ON SUBGROUPS OF THE DIXMIER GROUP AND CALOGERO-MOSER SPACES

. We describe the structure of the automorphism groups of algebras Morita equivalent to the ﬁrst Weyl algebra A 1 ( k ). In particular, we give a geometric presentation for these groups in terms of amalgamated products, using the Bass-Serre theory of groups acting on graphs. A key rˆole in our approach is played by a transitive action of the automorphism group of the free algebra k (cid:104) x,y (cid:105) on the Calogero-Moser varieties C n deﬁned in [5]. In the end, we propose a natural extension of the Dixmier Conjecture for A 1 ( k ) to the class of Morita equivalent algebras.

Combining (2) and (3), we thus have the decomposition Aut k A 1 ∼ = A * U B , which completely describes the structure of Aut k A 1 as a discrete group (cf. [1]). The aim of the present paper is to generalize the above results to the case when A 1 is replaced by a noncommutative domain D, which is Morita equivalent to A 1 as a k-algebra. This question was originally raised by J. T. Stafford in [19] (see op. cit., p. 636). To explain why it is natural we recall that the algebras D are classified, up to isomorphism, by a single integer n ≥ 0 ; the corresponding isomorphism classes are represented by the endomorphism rings D n := End A1 (M n ) of the (fractional) ideals M n = x n A 1 + (y + nx −1 ) A 1 and can be realized geometrically as rings of global differential operators on some rational singular curves (see [14,6] and [9] for a detailed exposition). Thus, the Dixmier group Aut k A 1 = Aut k D 0 appears naturally as the first member in the family {Aut k D n : n ≥ 0} . Our aim is to describe the 'higher' groups in this family: specifically, to give a presentation of Aut k D n for arbitrary n ≥ 0 in terms of amalgamated products.
The present paper is mostly a research announcement: we focus here on explanation and motivation of our results; detailed proofs and computations will appear elsewhere.
2. Calogero-Moser spaces and the Makar-Limanov isomorphism. Recall that G 0 is the automorphism group of the free algebra k x, y preserving [x, y]. Now, for n > 0 , we introduce the groups G n geometrically, using a natural action of G 0 on the Calogero-Moser spaces where M n (k) denotes the space of n × n matrices with entries in k, and PGL n (k) operates on pairs of matrices (X, Y ) by simultaneous conjugation. The action of G 0 on C n is defined by where σ −1 (X) and σ −1 (Y ) are the noncommutative polynomials σ −1 (x) ∈ k x, y and σ −1 (y) ∈ k x, y evaluated at (X, Y ). In [21], G. Wilson showed that C n is a smooth irreducible affine variety of dimension 2n, equipped with a natural symplectic form: ω = tr(dX ∧ dY ) . Knowing that G 0 is generated by triangular transformations (1), it is easy to see that the action (8) is symplectic. Much less obvious is the fact that (8) is transitive for all n ≥ 0. This last fact was proven in [5] (Theorem 1.3), and it plays a crucial role in the present paper. Now, we define the groups G n to be the stabilizers of points of C n under the action (8): precisely, for each n ≥ 0, we fix a basepoint (X 0 , Y 0 ) ∈ C n , with

YURI BEREST, ALIMJON ESHMATOV, AND FARKHOD ESHMATOV
where E i,j stands for the elementary matrix with (i, j)-entry 1, and let The following result can be viewed as a generalization of the above-mentioned theorem of Makar-Limanov.
