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

1$\begingroup$ To answer your last question, HarishChandra bimodules for $\mathfrak g$ are $(\mathfrak g\times\mathfrak g,G)$modules. So they are in particular the HarishChandra modules associated to representations of $G(\mathbb C)$ thought of as a real Lie group. But there is much more to be said... $\endgroup$– Sam GunninghamCommented Feb 6, 2020 at 16:32

$\begingroup$ @SamGunningham Thanks a lot! So is there a notion of HC modules for more general algebras that are quantizations of symplectic resolutions? Or is there only a notion of HC bimodules which replaces it? Also, since you mentioned that there is a lot more to be said, can you hint what exactly that is? $\endgroup$– Yellow PigCommented Feb 6, 2020 at 16:39
1 Answer
Here is an answer from a mathematician who prefers me to post it here myself:
HarishChandra 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 Cmodule, 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, HarishChandra 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 BezrukavnikovEtingof 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 Walgebras 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 BradenProudfootWebster, https://arxiv.org/pdf/1208.3863.pdf Proposition 6.16 (later in that paper they show that wallcrossing functors are always tensoring with an appropriate HC bimodule, Proposition 6.23)