Journal of Lie Theory Volume 12 (2002) 483–493 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

Casimir Operators on Pseudodifferential Operators of Several Variables

Min Ho Lee

Communicated by Peter Olver

Abstract. We study the actions of the Lie algebra n𝔰𝔩(2, ℂ) of SL(2, ℂ)^{n} and the associated Casimir operator on the space of pseudodifferential operators of n variables. We describe the effect of the Casimir operator on a pseudodifferential operator in connection with the symbol map and construct an n𝔰𝔩(2, ℂ)-invariant lifting of the symbol map.

1. Introduction

It is well-known that pseudodifferential operators of a single variable play an important role in the theory of nonlinear integrable partial differential equations, also known as soliton equations, as well as in conformal field theory. The algebra of pseudodifferential operators generalizes the algebra of differential operators and admits various other important algebraic structures (see e.g. [6]). In a recent paper [3] (see also [14]), Cohen, Manin and Zagier investigated connections among pseudodifferential operators, modular forms, and formal power series called Jacobi-like forms. Among other things, given a discrete subgroup Γ of SL(2, ℝ) acting on the Poincaré upper half plane H as usual, they constructed a natural correspondence between Γ-invariant pseudodifferential operators on H and sequences of modular forms for Γ. Since the product of two Γ-invariant pseudodifferential operators are again Γ-invariant, such a correspondence determines a family of noncommutative products of modular forms known as the Rankin-Cohen brackets (cf. [2], [12]). The construction of the Rankin-Cohen brackets can also be extended to the case of Siegel modular forms as was done in [1] and [4]. On the other hand, in [10] Olver and Sanders studied highly interesting connections of the Rankin-Cohen brackets for modular forms with various topics in pure and applied mathematics including transvectants, the Heisenberg group, solitons, Hirota operators and coherent states (see also [8], [9]).

Pseudodifferential operators of several variables were systematically introduced recently by Parshin in [11], where he discussed some of their algebraic structures as well as their role in soliton theory. As is expected, the close link between pseudodifferential operators and modular forms can be extended to the case of sev-

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag eral variables. Indeed, the group SL(2, ℂ)^{n} acts on the space of pseudodifferential operators of n variables, and a pseudodifferential operator on the n-fold product H^{n} of the Poincaré upper half plane H invariant under the action of a discrete subgroup Γ_{n} ⊂ SL(2, ℝ)^{n} can be identified with a sequence of Hilbert modular forms for Γ_{n}, and the Rankin-Cohen brackets for Hilbert modular forms can also be constructed (see [7]).

In this paper we study the actions of the Lie algebra n𝔰𝔩(2, ℂ) of the Lie group SL(2, ℂ)^{n} and the associated Casimir operator on the space of pseudodifferential operators of n variables. We describe the effect of the Casimir operator on a pseudodifferential operator in connection with the symbol map, which associates the coefficient of the highest order term to each pseudodifferential operator. We also construct an n𝔰𝔩(2, ℂ)-equivariant lifting of the symbol map.

2. Pseudodifferential operators of several variables

In this section we review some of the properties of pseudodifferential operators of several variables introduced by Parshin [11]. Let (z_{1}, …, z_{n}) be the standard coordinate system for ℂ^{n}, and let ∂_{1}, …, ∂_{n} be the associated partial differentiation operators given by

∂_{1} = \frac{∂}{∂z_{1}} , …, ∂_{n} = \frac{∂}{∂z_{n}} .

We denote by F the ring of complex-valued C^{∞} functions f(z) = f(z_{1}, …, z_{n}) on ℂ^{n}.

For convenience we often use the multi-index notation throughout the paper. Thus, given α = (α_{1}, …, α_{n}) ∈ ℤ^{n} and u = (u_{1}, …, u_{n}) ∈ ℂ^{n}, we have

∂^{α} = ∂_{1}^{α_{1}} ⋯ ∂_{n}^{α_{n}}, u^{α} = u_{1}^{α_{1}} ⋯ u_{n}^{α_{n}}.  (1)

If β = (β_{1}, …, β_{n}) ∈ ℤ_{+}^{n} with ℤ_{+} denoting the set of nonnegative integers, we write

β! = β_{1}! ⋯ β_{n}!, \binom{α}{β} = \binom{α_{1}}{β_{1}} ⋯ \binom{α_{n}}{β_{n}},

where for 1 ≤ i ≤ n we have

\binom{α_{i}}{0} = 1, \binom{α_{i}}{β_{i}} = \frac{α_{i}(α_{i} − 1) ⋯ (α_{i} − β_{i} + 1)}{β_{i}!}

for β_{i} > 0. Furthermore, for μ = (μ_{1}, …, μ_{n}), ν = (ν_{1}, …, ν_{n}) ∈ ℤ^{n} we write μ ≤ ν if μ_{i} ≤ ν_{i} for each i = 1, …, n, and also write c = (c, …, c) ∈ ℤ^{n} if c ∈ ℤ.

