= Solution
Work over $\mathbb C$ in characteristic zero. The second <Weyl algebra> has generators $x_1,x_2,\partial_1,\partial_2$ and relations $[\partial_i,x_j]=\delta_{ij}$, $[x_i,x_j]=[\partial_i,\partial_j]=0$. Its <Bernstein filtration> gives every generator degree one. The ordered monomials $x_1^{a_1}x_2^{a_2}\partial_1^{b_1}\partial_2^{b_2}$ form a basis, so
$$
\operatorname{gr}A_2\cong\mathbb C[x_1,x_2,\xi_1,\xi_2],\qquad
\dim F_rA_2=\binom{r+4}{4}.
$$
For a nonzero finitely generated module, choose a finite-dimensional generating subspace $M_0$ and put $M_r=F_rA_2M_0$. Its graded module is finite over this polynomial ring. The <Hilbert polynomial> theorem gives an eventual polynomial $\dim M_r$, whose degree $d(M)$ is the <Bernstein growth dimension of a Weyl algebra module>, equal to its <Gelfand–Kirillov dimension of a module>. Two choices of finite generators bound each other after fixed filtration shifts, proving independence of the choice. The <characteristic variety of a Weyl algebra module> is the support of this graded module; its algebraic dimension is $d(M)$. In particular $d(M)$ is an integer at most four.
Here is a growth proof of the lower bound, rather than just quoting the <Bernstein inequality for Weyl algebra modules>. A nonzero $u\in F_rA_2$ can be turned into a nonzero scalar by at most $r$ successive commutators with the four generators: choose a monomial of maximal total degree, differentiate it to a constant using $[\partial_i,\cdot]$ and $[x_i,\cdot]$, and all different monomials are killed by that derivative multi-index. This uses the ordered-monomial expressions and characteristic zero. Expanding the iterated commutator expresses that scalar as $\sum_jv_juw_j$ with $v_j,w_j\in F_rA_2$.
If $u$ acted as zero on $M_r$, every $uw_jM_0$ would be zero, so the scalar would kill the nonzero $M_0$, a contradiction. Therefore the action map
$$
F_rA_2\hookrightarrow\operatorname{Hom}_\mathbb C(M_r,M_{2r})
$$
is injective. This is the <faithful finite-step action of a Weyl algebra>. Comparing growth in
$$
\binom{r+4}{4}\le(\dim M_r)(\dim M_{2r})
$$
gives $4\le2d(M)$. Thus \b[the only possible dimensions for nonzero modules are $\boxed{d(M)\in\{2,3,4\}}$.]
All three values occur in cyclic modules. On $\mathbb C[x_1,x_2]$, let $x_i$ multiply and $\partial_i$ differentiate. The module is generated by $1$, its filtered pieces are precisely polynomials of degree at most $r$, and $\dim M_r=\binom{r+2}{2}$. Hence $\boxed{d(\mathbb C[x_1,x_2])=2}$; equivalently it is $A_2/(A_2\partial_1+A_2\partial_2)$.
For $M=A_2/A_2\partial_2$, the ordered basis shows that the left ideal $A_2\partial_2$ is spanned by monomials with positive $\partial_2$ exponent. The remaining basis is $x_1^{a_1}x_2^{a_2}\partial_1^{b_1}$. Its degree-$\le r$ piece has dimension $\binom{r+3}{3}$, so $\boxed{d(A_2/A_2\partial_2)=3}$. Finally the regular module has the four-variable ordered basis above, giving $\boxed{d(A_2)=4}$.
The dimension-two modules are the <holonomic Weyl algebra modules>. Exact sequences satisfy $d(M)=\max\{d(M'),d(M'')\}$, by using compatible good filtrations and the resulting Hilbert polynomials. Therefore nonzero subquotients of a holonomic module remain holonomic. Their positive integer multiplicities, namely $2!$ times the leading Hilbert-polynomial coefficient, add in exact sequences. Every nonzero successive factor consumes at least one unit of multiplicity, bounding the length of a submodule chain; hence holonomic modules have finite composition length. The polynomial module above is even simple: differentiating a nonzero polynomial to a constant and then multiplying by arbitrary polynomials generates the whole module. In contrast the regular and dimension-three examples exhibit the larger growth allowed for nonholonomic modules. These examples and the growth bound explain both the range and its representation-theoretic significance.
Back to article page
