See here for the property of etale cohomology of quotient variety. The key point is that when we reduce to , there may be cohomology groups, but they are all killed by , and hence stay so after passing to the limit over , so that when inverting , they disappear.
We would like to show geometric conjugacy class of ( is a -stable maximal torus and is a character of ) gives a partition of , where we say and are geometrically conjugate if there exists some such that and are -conjugate. The problem is that even they could still share irreducible characters since they are virtual characters.
Theorem (DM 13.3) If is a common irreducible constituent of and , then the pairs and are geometrically conjugate.
Proof: Let be the conjugate representation (isomorphic to the dual representation) and be the right -representation obtained from . The assumption implies that occurs in . Consider the variety where the notation means we quotient by the action . By Kunneth we know that the tensor product of cohomology is isomorphic to the cohomology of this variety as a -module-.
To understand the cohomology of this variety, we first note that this is isomorphic to , where the action is mapped to . The isomorphism is given by .
We further decompose the variety into where runs through the set of intertwiner from to (more precisely where ). For this we use the Bruhat decomposition .
Next we replace and by its twist by . More precisely, we introduce the variables , then
The actoion of now transfers to an action of . The benefit of doing this is this resembles more the transversal appearing in Mackey Formula. Now we can forget and redefine , and . We are going to use the decomposition .
Introduce There is an affine fibration given by , so the cohomology of is isomorphic to that of as a -module- if we equip with the action .
The idea now is to find a big enough torus commuting with the action of , since we know the vitural representation and are isomorphic. A natural idea is to take and , but for the action of to take to itself, it is easy to check that this happens iff . If we write , then since are commutative and (resp. ) is -stable (resp. -stable), we can check this is the same as .
We compute that , where is such that (and for this we also have and ).
https://www.math.columbia.edu/~chaoli/docs/LusztigClassification.html#ref-1