Definition 2.1. A pseudodifferential operator of n variables is a formal series of the form

L = ∑_{α≤ν} f_{α}(z)∂^{α}  (2)

for some ν ∈ ℤ^{n}, where z = (z_{1}, …, z_{n}) ∈ ℂ^{n} and f_{α} ∈ F for all α ≤ ν. We shall denote by ΨDO the complex vector space consisting of all pseudodifferential operators of n variables.

Definition 2.2. (i) The order of an element L ∈ ΨDO, denoted by ord(L), is the smallest integer r such that L = ∑_{i≤r} a_{i}∂_{n}^{i} with a_{r} ≠ 0.

(ii) The highest term of L ∈ ΨDO, denoted by HT(L), is the term in L defined inductively by

HT(L) = (HT(a_{r}))∂_{n}^{r}

for L = ∑_{i≤r} a_{i}∂_{n}^{i} with ord(L) = r.

If the highest term of L is of the form HT(L) = f(z)∂_{1}^{η_{1}} ⋯ ∂_{n}^{η_{n}}, then we set

ν(L) = (η_{1}, …, η_{n}) ∈ ℤ^{n}.

We denote by ≺ the lexicographic type of order on ℤ^{n} such that

ν(L) = (η_{1}, …, η_{n}) ≺ 0

if and only if

η_{n} < 0, or η_{n} = 0 and η_{n−1} < 0, or …, etc.

and use ⪯ to mean ≺ or =. Given an element ω = (ω_{1}, …, ω_{n}) ∈ ℤ^{n}, we consider the subspaces ΨDO_{ω} and ΨDO_{ω}^{∗} of ΨDO defined by

ΨDO_{ω} = {L ∈ ΨDO | ν(L) ⪯ ω}, ΨDO_{ω}^{∗} = {L ∈ ΨDO | ν(L) ≺ ω}.

Let Ξ_{ω} be the symbol map sending a pseudodifferential operator L ∈ ΨDO_{ω} to the coefficient of its highest term, that is, Ξ_{ω}(L) = f_{ω}(z) if HT(L) = f_{ω}(z)∂^{ω}. Then we see that the kernel of Ξ_{ω} is ΨDO_{ω}^{∗}, and therefore we obtain a short exact sequence

0 → ΨDO_{ω}^{∗} → ΨDO_{ω} ⟶^{Ξ_{ω}} F → 0  (3)

of complex vector spaces.

3. Casimir operators

Let 𝔰𝔩(2, ℂ) be the Lie algebra of the Lie group SL(2, ℂ), and let {X, Y, H} be the standard basis for 𝔰𝔩(2, ℂ) given by

X = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) , Y = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) , H = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix})  (4)

satisfying

[H, X] = 2X, [H, Y] = −2Y, [X, Y] = H.

We denote by n𝔰𝔩(2, ℂ) the direct sum of n copies of 𝔰𝔩(2, ℂ), which is the Lie algebra of SL(2, ℂ)^{n}. For 1 ≤ i ≤ n let

ε_{i} : 𝔰𝔩(2, ℂ) → n𝔰𝔩(2, ℂ)  (5)

be the natural inclusion map sending an element of 𝔰𝔩(2, ℂ) to the i-th component of n𝔰𝔩(2, ℂ), and set

X_{i} = ε_{i}(X), Y_{i} = ε_{i}(Y), H_{i} = ε_{i}(H).

Then we see that the set

{X_{i}, Y_{i}, H_{i} | 1 ≤ i ≤ n}  (6)

is a basis for n𝔰𝔩(2, ℂ). Let End(ΨDO) be the space of complex linear endomorphisms of ΨDO, and define the complex linear map σ : n𝔰𝔩(2, ℂ) → End(ΨDO) by

σ(X_{i}) = \sqrt{−1}z_{i}^{2}∂_{i}, σ(Y_{i}) = \sqrt{−1}∂_{i}, σ(H_{i}) = 2z_{i}∂_{i}  (7)

for 1 ≤ i ≤ n. As usual End(ΨDO) has the structure of a complex Lie algebra whose bracket operation is given by

[ψ_{1}, ψ_{2}] = ψ_{1}ψ_{2} − ψ_{2}ψ_{1}

for all ψ_{1}, ψ_{2} ∈ ΨDO. We denote this Lie algebra by 𝔤𝔩(ΨDO).