To construct the isomorphism of Theorem 1, we first note that the groups Aut k D n can be naturally identified with subgroups of Aut k D 0 . To be precise, let Pic k D denote the (noncommutative) Picard group of a k-algebra D. By definition, Pic k D is the group of k-linear Morita equivalences of the category of Dmodules; its elements can be represented by the isomorphism classes of invertible D-bimodules [P ] (see, e.g., [3], Ch. II, Sect. 5). There is a natural group homomorphism ω D : Aut k D → Pic k D , taking σ ∈ Aut k D to the class of the bimodule [ 1 D σ ], and if D is a ring Morita equivalent to D, with progenerator M , then there is a group isomorphism α M : Pic k D ∼ → Pic k D given by Thus, in our situation, for each n ≥ 0, we have the following diagram (10) where the vertical map α Mn is an isomorphism and the two horizontal maps are injective. Moreover, since D 0 = A 1 , a theorem of Stafford (see [19], Theorem 4.7) implies that ω D0 is actually an isomorphism. Inverting this isomorphism, we define the embedding i n : Aut k D n → Aut k D 0 , which makes (10) a commutative diagram. Now, we have group homomorphisms where the first map is the canonical inclusion and the second is Makar-Limanov's isomorphism (2). The claim of Theorem 1 is that the image of i n coincides with the image of G n in Aut k A 1 : this gives the required isomorphism G n ∼ → Aut k D n . Theorem 1 is a consequence of the main results of [5]. In fact, it is shown in [5] that there is a natural G 0 -equivariant bijection between n≥0 C n and the space of isomorphism classes of right ideals of A 1 . This bijection can be described explicitly as follows (see [4]). A point of C n is represented by a pair of linear endomorphisms (X, Y ) of k n satisfying the condition that [X, Y ] + I n has rank 1.
The assignment (X, Y ) → M (X, Y ) induces a map from C n to the set of isomorphism classes of right ideals of A 1 ; amalgamating such maps for all n yields the required bijection. Notice that M (X 0 , Y 0 ) = M n for X 0 and Y 0 given by (9), so our basepoints (X 0 , Y 0 ) ∈ C n correspond precisely to the classes of the ideals M n whose endomorphism rings are the algebras D n .
3. G n as a fundamental group. We will use Theorem 1 to give a geometric presentation for the groups Aut k D n . To this end, we associate to each space C n a graph Γ n consisting of orbits of certain subgroups of G 0 and identify G n with the fundamental group π 1 (Γ n , * ) of a graph of groups Γ n defined by stabilizers of those orbits in Γ n . The Bass-Serre theory of groups acting on graphs will then give an explicit formula for π 1 (Γ n , * ) in terms of generalized amalgamated products.
To state our results in precise terms we recall the notion of a graph of groups and its fundamental group, following Serre's classic book [17]. The readers unfamiliar with Bass-Serre theory are recommended to skim §5 of Chapter I of [17].
A graph of groups Γ = (Γ, G) consists of the following data: (1) a connected oriented graph Γ with vertex set V = V (Γ), edge set E = E(Γ) and incidence maps i, t : E → V , (2) a group G a assigned to each vertex a ∈ V , (3) a group G e assigned to each edge e ∈ E, (4) a pair of injective group homomorphisms α e : G e → G i(e) and β e : G e → G t(e) defined for each e ∈ E. Associated to Γ is the path group π(Γ) , which is given by the presentation where ' * ' stands for the free product (= coproduct in the category of groups) and E for the free group with basis set E = E(Γ). Now, choosing a maximal tree T in Γ, we define π 1 (Γ, T ) , the fundamental group of Γ relative to T , as a quotient of π(Γ) by 'contracting the edges of T to a point': precisely, For different maximal trees T ⊆ Γ, the groups π 1 (Γ, T ) are isomorphic. Moreover, if Γ is trivial (i. e. G a = {1} for all a ∈ V ), then π 1 (Γ, T ) is isomorphic to the usual fundamental group π 1 (Γ, a 0 ) of the graph Γ viewed as a CW-complex. In general, π 1 (Γ, T ) can be also defined in a topological fashion by introducing an appropriate notion of path and homotopy equivalence of paths in Γ. When its underlying graph is a tree (Γ = T ), Γ can be viewed as a directed system of groups {G i(e) αe ←− G e βe −→ G t(e) } indexed by the edges of T . In this case, the fundamental group π 1 (Γ, T ) is given by the inductive limit lim − → Γ , which is called the tree product of groups {G a } amalgamated by {G e } along T . For example, if T is a segment with V (T ) = {0, 1} and E(T ) = {e} , the tree product is the usual amalgamated free product G 0 * Ge G 1 . In general, abusing notation, we will denote the tree product by Now, we return to our situation. To define the graph Γ n we take the subgroups A, B and U of G 0 defined by the transformations (4), (5) and (6), respectively. Restricting the action of G 0 on C n to these subgroups, we let Γ n be the oriented bipartite graph, with vertex and edge sets (12) V (Γ n ) := (A\C n ) (B\C n ) , E(Γ n ) := U \C n , and the incidence maps E(Γ n ) → V (Γ n ) given by the canonical projections i : U \C n → A\C n and t : U \C n → B\C n . Since the elements of A and B generate G 0 and G 0 acts transitively on each C n , the graph Γ n is connected. Now, on each orbit in A\C n and B\C n , we choose a basepoint and elements σ A ∈ G 0 and σ B ∈ G 0 moving these basepoints to the basepoint (X 0 , Y 0 ) of C n .
