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Let $G$ be a reductive algebraic group (say $GL_n$) and $[\cdot,\cdot]: G \times G \to G$ the commutator map taking $(g,h) \mapsto ghg^{-1} h^{-1}$. Note that $[\cdot,\cdot]$ and therefore $R[\cdot,\cdot]_* \mathbb{Q}_{G \times G}$ is $G$-conjugation equivariant. So I want to think of it as being like a class function.

Is there a description of $R[\cdot,\cdot]_* \mathbb{Q}_{G \times G}$ in terms of character sheaves?

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If $G$ is over a finite field, then the trace of Frobenius of this complex is just a class function. Applied to an element of the group, this class function just counts the number of pairs of elements whose commutator is that element. This class function exists in every finite group , and can always be written in terms of characters. How to write it in terms of characters is a purely group-theoretic problem. Do you know how to solve that problem? –  Will Sawin Jun 12 '13 at 14:25
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Sure, but that only tells you at best the semisimplification of the perverse cohomology sheaves of the complex... –  Vivek Shende Jun 12 '13 at 16:44

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I don't really have an answer to this, but I have a number of comments that became too long.

First let me give some more context, to explain where I'm coming from: The quotient of $G\times G$ by the diagonal conjugation action parameterizes local systems on a punctured torus, and the commutator map $R$ describes the restriction of a local system to a circle around the puncture. Thus the complex $R_\ast \mathbb Q_{G\times G}$ encodes the cohomology of twisted moduli stacks of local systems on the torus (for example, the costalk at the identity element of $G$ is given by Borel-Moore chains on the moduli of local systems on a torus). More generally, one can consider local systems on any punctured surface together with the restriction to the monodromy around the puncture and obtain a similar complex.

As Will Sawin hinted at, the finite group analogue of this problem has a nice solution, sometimes referred to as the Frobenius formula. For example, Hausel and Rodriguez-Villegas use this formula in for the finite group $G(\mathbb F_q)$ to obtain information about the cohomology (and mixed Hodge structure) of the moduli of $G$ local systems. This can be related to your question via the "trace of Frobenius" operation.

I interpret your question as essentially asking for the categorified analogue of the Frobenius formula.

To see the kind of thing that could happen, it might be helpful to go through a simple

Example: $G=T=\mathbb C^\times$.

In this case, the commutator map $R$ is just the constant map at the identity, and thus the sheaf $R_\ast \mathbb Q_{T\times T}$ is supported at the identity of $T$. Character sheaves on $T$ are local systems. Thus you are essentially asking: "How to express the skyscraper sheaf at the identity in terms of local systems". Just as for finite abelian groups, the delta function at 1 is the sum of all characters (Plancherel formula), one can think of the skyscraper sheaf as an "integral" of local systems of all monodromies (this can be made precise via the Mellin transform for $D$-modules). One thing to note about this example is that the sheaf $R_\ast \mathbb Q$ is continuous in the central character parameter - it has no hope of being a direct sum of character sheaves (or extension etc.).

The answer to this question for general reductive groups, is the subject of ongoing work of myself with David Ben-Zvi and David Nadler. The idea is as follows: Pick a $\lambda \in \mathfrak h ^\ast$ (where $\mathfrak h$ is the Lie algebra of the torus of $G$). One can look at the projection $(R_\ast \mathbb Q)^\lambda$ of the sheaf $R_\ast \mathbb Q$ into the category of character sheaves (meaning the triangulated category generated by character sheaves) with central character $\lambda$. This complex has an interpretation in terms of the Character Theory topological field theory $\chi ^\lambda$ of Ben-Zvi and Nadler. More precisely, it is the object associated with the punctured torus, considered as a cobordism from $\emptyset$ to $S^1$. One consequence of this is that $(R_\ast \mathbb Q)^\lambda$ can be computed in terms of the combinatorics of the Hecke Category/Character Sheaves (it is not clear to me how to do this at the moment though). Sorry, I can't give a reference...

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Indeed, the H-RV formula is of course exactly why I was asking, recently I was wondering if it would admit some easy categorification which saw the weights via character sheaves. That's a pretty helpful explanation... –  Vivek Shende Jun 13 '13 at 1:58
    
I meant, "saw more than just the weights" –  Vivek Shende Jun 13 '13 at 1:58

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