Lemma 3.1. The linear map σ given by (7) determines a Lie algebra homomorphism from n𝔰𝔩(2, ℂ) to 𝔤𝔩(ΨDO).

Proof. It suffices to check the condition for the basis elements for n𝔰𝔩(2, ℂ) in (6). Using (7), for each i we obtain

[σ(H_{i}), σ(X_{i})] = [2z_{i}∂_{i}, \sqrt{−1}z_{i}^{2}∂_{i}]
= 2\sqrt{−1}(z_{i}(2z_{i}∂_{i} + z_{i}^{2}∂_{i}^{2}) − z_{i}^{2}(∂_{i} + z_{i}∂_{i}^{2}))
= 2\sqrt{−1}z_{i}^{2}∂_{i} = 2σ(X_{i}) = σ([H_{i}, X_{i}]),
[σ(H_{i}), σ(Y_{i})] = [2z_{i}∂_{i}, \sqrt{−1}∂_{i}]
= 2\sqrt{−1}(z_{i}∂_{i}^{2} − ∂_{i} − z_{i}∂_{i}^{2})
= −2\sqrt{−1}∂_{i} = −2σ(Y_{i}) = σ([H_{i}, Y_{i}]),
[σ(X_{i}), σ(Y_{i})] = [\sqrt{−1}z_{i}^{2}∂_{i}, \sqrt{−1}∂_{i}]
= −(z_{i}^{2}∂_{i}^{2} − 2z_{i}∂_{i} − z_{i}^{2}∂_{i}^{2})
= 2z_{i}∂_{i} = σ(H_{i}) = σ([X_{i}, Y_{i}]),

and hence the lemma follows.

By Lemma 3.1 the composition of σ with the adjoint representation of the Lie algebra 𝔤𝔩(ΨDO) determines a representation of the Lie algebra n𝔰𝔩(2, ℂ) in the complex vector space ΨDO. The associated Casimir element C is given by

C = ∑_{i=1}^{n}(σ(H_{i})^{2}/2 + σ(X_{i})σ(Y_{i}) + σ(Y_{i})σ(X_{i}))  (8)

(see e.g. [5, §6.2]). For 1 ≤ i ≤ n and 1 ≤ j ≤ 3 we set

L_{i,j} = z_{i}^{j−1}∂_{i},  (9)

which we regard as operators acting on ΨDO by commutation. Then by (7) and (8) we see that the Casimir operator can be written in the form

C = ∑_{i=1}^{n}(2L_{i,2}^{2} − L_{i,1}L_{i,3} − L_{i,3}L_{i,1})  (10)

and that C(ΨDO_{η}) ⊂ ΨDO_{η} for each η ∈ ℤ^{n}.

Theorem 3.2. Given an element ψ = ∑_{ν≥0} f_{ν}∂^{η−ν} ∈ ΨDO_{η} with

η = (η_{1}, …, η_{n}) ∈ ℤ^{n}

we have

Ξ_{η}(Cψ) = 2 ∑_{i=1}^{n} η_{i}(η_{i} + 1)Ξ_{η}(ψ),

where Ξ_{η} is the symbol map in (3).

Proof. First, we consider an element of the form h∂^{ω} ∈ ΨDO with ω = (ω_{1}, …, ω_{n}). Then for each i ∈ {1, …, n} we have

L_{i,1}(h∂^{ω}) = [∂_{i}, h∂^{ω}] = ∂_{i}(h∂^{ω}) − h∂^{ω}∂_{i}
= (∂_{i}h)∂^{ω} + h∂^{ω+e_{i}} − h∂^{ω+e_{i}} = (∂_{i}h)∂^{ω}
L_{i,2}(h∂^{ω}) = [z_{i}∂_{i}, h∂^{ω}] = z_{i}∂_{i}(h∂^{ω}) − h∂^{ω}(z_{i}∂_{i})
= z_{i}(∂_{i}h)∂^{ω} + z_{i}h∂^{ω+e_{i}} − hz_{i}∂^{ω+e_{i}} − ω_{i}h∂^{ω} = (z_{i}(∂_{i}h) − ω_{i}h)∂^{ω}
L_{i,3}(h∂^{ω}) = [z_{i}^{2}∂_{i}, h∂^{ω}] = z_{i}^{2}∂_{i}(h∂^{ω}) − h∂^{ω}(z_{i}^{2}∂_{i})

= z_{i}^{2}(∂_{i}h)∂^{ω} + z_{i}^{2}h∂^{ω+e_{i}} − h(z_{i}^{2}∂^{ω+e_{i}} + 2ω_{i}z_{i}∂^{ω} + ω_{i}(ω_{i} − 1)∂^{ω−e_{i}})