Next, on each U -orbit O U ∈ U \C n , we also choose a basepoint and an element σ U ∈ G 0 moving this basepoint to (X 0 , Y 0 ) such that σ U ∈ σ A A ∩ σ B B , where σ A and σ B correspond to the (unique) A-and B-orbits containing O U . Then, we assign to the vertices and edges of Γ n the stabilizers A σ = G n ∩ σAσ −1 , B σ = G n ∩ σBσ −1 , U σ = G n ∩ σU σ −1 of the corresponding elements σ in the graph of right cosets of G 0 under the action of G n . These data together with natural group homomorphisms α σ : U σ → A σ and β σ : U σ → B σ define a graph of groups Γ n over Γ n , and its fundamental group π 1 (Γ n , T ) relative to a maximal tree T ⊆ Γ n has canonical presentation, cf. (11): .
In (13), the amalgam (A σ * Uσ B σ * . . .) stands for the tree product taken along the edges of T , while E(Γ n \ T ) denotes the free group based on the set of edges of Γ n in the complement of T . Our main result is the following Theorem 2. For each n ≥ 0, the group G n is isomorphic to π 1 (Γ n , T ) . In particular, G n has an explicit presentation of the form (13).
Theorems 1 and 2 reduce the problem of describing the groups Aut k D n to a purely geometric problem of describing the structure of the orbit spaces of A and B and U on the Calogero-Moser varieties C n . Using the earlier results of [21] and [5] and some geometric invariant theory, one can obtain much information about these orbits (and hence about the groups G n ). In particular, the graphs Γ n can be completely described for small n; it turns out that Γ n is a finite tree for n = 0, 1, 2 (see examples below), but has infinitely many cycles for n ≥ 3.
4. The graphs Γ n and an adelic Grassmannian. We now explain the origin of the graphs Γ n by realizing them as quotient graphs of a certain 'universal' tree Γ, on which all the groups Aut k D n naturally act. Our construction of Γ is motivated by algebraic geometry: specifically, an application of the Bass-Serre theory in the theory of surfaces (see, e.g., [13], [23]). In that approach, the automorphism group of an affine surface S is described via its action on a tree, whose vertices correspond to certain (admissible) projective compactifications of S. Following a standard philosophy in noncommutative geometry (see, e.g., [20]), we may think of our algebra D as the coordinate ring of a 'noncommutative affine surface'; a 'projective compactification' of D is then determined by a choice of filtration. Thus, we will define Γ by taking as its vertices a certain class of filtrations on the algebra D. It turns out that these filtrations can be naturally parametrized by an infinitedimensional adelic Grassmannian Gr ad introduced in [22] and studied in [21,5,8] (in particular, we rely heavily on results of [8]). Our contruction is close in spirit to Serre's classic application of Bruhat-Tits trees for computing arithmetic subgroups of SL 2 (K) over the function fields of curves (see [17], Chap. II, § 2); however, we are not aware of a direct connection.
