[Paper Review] Dixmier Groups and Borel Subgroups

Yuri Berest, Alimjon Eshmatov+1 more|arXiv (Cornell University)|Jan 28, 2014

Advanced Topics in Algebra55 references3 citations

TL;DR

This paper studies the Dixmier groups $G_n$, defined as automorphism groups of rings of differential operators on rational singular curves of differential genus $n$, via their action on Calogero-Moser varieties $\mathcal{C}_n$. It establishes that the action of $G$ on $\mathcal{C}_n$ is doubly transitive for $n \geq 1$ and diagonally transitive on products of distinct $\mathcal{C}_{n_i}$, proving that stabilizers $G_n$ are maximal subgroups and providing explicit descriptions of Borel subgroups via conjugation actions on polynomial generators.

ABSTRACT

Let G be the group of symplectic (unimodular) automorphisms of the free associative algebra on two generators. A theorem of G.Wilson and the first author asserts that G acts transitively the Calogero-Moser spaces C_n for all n. We generalize this theorem in two ways: first, we prove that the action of G on C_n is doubly transitive, meaning that G acts transitively on the configuration space of (ordered) pairs of points in C_n; second, we prove that the diagonal action of G on the product of (any number of) copies of C_n is transitive provided the corresponding n's are pairwise distinct. In the second part of the paper, we study the isotropy subgroups G_n of G in C_n. We equip each G_n with the structure of an ind-algebraic group and classify the Borel subgroups of these ind-algebraic groups for all n. Our classification shows that every Borel subgroup of G (= G_0) is conjugate to the subgroup B of triangular (elementary) automorphisms; on the other hand, for n > 0, the conjugacy classes of Borel subgroups of G_n are parametrized by certain orbits of B in C_n. Our main result is that the conjugacy classes of non-abelian Borel subgroups of G_n correspond precisely to the B-orbits of the C^*-fixed points in C_n and thus, are in bijection with the partitions of n. We also prove an infinite-dimensional analogue of the classical theorem of R.Steinberg, characterizing the (non-abelian) Borel subgroups of G_n in purely group-theoretic terms. Together with our classification this last theorem implies that the G_n are pairwise non-isomorphic as abstract groups. Our study of the groups G_n is motivated by the fact that these are the automorphism groups of non-isomorphic simple algebras Morita equivalent to the Weyl algebra A_1(C). From this perspective, our results generalize well-known theorems of J.Dixmier and L.Makar-Limanov about the automorphism group of A_1(C).

Motivation & Objective

• To understand the structure and representation theory of the Dixmier groups $G_n = \mathrm{Aut}_{\mathbb{C}} D(X_n)$, where $X_n$ is a rational singular curve of differential genus $n$.
• To analyze the action of the group $G$ of symplectic automorphisms of the free algebra $R = \mathbb{C}\langle x,y \rangle$ on the Calogero-Moser varieties $\mathcal{C}_n$.
• To establish strong transitivity properties of this action, including double transitivity and diagonal transitivity on products of distinct $\mathcal{C}_{n_i}$, and to deduce maximality of stabilizers $G_n$.
• To classify the Borel subgroups of $G_n$ via conjugation of standard subgroups generated by transformations of the form $(x + \lambda y^{s-1}, y)$ and $(x, y + \lambda x^{s-1})$, using the geometry of $\mathcal{C}_n$.

Proposed method

• The group $G$ is identified with the automorphism group of the Weyl algebra $A_1(\mathbb{C})$, and $G_n$ is realized as the stabilizer of a point in the Calogero-Moser variety $\mathcal{C}_n$ under the $G$-action.
• Double transitivity is proven by analyzing the $G$-action on the configuration space $\mathcal{C}_n^{[2]}$ of ordered pairs of distinct points in $\mathcal{C}_n$, showing that $G$ acts transitively on this space.
• Diagonal transitivity on products $\mathcal{C}_{n_1} \times \cdots \times \mathcal{C}_{n_m}$ for distinct $n_i$ is established using the structure of the $G$-action and the geometry of the varieties.
• Borel subgroups are constructed via conjugation of standard subgroups $T \ltimes \{ \Psi_{q(y)} \}$ or $T \ltimes \{ \Phi_{q(x)} \}$, where $T$ is the torus of diagonal automorphisms.
• The classification relies on the correspondence between $T$-fixed points in $\mathcal{C}_n$ and integer partitions $\mu$ of $n$, with each partition yielding a distinct Borel subgroup.
• Explicit formulas for Borel subgroups are derived using conjugation by $\Psi_{-y^k}$ and $\Phi_{x^k}$, with the structure of the subsemigroup $S_\mu$ preserving the monomial set $R_\mu$ of the associated Weierstrass polynomial $W_\mu$.

Experimental results

Research questions

• RQ1 Is the action of the group $G$ on the Calogero-Moser variety $\mathcal{C}_n$ doubly transitive for $n \geq 1$?
• RQ2 Does the diagonal action of $G$ on $\mathcal{C}_{n_1} \times \cdots \times \mathcal{C}_{n_m}$ remain transitive when the $n_i$ are pairwise distinct?
• RQ3 Are the stabilizers $G_n$ of points in $\mathcal{C}_n$ maximal subgroups of $G$?
• RQ4 Can all Borel subgroups of $G_n$ be explicitly described in terms of conjugates of standard subgroups generated by transformations of the form $(x + \lambda y^{s-1}, y)$ and $(x, y + \lambda x^{s-1})$?
• RQ5 What is the precise structure of the subsemigroup $S_\mu \subset \mathbb{N}$ that preserves the monomial set $R_\mu$ associated with a partition $\mu$ of $n$?

Key findings

• The action of $G$ on $\mathcal{C}_n$ is doubly transitive for all $n \geq 1$, meaning $G$ acts transitively on the space of ordered pairs of distinct points in $\mathcal{C}_n$, with exactly two orbits: the diagonal and its complement.
• The diagonal action of $G$ on $\mathcal{C}_{n_1} \times \cdots \times \mathcal{C}_{n_m}$ is transitive for any pairwise distinct $n_1, \ldots, n_m$, implying strong global symmetry across different $\mathcal{C}_n$.
• The stabilizer $G_n$ of any point in $\mathcal{C}_n$ is a maximal subgroup of $G$, a consequence of the double transitivity of the action.
• For each partition $\mu$ of $n$, the Borel subgroup $B(\mu)$ of $G_n$ is isomorphic to $T \ltimes G_{\mu,y}$, where $G_{\mu,y}$ is generated by transformations $\Psi_{c y^{s-1}}$ with $s \in S_\mu$, $c \in \mathbb{C}$, and $S_\mu$ is the semigroup preserving the monomial set $R_\mu$ of the Weierstrass polynomial $W_\mu$.
• For $n=1$, the Borel subgroup is $B_{(1)} = T \ltimes \{ \Psi_{c y^k} \mid c \in \mathbb{C}, k \geq 1 \}$, corresponding to the full unipotent radical of the standard Borel.
• For $n=2$, there are two Borel subgroups: $B_{(2)} = T \ltimes \{ \Psi_{c y^k} \mid k \geq 2 \}$ and $B_{(1,1)} = T \ltimes \{ \Phi_{c x^k} \mid k \geq 2 \}$, reflecting different root systems from the two $T$-fixed points in $\mathcal{C}_2$.