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Let $G$ be a connected reductive algebraic group over (say) $\mathbb{C}$. The set $\hat{G}_u$ of isomorphism classes of unipotent irreducible character sheaves has some complicated classification in terms of two-sided cells in $W$ and so on... I can't understand from the literature:

Should it be correct that $\hat{G}_u$ is in bijection with the set of isomorphism classes of irreducible $G^{\vee}$-equivariant perverse sheaves on the nilpotent cone of the Langlands-dual group? Does it follow "synthetically" from the complicated classification? Should there be some equivalence of categories explaining this bijection?

Thanks

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  • $\begingroup$ Unipotent character sheaves are certain simple G-equivariant perverse sheaves on G. There are subtleties, but basically they are those whose singular support is contained in the nilpotent cone (via T^*G = G x Lie algebra). Probably the most useful modern perspective is arxiv.org/abs/0902.1493. This is not geometric, but rather categorical (Drinfeld centre of Hecke category). Because the Hecke categories of G and its dual are the same, unipotent character sheaves admit the same classification for $G$ and $G^\vee$. (This is far from obvious from the definitions.) $\endgroup$ Jun 2, 2018 at 19:12
  • $\begingroup$ @GeordieWilliamson Thanks, but can you comment regarding my question? The unipotent character sheaves that enter the Springer sheaf can be parametrized by Irr(W), and hence by some G-equivariant irr. loc. sys. on the nilp. cone (IC extensions of trivial sheaves on orbits). I wondered whether all unipotent character sheaves should be in bijection with all G-equivariant irr. loc. sys. on the nilp. cone. $\endgroup$
    – Sasha
    Jun 2, 2018 at 20:40
  • $\begingroup$ Ahh sorry, I missed this part of the question. The unipotent character sheaves that show up as summands of the direct image from the Grothendieck-Springer resolution (whose Fourier transform is the Springer sheaf) are so-called "principal series". These are the "easy" character sheaves. The others are more complicated, and appear in types other than A (the first interesting example is $Sp_4$). Thus the answer to the question in the second paragraph is "no" outside of type A. $\endgroup$ Jun 3, 2018 at 5:37
  • $\begingroup$ @GeordieWilliamson Thanks, but again, my question is a bit further; I think that the principal series indeed matches with IC extensions of CONSTANT local systems on the G-orbits of the nilp. cone (is it true?); My question is, would it be true that all unipotent character sheaves match with all G-eq. irr. perverse sheaves on the nilp. cone? $\endgroup$
    – Sasha
    Jun 3, 2018 at 6:36
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    $\begingroup$ The only classification I know is via 2-sided cells: to each two sided cell $c$ is associate a finite group $A(c)$ (I think it is called "Lusztig's canonical quotient"), and character sheaves belonging to this cell are in bijection with simple conjugation equivariant sheaves on $A(c)$. $\endgroup$ Jun 4, 2018 at 5:52

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