= (z_{i}^{2}(∂_{i}h) − 2ω_{i}z_{i}h)∂^{ω} − ω_{i}(ω_{i} − 1)h∂^{ω−e_{i}},

where e_{i} denotes the element of ℤ^{n} with 1 in the i-th entry and 0 elsewhere. Thus we see that

L_{i,2}^{2}(h∂^{ω}) = z_{i}(∂_{i}(z_{i}(∂_{i}h) − ω_{i}h))∂^{ω} − ω_{i}(z_{i}(∂_{i}h) − ω_{i}h)∂^{ω}
= z_{i}((∂_{i}h) + z_{i}(∂_{i}^{2}h) − ω_{i}(∂_{i}h))∂^{ω} − ω_{i}z_{i}(∂_{i}h)∂^{ω} + ω_{i}^{2}h∂^{ω}
= z_{i}(∂_{i}h)∂^{ω} + z_{i}^{2}(∂_{i}^{2}h)∂^{ω} − ω_{i}z_{i}(∂_{i}h)∂^{ω} − ω_{i}z_{i}(∂_{i}h)∂^{ω} + ω_{i}^{2}h∂^{ω},
L_{i,1}L_{i,3}(h∂^{ω}) = (∂_{i}(z_{i}^{2}(∂_{i}h) − 2ω_{i}z_{i}h))∂^{ω} − ω_{i}(ω_{i} − 1)(∂_{i}h)∂^{ω−e_{i}}
= (2z_{i}(∂_{i}h) + z_{i}^{2}(∂_{i}^{2}h) − 2ω_{i}h − 2ω_{i}z_{i}(∂_{i}h))∂^{ω}
− ω_{i}(ω_{i} − 1)(∂_{i}h)∂^{ω−e_{i}},
L_{i,3}L_{i,1}(h∂^{ω}) = (z_{i}^{2}(∂_{i}^{2}h) − 2ω_{i}z_{i}(∂_{i}h))∂^{ω} − ω_{i}(ω_{i} − 1)(∂_{i}h)∂^{ω−e_{i}}.

Using these relations and (10), it follows that

C(h∂^{ω}) = 2 ∑_{i=1}^{n}(ω_{i}(ω_{i} + 1)h∂^{ω} + ω_{i}(ω_{i} − 1)(∂_{i}h)∂^{ω−e_{i}}).

Thus, if ψ = ∑_{ν≥0} f_{ν}∂^{η−ν} ∈ ΨDO_{η}, we have

Cψ − 2 ∑_{i=1}^{n} η_{i}(η_{i} + 1)ψ = 2 ∑_{ν≥0} ∑_{i=1}^{n}((η_{i} − ν_{i})(η_{i} − ν_{i} + 1) − η_{i}(η_{i} + 1))f_{ν}∂^{η−ν}

+ 2 ∑_{ν≥0} ∑_{i=1}^{n}(η_{i} − ν_{i})(η_{i} − ν_{i} − 1)(∂_{i}f_{ν})∂^{η−ν−e_{i}}

= 2 ∑_{i=1}^{n} ∑_{ν≥e_{i}} ν_{i}(ν_{i} − 2η_{i} − 1)f_{ν}∂^{η−ν}

+ 2 ∑_{ν≥0} ∑_{i=1}^{n}(η_{i} − ν_{i})(η_{i} − ν_{i} − 1)(∂_{i}f_{ν})∂^{η−ν−e_{i}}.

Hence we obtain

Ξ_{η}(Cψ − 2 ∑_{i=1}^{n} η_{i}(η_{i} + 1)ψ) = 0,

and therefore the theorem follows.

Remark 3.3. Given η = (η_{1}, …, η_{n}) ∈ ℤ^{n}, if ΨDO_{η}^{∗} is as in (3), we have C(ΨDO_{η}^{∗}) ⊂ ΨDO_{η}^{∗}; hence the Casimir operator C acts on the quotient space ΨDO_{η}/ΨDO_{η}^{∗} by

C(ΨDO_{η}^{∗} + ψ) = ΨDO_{η}^{∗} + Cψ  (11)

for all ψ ∈ ΨDO_{η}. Using (11) and Theorem 3.2, we see that

C(ΨDO_{η}^{∗} + ψ) = ΨDO_{η}^{∗} + Ξ_{η}(Cψ)

= ΨDO_{η}^{∗} + 2 ∑_{i=1}^{n} η_{i}(η_{i} + 1)Ξ_{η}(ψ)

= 2 ∑_{i=1}^{n} η_{i}(η_{i} + 1)(ΨDO_{η}^{∗} + ψ),

and therefore it follows that the Casimir operator C acts on ΨDO_{η}/ΨDO_{η}^{∗} as multiplication by 2 ∑_{i=1}^{n} η_{i}(η_{i} + 1).

