proved that they satisfy an other orthogonality relation (we call the second orthogonality relation). In this note, we explain how the main results of our paper (Laumon and Letellier.: Forum Math. Sigma 11, 2023) provide a geometric understanding of these two orthogonality relations and how we can see geometrically that the two orthogonality relations are in fact equivalent, i.e. one can be obtained from the other one and vice-versa. In this geometric approach the language of stacks is essential.
