I would ask this question in a comment but I don't have enough reputation to comment yet. So I am studying the paper of Mirkovic and Vilonen "Characteristic Varieties of character sheaves" and I am trying to understand Lusztig's construction of $K_w^{\mathcal{L}}$. In a comment in How to understand character sheaves it was mentioned that if $G=GL(2)$, $\mathcal{L}$ is trivial and $w$ is the nontrivial element of the Weyl group, $K_w^{\mathcal{L}}$ under function sheaf correspondance is $q[Triv]-[St]$. I'm trying to apply the definitions but I am kind of stuck. Could someone explicitly explain what do you get by Lusztig's construction in this simple example?
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1$\begingroup$ Do these notes help? people.mpim-bonn.mpg.de/geordie/Ostrik.pdf I think Example 1.6 is pretty close to what you want... $\endgroup$– Geordie WilliamsonCommented Nov 14, 2019 at 17:05
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$\begingroup$ They do very much! So please tell me if I am correct but the fact that characters are compatible with Grothendieck's six operations should immediately imply the calculation by the character of the line bundle. But there are many points which are not clear to me for example: In 1.3, how is not $n^*E$ not always a sum of constant sheaves? Could you give me an example? Also, how did we get these characteristic functions in example 1.6? $\endgroup$– Ioannis ZolasCommented Nov 18, 2019 at 0:29
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$\begingroup$ Also, even if we use the characteristic function for the finite field case, what do we do in the complex case (this is of greater interest for me right now), where we don't have a Frobenius? Isn't there some way to understand $K_w^{\mathcal{L}}$ straight from the definition? $\endgroup$– Ioannis ZolasCommented Nov 18, 2019 at 1:09
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