4. The lifting map

Since F may be regarded as a subspace of ΨDO, the action of n𝔰𝔩(2, ℂ) on ΨDO determined by σ in (5) induces an action of n𝔰𝔩(2, ℂ) on F. Relative to such actions we see easily that the symbol map Ξ_{η} : ΨDO_{η} → F with η ∈ ℤ^{n} is n𝔰𝔩(2, ℂ)-equivariant, that is,

Ξ_{η}(σ(W)ψ) = σ(W)Ξ_{η}(ψ)

for all ψ ∈ ΨDO_{η} and W ∈ n𝔰𝔩(2, ℂ). In this section we construct a lifting of the symbol map Ξ_{η} that is n𝔰𝔩(2, ℂ)-equivariant.

Via the inclusion map ε_{i} : 𝔰𝔩(2, ℂ) → n𝔰𝔩(2, ℂ) in (5), the representation σ of n𝔰𝔩(2, ℂ) on ΨDO induces the representation σ_{i} = σ ∘ε_{i} of 𝔰𝔩(2, ℂ) in ΨDO for each i ∈ {1, …, n}. If X, Y, H and X_{i}, Y_{i}, H_{i} are as in (4) and (6), the Casimir element C_{i} associated to σ_{i} is given by

C_{i} = σ_{i}(H)^{2}/2 + σ_{i}(X)σ_{i}(Y) + σ_{i}(Y)σ_{i}(X)  (12)
= σ(H_{i})^{2}/2 + σ(X_{i})σ(Y_{i}) + σ(Y_{i})σ(X_{i})
= 2L_{i,2}^{2} − L_{i,1}L_{i,3} − L_{i,3}L_{i,1},

where the L_{i,j} are as in (9) for 1 ≤ j ≤ 3.

Lemma 4.1. Given i ∈ {1, …, n}, let C_{i} be the Casimir operator in (12), and let ψ = ∑_{ν≥0} f_{ν}∂^{η−ν} ∈ ΨDO_{η} with η ∈ ℤ^{n}. Then we have

(C_{i} − 2η_{i}(η_{i} + 1))ψ = 2 ∑_{ν≥e_{i}}(ν_{i}(ν_{i} − 2η_{i} − 1)f_{ν}
+ (η_{i} − ν_{i} + 1)(η_{i} − ν_{i})(∂_{i}f_{ν−e_{i}}))∂^{η−ν}.  (13)

Proof. Let ψ = ∑_{ν≥0} f_{ν}∂^{η−ν} ∈ ΨDO_{η} with η ∈ ℤ^{n}. Using the expressions of L_{i,j} used in the proof of Theorem (3.2), we see that

C_{i}ψ = ∑_{ν≥0}(2(η_{i} − ν_{i})(η_{i} − ν_{i} + 1)f_{ν}∂^{η−ν}
+ 2(η_{i} − ν_{i})(η_{i} − ν_{i} − 1)(∂_{i}f_{ν})∂^{η−ν−e_{i}})

= 2 ∑_{ν≥0}(η_{i} − ν_{i})(η_{i} − ν_{i} + 1)f_{ν}∂^{η−ν}

+ 2 ∑_{ν≥e_{i}}(η_{i} − ν_{i} + 1)(η_{i} − ν_{i})(∂_{i}f_{ν−e_{i}})∂^{η−ν}

= 2η_{i}(η_{i} + 1) ∑_{ν≥0} f_{ν}∂^{η−ν}

+ 2 ∑_{ν≥e_{i}}((η_{i} − ν_{i})(η_{i} − ν_{i} + 1) − η_{i}(η_{i} + 1))f_{ν}∂^{η−ν}

+ 2 ∑_{ν≥e_{i}}(η_{i} − ν_{i} + 1)(η_{i} + ν_{i})(∂_{i}f_{ν−e_{i}})∂^{η−ν}

= 2η_{i}(η_{i} + 1)ψ + 2 ∑_{ν≥e_{i}}(ν_{i}(ν_{i} − 2η_{i} − 1)f_{ν}

+ (η_{i} − ν_{i} + 1)(η_{i} + ν_{i})(∂_{i}f_{ν−e_{i}}))∂^{η−ν}.

Hence the lemma follows.