We begin by briefly recalling the definition of Gr ad . Let k[z] be the polynomial ring in one variable z. For each λ ∈ k , we choose a λ-primary subspace in k[z]; that is, a k-linear subspace V λ ⊆ k[z] containing a power of the maximal ideal m λ at λ. We suppose that V λ = k[z] for all but finitely many λ's. Let V = λ V λ (such a subspace V is called primary decomposable in k[z]) and, finally, let where n λ is the codimension of V λ in k[z] . By definition, Gr ad k consists of all subspaces W ⊂ k(z) obtained in this way. For each W ∈ Gr ad k we set Taking Spec of A W then gives a rational curve X, the inclusion A W → k[z] corresponds to normalization π : A 1 k → X (which is set-theoretically a bijective map), and the A W -module W defines a rank 1 torsion-free coherent sheaf L over X . In this way, the points of Gr ad k correspond bijectively to isomorphism classes of triples (π, X, L) (see [22]). Now, following [5], for W ∈ Gr ad k we define 1 where k(z)[∂ z ] is the ring of rational differential operators in the variable z. This last ring carries two natural filtrations: the standard filtration, in which both generators z and ∂ z have degree 1, and the differential filtration, in which deg ( (15). Observe that the group Aut k D acts naturally on the set Gr ad k (D) (by composition), and this action induces an action of Aut k D on the graph Γ via (15). We write Aut k (D)\Γ for the corresponding quotient graph.
Theorem 3. (a) Γ is a tree, which is independent of D (up to isomorphism).
(b) For each n ≥ 0, the graph Aut k (D n )\Γ is naturally isomorphic to Γ n . 1 In geometric terms, D(W ) can be thought of as the ring D L (X) of twisted differential operators on X with coefficients in L. 2 More generally, we may think of Gr ad k as a groupoid, in which the objects are the W 's and the arrows are given by the algebra isomorphisms D(W ) → D(W ). For D = D(W ), the set Gr ad k (D) is then a costar in Gr ad k , consisting of all arrows with target at W . In [8], this set was denoted by Grad D .
Theorem 3 can be viewed as a generalization of the main results of [8]. Indeed, this last paper is concerned with a description of the maximal abelian ad-nilpotent (mad) subalgebras of D n : its main theorems (Theorem 1.5 and Theorem 1.6) say that the space Mad(D n ) of all mad subalgebras of D n is independent of D n and its quotient modulo the natural action of Aut k D n is isomorphic to the orbit space B\C n . Now, every admissible filtration of type B determines a mad subalgebra of D n , which is simply the degree zero component of that filtration. (Indeed, by definition, a type B filtration comes through an isomorphism from the usual differential filtration on some D(W ), but the degree zero component of the differential filtration is just A W , which is certainly a mad subalgebra of D(W ).) Thus, we have a well-defined map P B (D n ) → Mad(D n ) . This map is injective, since each type B filtration is maximal ad-nilpotent and hence determined by its degree zero component. On the other hand, Theorem 1.4 of [8] says that every mad subalgebra of D n arises from some W ∈ Gr ad : this implies the surjectivity of the above map. Summing up, we have a natural bijection P B (D n ) ∼ = Mad(D n ), which is equivariant under the action of Aut k D n . This means that P B (D n ) does not depend on D n , which is part of Theorem 3(a), and which is part of Theorem 3(b). The main part of Theorem 3 does not follow directly from the results of [8], although its proof relies on techniques of that paper.
In the end, we should mention that, for D = A 1 (k), our construction of the tree Γ agrees with the one given in [1].

Examples.
We now look at the graphs Γ n and groups G n for small n. For n = 0, the space C 0 is just a point, and so are a fortiori its orbit spaces. The graph Γ 0 is thus a segment, and the corresponding graph of groups Γ 0 is given by [ A U −→ B ] . Formula (13) then says that G 0 = A * U B , which agrees, of course, with Makar-Limanov's isomorphism (3), and G 0 is generated by its subgroups G 0,x := { Φ p ∈ G 0 : p ∈ k[x] } , G 0,y := { Ψ q ∈ G 0 : q ∈ k[y] } , which is the Dixmier result cited in the Introduction.