What are Harish-Chandra bimodules used for?

Question

There are many recent papers on classification of Harish-Chandra bimodules for rational Cherednik algebras and, more generally, non-commutative algebras which are quantizations of symplectic singularities (Losev). What is the meaning of Harish-Chandra bimodules in terms of representation theory of the underlying algebra/its category O? Are Harish-Chandra bimodules related to the classical notion of Harish-Chandra modules?

Answer 1

Here is an answer from a mathematician who prefers me to post it here myself:

Harish-Chandra bimodules make sense in a very wide context. Take two filtered algebras A, A' that quantize the same commutative algebra $C$, and fix isomorphisms ${\rm gr} A \to C$, ${\rm gr} A^{'} \to C$. Then one can make sense of the definition of a $HC (A, A^{'})$-bimodule. These are (A, A')-bimodules, say B, that admit a filtration such that \gr B is a finitely generated C-module, meaning that the left and right actions on C coincide. It is not hard to see that if A, A' are $U(g)$ for a simple Lie algebra g, this coincides with the notion of HC bimodule that I alluded above.

In the context of symplectic singularities, note that you need to have a Hamiltonian $\mathbb C^*$-action to define the category O. Such an action does not always exist (e.g. for Kleinian singularities outside of type A). In this sense, HC bimodules are a substitute for the category O. See for example Ginzburg https://arxiv.org/pdf/0807.0339.pdf

When you do have categories O, HC bimodules give, via tensor product, functors between categories O for different quantization parameters. For example, projective functors in Lie theory are a special case of tensoring with a HC $U(g)$-bimodule. In this sense, HC bimodules also generalize the notion of projective functors. Translation functors for Cherednik algebras are a special case of this. I must warn, however, that tensoring with a HC bimodule is in general a very bad functor -- it can kill many things and it is not exact. Nevertheless, these functors were used by Losev to construct derived equivalences between categories O for Cherednik algebras https://arxiv.org/pdf/1406.7502.pdf

Also, Harish-Chandra bimodules are much more sensitive to the quantization parameter than the category O is. Category O always has the same number of simples = number of fixed points under Hamiltonian torus action. This is far from being true for HC bimodules. For example, for type A Cherednik algebras the quantization parameter is a complex number $c$ (I apologize if I am overexplaining, I don't know how familiar you are with these). If $c$ is not a rational number with denominator $1 < d \leq n$ ($n =$ rank of symmetric group) then the category O is semisimple and equivalent to reps of $S_n$. This is not true for the category of HC $H_{c}$-bimodules. For these parameters, the category is still semisimple, but it is only equivalent to reps of $S_n$ when c is an integer. Otherwise, it is equivalent to Vec. In this sense, HC bimodules detect how integral the parameter is. See https://arxiv.org/pdf/1409.5465.pdf Theorem 1.1 for the case of rational Cherednik algebras (the subgroup $W_{c}$ essentially detects how far c is from being integral). This was generalized by Losev to symplectic singularities in https://arxiv.org/pdf/1810.07625.pdf

One more thing, the simplest example of a HC $A$-bimodule is the regular bimodule. So one can use HC bimodules to answer questions about, for example, ideals in $A$ (usually these techniques come from constructing restriction functors for HC bimodules, similar to the Bezrukavnikov-Etingof functors for category O and applying them to the regular bimodule). This was used by Losev for Cherednik algebras in https://arxiv.org/pdf/1001.0239.pdf (see Thms 1.3.1 and 5.8.1) and for finite W-algebras in https://arxiv.org/pdf/0807.1023.pdf

Finally, in the context of symplectic resolutions it is believed that HC bimodules should categorify the homology of the generalized Steinberg variety. This is of course not true in general (even for Cherednik algebras for the reasons above -- for some parameters there are simply not enough irreducibles) but it should be true for integral parameters, for an appropriate notion of integral. See Braden-Proudfoot-Webster, https://arxiv.org/pdf/1208.3863.pdf Proposition 6.16 (later in that paper they show that wall-crossing functors are always tensoring with an appropriate HC bimodule, Proposition 6.23)