Given ω ≥ 0 with ω ≠ 0, we define the linear maps L_{ω} : F → ΨDO_{ω}, L_{−ω} : F → ΨDO_{−ω}, and L_{0} : F → F by

L_{ω}(f) = \frac{ω!(ω − 1)!}{(2ω)!} ∑_{0≤ν≤ω−1} \frac{(2ω − ν)!}{ν!(ω − ν)!(ω − ν − 1)!}(∂^{ν}f)∂^{ω−ν},  (14)
L_{0}(f) = f,  (15)
L_{−ω}(f) = \binom{2ω − 1}{ω} ∑_{ν≥0}(−1)^{|ν|} \frac{(ν + ω)!(ν + ω − 1)!}{ν!(ν + 2ω − 1)!}(∂^{ν}f)∂^{−ω−ν},  (16)

for all f ∈ F. Then we see easily that

Ξ_{η} ∘ L_{η}(f) = f

for each f ∈ F and η ∈ ℤ^{n}; hence L_{η} is a lifting of Ξ_{η}.

Example 4.2. We consider the case, where n = 2, z_{1} = z, z_{2} = w and ω = (2, 3) ∈ ℤ_{+}^{2}. Then, given f = f(z, w) ∈ F, by (14) we have

L_{ω}(f) = \frac{2!3!1!2!}{4!6!} ∑_{i=0}^{1} ∑_{j=0}^{2} \frac{(4 − i)!(6 − j)!}{i!j!(2 − i)!(3 − j)!(1 − i)!(2 − j)!} \frac{∂^{i+j}f}{∂z^{6}∂w^{j}} (z, w)∂_{z}^{2−i}∂_{w}^{3−j}

= f(z, w)∂_{z}^{2}∂_{w}^{3} + \frac{∂f}{∂w}(z, w)∂_{z}^{2}∂_{w}^{2} + \frac{1}{5} \frac{∂^{2}f}{∂w^{2}} (z, w)∂_{z}^{2}∂_{w} + \frac{1}{2} \frac{∂f}{∂z} (z, w)∂_{z}∂_{w}^{3}

+ \frac{1}{2} \frac{∂^{2}f}{∂z∂w}(z, w)∂_{z}∂_{w}^{2} + \frac{1}{10} \frac{∂^{3}f}{∂z∂w^{2}} (z, w)∂_{z}∂_{w},

where ∂_{z} = ∂/∂z and ∂_{w} = ∂/∂w. Thus L_{ω}(f) is in fact a differential operator rather than a pseudodifferential operator. On the other hand, using (16) and

\binom{2ω − 1}{ω} = \binom{(3, 5)}{(2, 3)} = \binom{3}{2}\binom{5}{3} = 30,

we see that L_{−ω}(f) is a pseudodifferential operator given by

L_{−ω}(f) = 30 ∑_{i=0}^{∞} ∑_{j=0}^{∞}(−1)^{i+j} \frac{(i + 2)!(j + 3)!i + 1)!(j + 2)!}{i!j!(i + 3)!(j + 5)!} \frac{∂^{i+j}f}{∂z^{6}∂w^{j}} (z, w)∂_{z}^{−i}∂_{w}^{−j}.

Theorem 4.3. For each η ∈ ℤ^{n} the linear map L_{η} : F → ΨDO_{η} given by (14) is n𝔰𝔩(2, ℂ)-equivariant, that is,

L_{η}(σ(W)f) = σ(W)L_{η}(f)  (17)

for all W ∈ n𝔰𝔩(2, ℂ) and f ∈ F, where σ is as in (7).

Proof. We shall show that a lifting L_{η} of Ξ_{η} satisfying (17) must be given by the formulas (14), (15), and (16). If L_{η} is such a map, using the embeddings ε_{i} : 𝔰𝔩(2, ℂ) → n𝔰𝔩(2, ℂ), we see that L_{η} is 𝔰𝔩(2, ℂ)-equivariant via σ_{i} = σ ∘ ε_{i} for each i ∈ {1, …, n}. Using Lemma 4.1 and an argument similar to the one in Remark 3.3, we also see that the Casimir operator C_{i} operates on the quotient space ΨDO_{η}/ΨDO_{η}^{∗} as multiplication by 2η_{i}(η_{i} + 1). Since F is isomorphic to ΨDO_{η}/ΨDO_{η}^{∗}, we have

C_{i}(L_{η}(f)) = L_{η}(C_{i}f) = 2η_{i}(η_{i} + 1)L_{η}(f)  (18)

for 1 ≤ i ≤ n and f ∈ F. Let ω ≥ 0 with ω ≠ 0, and set

L_{ω}(f) = ∑_{ν≥0} f_{ν}∂^{ω−ν}.

