Higher Heisenberg Lie algebras and metaplectic representations

Jan Nekovář

0. Introduction

Theta functions and, more generally, modular forms of half-integral weight can be interpreted as automorphic forms on the metaplectic group, a two-fold covering of the adelic symplectic group. Representation theoretical foundations of this theory are provided by the Weil representation [We], a unitary representation of a semi-direct product of the metaplectic group with an adelic Heisenberg group. This representation can be constructed from scratch using the Stone-von Neumann uniqueness theorem for unitary representations of the Heisenberg group.

Higher metaplectic groups are non-trivial central extensions

1 ⟶ μ_{n} ⟶ G̃ ⟶ G(𝔸_{K}) ⟶ 1,

where G is a classical group such as GL(r) or SL(r) and K a number field containing μ_{n}. The extension G̃ splits over G(K); this fact is equivalent to the n-th power reciprocity law over K. It is known that the case r = n is somewhat special: there is a representation of G̃ analogous to (the even part of) the Weil representation, which was studied by Kazhdan-Patterson [KP]. One of the main open questions of the theory is whether this representation (more precisely, its direct sum with n − 1 other representations, which are analogues of the odd part of the Weil representation) admits an action of a suitable higher Heisenberg group.

In this article we show that this is true on the level of Lie algebras. We define the “Heisenberg Lie algebra of level n” h(n) to be the quotient of the free Lie algebra on an n-dimensional vector space V by relations

(ad(x))^{n}y = 0 (x, y ∈ V);

for n = 2 this is the usual Heisenberg Lie algebra with basis P, Q, 1. Our main result states that the Weil-like representations of the Lie algebra sl(n) extends naturally to a representation of the semi-direct product of h(n) with sl(n) (if we work with Lie algebras over a field F of characteristic zero, then the action of h(n) depends on a parameter in F^{∗}/F^{∗n}). For n = 3, h(3) acts through its finite-dimensional quotient G_{2}, as observed first by Bernstein (unpublished).

1. Representations of sl(2)

(1.1) The Weil representation of sl(2)

Let F be a field of characteristic zero. The operators Q = t and P = d/dt acting on the polynomial ring F [t] generate the Heisenberg Lie algebra h(2) with basis P, Q, 1 = [P, Q]. The Weil representation of sl(2) = sl(2, F) in its simplest form acts on F [t] by the following formulas:

X = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) ↦ −\frac{Q^{2}}{2} , Y = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) ↦ \frac{P^{2}}{2} , H = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ↦ \frac{P Q + QP}{2}

Taken together, the six operators 1, P, Q, −Q^{2}/2, P^{2}/2, (P Q + QP)/2 define a representation of the semidirect product h(2) ⋊ sl(2) on F [t] (sl(2) acts trivially on 1 and by the standard two-dimensional representation on the span of P, Q). The action of h(2) on F [t] is irreducible. As an sl(2)-module, F [t] is a direct sum

F [t] = F [t^{2}] ⊕ tF [t^{2}],

where F [t^{2}] (resp. tF [t^{2}]) is an irreducible sl(2)-module of lowest weight \frac{1}{2} (resp. \frac{3}{2}) with a lowest weight vector 1 (resp. t). This follows from Y 1 = Y t = 0 and H1 = \frac{1}{2}, Ht = \frac{3}{2}t.

(1.2) The space V (μ) ⊕ V (μ + 1)

Let us consider slightly more general representations of sl(2). For μ ∈ F, denote by V (μ) the lowest weight Verma module (not necessarily irreducible) with lowest weight μ + 1. Then

\begin{aligned}V (μ) & = ⨁_{n=0}^{∞} F · X^{n}u, Y u = 0, Hu = (μ + 1)u \\ V (μ + 1) & = ⨁_{n=0}^{∞} F · X^{n}v, Y v = 0, Hv = (μ + 2)v. \end{aligned}

Consider the direct sum U(μ) = V (μ) ⊕ V (μ + 1). We are going to construct non-zero operators A, B : U(μ) ⟶ U(μ) analogous to P, Q : U(−\frac{1}{2}) ⟶ U(−\frac{1}{2}) in the Weil representation:

(i) A (resp. B) should increase (resp. decrease) weights by one, i.e.

A⟨X^{n}u⟩ ⊆ ⟨X^{n}v⟩, A⟨X^{n}v⟩ ⊆ ⟨X^{n+1}u⟩, B⟨X^{n}u⟩ ⊆ ⟨X^{n−1}v⟩, B⟨X^{n}v⟩ ⊆ ⟨X^{n}u⟩.

