Algebras Morita equivalent with the Weyl Algebra and its smash products with a finite group

Question

My question os motivated, naturally, by the problem of classifying symplectic reflection algebras up to Morita equivalence (a classical reference for rational Cherednik algebras is Y. Berest, P. Etingof, V. Ginzburg, "Morita equivalence of Cherednik algebras", MR2034924; the most up do date work in this subject I know of is I. Losev, Derived equivalences for Symplectic reflection algebras, https://arxiv.org/abs/1704.05144);

and also the problem of understading rings of differential operators on irreducilbe affine complex varieties $X$ up to Morita equivalence (a nice discussion of this lovely problem in the intersection of ring theory and algebraic geometry can be found in Y. Berest, G. Wilson, "Differential isomorphism and equivalence of algebraic varieties", MR2079372)

Given that, my questions are:

(Question 1): What are the more general known conditions on a symplectic reflection algebra $H_{1,c}(V,\Gamma)$ that imples it is Morita equivalent to $\mathcal{D}(V) \rtimes \Gamma$?

(Question 2): What are the recent developments made in the study of equivalence of rings of differential operators up to Morita equivalence (and in particular Morita equivalent to the Weyl algebra) since Berest, Wilson [op. cit.]?

(Question 3): Etingof in "Cherednik and Hecke algebras of varieties with a finite group action", MR3734656, introduces more general versions of rational Cherednik algebras and discuss the possibility of extending the results in Y. Berest, O. Chalykh, Quasi-invariants of complex reflection groups ,MR 2801407, in this setting. So, being optmistic, one hipothetically could obtain results similar as those discussed in Berest, Etingof, Ginzburg [op. cit] regarding Morita equivalence of these generalized rational Cherednik algebras with smash products of rings with differential operatos with a finite groups. Has this line of inquiry lead to results relevant to this discussion so far?

(Question 4): This is totally unrelated to the previous questions. It is more of a very open question in ring theory: are there interesting simple Noetherian algebras, coming from another areas than those above, which are Morita equivalent to a Weyl algebra or a smash product of it with a finite group?

Answer 1

I do not know much on recent developments related to the first three questions asked. However, i know of some old results related mainly to the fourth question:

If $A_1$ is the Weyl algebra over an alg closed field of zero char, with the two generators denoted $p,q$ and $I$ is a non-zero, right ideal, then $M_2(End_{A_1}(I))\cong M_2(A_1)$ and $A_1$ is Morita equivalent to $End_{A_1}(I)$. Furthermore, these algebras are not generally isomorphic: Pick for example $I=p^2A_1+(pq+1)A_1$. Its endomorphism ring is isomorphic to $\{x\in Q|xI\subseteq I\}$, where $Q$ is the quotient division ring of $A_1$. This is not isomorphic to $A_1$ but it is Morita equivalent to it. If you are interested in this example, this is presented in An example of a ring Morita equivalent to the Weyl algebra $A_1$, S.P. Smith, J. of Alg, 73, 552 (1981).

Another result which may be of interest -regarding your fourth question- is that:

If the semigroup $k\Lambda$ has the same quotient field with $k[t]$, then $D(K)$ is Morita equivalent to $A_1$.

Here $K$ stands for certain subalgebras of $k[t]$ and $D(K)$ for the ring of differential operators on $K$. This is shown in: Some rings of differential operators which are Morita equivalent to $A_1$, Ian Musson, Proc. of the Am. Math. Soc., 98, 1, 1986

Finally, if you are interested in examples involving smash products with finite group algebras, i do not have some readily available but i think it is natural to look for such in the graded version of Morita equivalence.

I hope these are of some interest to the OP. Sorry in advance if these are too old and you are already aware of them.

P.S.: One more thing which might be of some interest with respect to the second question: The article Rings graded equivalent to the Weyl algebra, J. of Alg., vol. 321, 2, 2009, generalizes some results of Y. Berest, G. Wilson and Stafford, in the setting of graded module categories. (also i think this article is the first -although i am not sure- which introduces the terminology "graded Morita equivalence")