If ν ≥ e_{i}, then by (13) and (18) we have

ν_{i}(ν_{i} − 2ω_{i} − 1)f_{ν} + (ω_{i} − ν_{i} + 1)(ω_{i} − ν_{i})(∂_{i}f_{ν−e_{i}}) = 0.

Hence we obtain

f_{ν} = −\frac{(ω_{i} − ν_{i} + 1)(ω_{i} − ν_{i})}{ν_{i}(ν_{i} − 2ω_{i} − 1)}∂_{i}f_{ν−e_{i}}

for all ν ≥ e_{i}. Thus, if ν_{i} ≤ ω_{i} − 1, by iteration we have

f_{ν} = (−1)^{ν_{i}} \frac{(ω_{i} − ν_{i} + 1) ⋯ ω_{i} · (ω_{i} − ν_{i}) ⋯ (ω_{i} − 1)}{ν_{i}!(ν_{i} − 2ω_{i} − 1) ⋯ (−2ω_{i})}∂_{i}^{ν_{i}}f_{ν−ν_{i}e_{i}}

= \frac{(ω_{i}!/(ω_{i} − ν_{i})!)((ω_{i} − 1)!/(ω_{i} − ν_{i} − 1)!)}{ν_{i}!(2ω_{i})!/(2ω_{i} − ν_{i})!}∂_{i}^{ν_{i}}f_{ν−ν_{i}e_{i}}

= \frac{ω_{i}!(ω_{i} − 1)!}{(2ω_{i})!} · \frac{(2ω_{i} − ν_{i})!}{ν_{i}!(ω_{i} − ν_{i})!(ω_{i} − ν_{i} − 1)!}∂_{i}^{ν_{i}}f_{ν−ν_{i}e_{i}}.

If ν_{i} > ω_{i} − 1, then we have f_{ν} = 0. Hence we obtain

f_{ν} = \frac{ω!(ω − 1)!}{(2ω)!} · \frac{(2ω − ν)!}{ν!(ω − ν)!(ω − ν − 1)!}∂^{ν}f_{0}

if 0 ≤ ν ≤ ω − 1, and f_{ν} = 0 otherwise. Thus it follows that

L_{ω}(f) = \frac{ω!(ω − 1)!}{(2ω)!} ∑_{0≤ν≤ω−1} \frac{(2ω − ν)!}{ν!(ω − ν)!(ω − ν − 1)!}(∂^{ν}f_{0})∂^{ω−ν}.

However, since L_{ω} is a lifting of Ξ_{ω}, we have f_{0} = Ξ_{ω}(L_{ω}f) = f, and therefore we obtain (14). On the other hand, if L_{−ω}(f) = ∑_{ν≥0} h_{ν}∂^{−ω−ν} ∈ ΨDO_{−w}, then by (13) and (18) we have

ν_{i}(ν_{i} + 2ω_{i} − 1)h_{ν} + (ω_{i} + ν_{i} − 1)(ω_{i} + ν_{i})(∂_{i}h_{ν−e_{i}}) = 0

for 1 ≤ i ≤ n and f ∈ F. Hence we obtain

h_{ν} = −\frac{(ω_{i} + ν_{i} − 1)(ω_{i} + ν_{i})}{ν_{i}(ν_{i} + 2ω_{i} − 1)}∂_{i}h_{ν−e_{i}}

for all ν ≥ e_{i}. Thus by iteration we see that

h_{ν} = (−1)^{ν_{i}} \frac{((ν_{i} + ω_{i})!/ω_{i}!)((ν_{i} + ω_{i} − 1)!/(ω_{i} − 1)!)}{ν_{i}!(ν_{i} + 2ω_{i} − 1)!/(2ω_{i} − 1)!}∂_{i}^{ν_{i}}h_{ν−ν_{i}e_{i}}

= (−1)^{ν_{i}} \binom{2ω_{i} − 1}{ω_{i}} \frac{(ν_{i} + ω_{i})!(ν_{i} + ω_{i} − 1)!}{ν_{i}!(ν_{i} + 2ω_{i} − 1)!}∂_{i}^{ν_{i}}h_{ν−ν_{i}e_{i}}.

Applying this for each i, we obtain

h_{ν} = (−1)^{|ν|}\binom{2ω − 1}{ω}\frac{(ν + ω)!(ν + ω − 1)!}{ν!(ν + 2ω − 1)!}∂^{ν}h_{0}.

Hence it follows that

L_{−ω}(f) = \binom{2ω − 1}{ω} ∑_{ν≥0}(−1)^{|ν|} \frac{(ν + ω)!(ν + ω − 1)!}{ν!(ν + 2ω − 1)!}(∂^{ν}h_{0})∂^{−ω−ν},

and therefore we obtain (16) by using h_{0} = Ξ_{−ω}(L_{−ω}(f)) = f. Since a lifting L_{0} with L_{0}(F) ⊂ F should obviously be the identity map, the proof of the theorem is complete.

Remark 4.4. Let Γ be a discrete subgroup of SL(2, ℂ)^{n}. Then for each η ∈ ℤ^{n} the exact sequence in (3) induces the short exact sequence

0 → ΨDO_{η}^{∗}^{Γ} → ΨDO_{η}^{Γ} ⟶^{Ξ_{η}} F^{Γ} → 0,  (19)

where (·)^{Γ} denotes the subset of Γ-fixed elements. By Theorem 4.3 the linear map L_{η} : F → ΨDO_{η} is also equivariant with respect to the actions of the Lie group SL(2, ℂ)^{n} of the Lie algebra n𝔰𝔩(2, ℂ) on F and on ΨDO_{η}. Thus it follows that the short exact sequence (19) splits. If η ≥ 0, then F^{Γ} is the space of Hilbert modular forms for Γ of weight −2η, which was considered in [7].

5. Concluding remarks

We have discussed the action of the Lie group SL(2, ℝ)^{n} on pseudodifferential operators of n variables in terms of the corresponding Lie algebra action. As was described in the introduction, if Γ_{n} is a discrete subgroup of SL(2, ℝ)^{n}, each Γ_{n}-invariant pseudodifferential operators of n variables can be identified with a sequence of Hilbert modular forms, which are essentially modular forms of several variables, for Γ_{n}. Using this correspondence and the role of pseudodifferential operators in soliton theory, we see that there is at least an indirect link between soliton equations and modular forms. In fact, Olver and Sanders [10] also discussed a connection between the Rankin-Cohen brackets for modular forms and solitons via Hirota operators. It would be interesting to search for more direct connections between modular forms and soliton equations and their solutions.

In another direction, we can consider nonholomorphic modular forms. In a recent monograph [13], Unterberger, among other things, discussed the Rankin-Cohen brackets for nonholomorphic modular forms and their relation with quantization theory through harmonic analysis. It might be worth studying actions of SL(2, ℝ)^{n} or its Lie algebra on pseudodifferential operators of n variables with nonholomorphic coefficients and exploring the possibility of relating them with nonholomorphic modular forms.

References

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[2] Cohen, H., Sums involving the values at negative integers of L functions of quadratic characters, Math. Ann. 217 (1977), 81–94.

[3] Cohen, P., Y. Manin, and D. Zagier, Automorphic pseudodifferential operators, Algebraic aspects of nonlinear systems, Birkhäuser, Boston, 1997, pp. 17–47.

[4] Eholzer, W., and T. Ibukiyama, Rankin-Cohen type differential operators for Siegel modular forms, Internat. J. Math. 9 (1998), 443–463.

[5] Humphreys, J., “Introduction to Lie Algebras and Representation Theory, ” Springer-Verlag, Heidelberg, 1972.

[6] Khesin, B., and I. Zakharevich, Poisson-Lie group of pseudodifferential symbols, Comm. Math. Phys. 171 (1995), 475–530.

[7] Lee, M, H., Hilbert modular pseudodifferential operators, Proc. Amer. Math. Soc. (2001), 3151–3160.

[8] Olver, P., “Equivalence, Invariants, and Symmetry, ” Cambridge Univ. Press, Cambridge, 1995.

[9] —, “Classical Invariant Theory, ” Cambridge Univ. Press, Cambridge, 1999.

[10] Olver, P., and J. Sanders, Transvectants, modular forms, and the Heisenberg algebra, Adv. in Appl. Math. 25 (2000), 252–283.

[11] Parshin, A., On a ring of formal pseudo-differential operators, Proc. Steklov Inst. Math. 224 (1999), 266–280.

[12] Rankin, R., The construction of automorphic forms from the derivatives of a given form, J. Indian Math. Soc. 20 (1956), 103–116.

[13] Unterberger, A., “Quantization and Non-Holomorphic Modular Forms, ” Lecture Notes in Math. 1742, Springer-Verlag, Heidelberg, 2000.

[14] Zagier, D., Modular forms and differential operators, Proc. Indian Acad. Sci. Math. Sci. 104 (1994), 57–75.

Min Ho Lee Department of Mathematics University of Northern Iowa Cedar Falls, Iowa 50614 U. S. A. lee@math.uni.edu

Received May 15, 2001 and in final form August 6, 2001