(ii) The vector space V = ⟨A, B⟩ ⊂ End(U(μ)) is stable under the adjoint action of the image of sl(2) in End(U(μ)) and defines the standard two-dimensional representation of sl(2):

\begin{aligned}[X, A] & = 0, \\ [X, B] & = A, \end{aligned}  \begin{aligned}[H, A] & = A, \\ [H, B] & = −B, \end{aligned}  \begin{aligned}[Y, A] & = B \\ [Y, B] & = 0. \end{aligned}

These conditions imply that

(1.2.1)  A : X^{n}u ↦ a_{1}X^{n}v,  X^{n}v ↦ a_{2}X^{n+1}u  (n ≥ 0)

for some a_{1}, a_{2} ∈ F. The standard formulas for the action of sl(2) on U(μ)

\begin{matrix}X : X^{n}u & ↦ X^{n+1}u \\ Y : X^{n}u & ↦ −n(μ + n)X^{n−1}u \\ H : X^{n}u & ↦ (μ + 2n + 1)X^{n}u \end{matrix}  \begin{matrix}X : X^{n}v & ↦ X^{n+1}v \\ Y : X^{n}v & ↦ −n(μ + n + 1)X^{n−1}v \\ H : X^{n}v & ↦ (μ + 2n + 2)X^{n}v \end{matrix}

together with [Y, A] = B then imply

(1.2.2)  B : X^{n}u ↦ −na_{1}X^{n−1}v,  X^{n}v ↦ −(μ + n + 1)a_{2}X^{n}u  (n ≥ 0).

A short calculation shows that A and B defined by the formulas (1.2.1–2) satisfy (i) and (ii). We can renormalize the lowest weight vectors by u ↦ t_{0}u, v ↦ t_{1}v and the operator A ↦ t_{2}A (t_{i} ∈ F^{∗}); the pair (a_{1}, a_{2}) is then replaced by (a_{1}t_{0}^{−1}t_{1}t_{2}, a_{2}t_{0}t_{1}^{−1}t_{2}). Assuming that a_{i} ≠ 0, the renormalization leaves invariant a_{1}a_{2} (mod F^{∗2}) ∈ F^{∗}/F^{∗2}. Replacing F by F (\sqrt{a_{1}a_{2}}) and choosing suitable t_{i}, we can (and will) assume that a_{1} = a_{2} = 1, i.e.

\begin{matrix}A : X^{n}u & ↦ X^{n}v \\ B : X^{n}u & ↦ −nX^{n−1}v \end{matrix}  \begin{matrix}X^{n}v & ↦ X^{n+1}u \\ X^{n}v & ↦ −(μ + n + 1)X^{n}u \end{matrix}

Note that A^{2} = X, B^{2} = −Y, AB + BA = −H.

(1.3) The Lie algebra L_{μ}

Denote by L_{μ} ⊂ End(U(μ)) the Lie subalgebra generated by A and B. One computes

\begin{matrix}[A, B] : X^{n}u & ↦ (μ + 1)X^{n}u \\ [A, [A, B]] : X^{n}u & ↦ (2μ + 1)X^{n}v \\ [B, [A, B]] : X^{n}u & ↦ −(2μ + 1)nX^{n−1}v \end{matrix}  \begin{matrix}X^{n}v & ↦ −μX^{n}v \\ X^{n}v & ↦ −(2μ + 1)X^{n+1}u \\ X^{n}v & ↦ (2μ + 1)(μ + n + 1)X^{n}u \end{matrix}

These formulas show a dichotomy in the behaviour of L_{μ}: Special case μ = −\frac{1}{2} (corresponding to the Weil representation): [A, B] = \frac{1}{2} · I acts on U(−\frac{1}{2}) as a scalar and L_{− \frac{1}{2}} = h(2) is the Heisenberg Lie algebra.

Generic case μ ≠ −\frac{1}{2}: denote by π_{i} (i = 0, 1) the projection of U(μ) on V (μ + i) along V (μ + 1 − i). We have [A, B] = (μ + 1)π_{0} − μπ_{1}, hence

π_{0} = \frac{μI + [A, B]}{2μ + 1}, π_{1} = \frac{(μ + 1)I − [A, B]}{2μ + 1}.

A short calculation shows that operators π_{i}A, π_{i}B, π_{i}X, π_{i}Y, π_{i}H (i = 0, 1) are all contained in L_{μ} and that (ad(π_{0}X))^{n}(π_{0}B) ≠ 0 for all n ≥ 0. This implies that dim(L_{μ}) = ∞ for μ ≠ −\frac{1}{2}.

(1.4) The Casimir operator

What is so special about the weight μ = −\frac{1}{2}? The Casimir operator

Ω = XY + Y X + \frac{H^{2}}{2} = 2XY + \frac{H^{2}}{2} − H = 2Y X + \frac{H^{2}}{2} + H

acts on V (μ) (resp. V (μ + 1)) by the scalar

\frac{(μ + 1)^{2}}{2} − (μ + 1) = \frac{(μ + 1)(μ − 1)}{2} (resp. \frac{(μ + 2)μ}{2}) .

As

\frac{(μ + 2)μ}{2} − \frac{(μ + 1)(μ − 1)}{2} = μ + \frac{1}{2},

we see that

Ω acts on U(μ) as a scalar ⟺ μ = −\frac{1}{2}

(in which case Ω = −\frac{3}{8} · I).

This observation gives an alternative approach to the study of L_{− \frac{1}{2}} that does not involve computation of iterated commutators in End(U(−\frac{1}{2})).

(1.5) A calculation in U(L(V) ⋊ sl(2))

Let L(V) be the free Lie algebra over V = ⟨A, B⟩. The action of A, B and sl(2) extends uniquely to a representation of the semi-direct product L(V) ⋊ sl(2) on U(μ). Assume, until the end of Sect. 1, that μ = −\frac{1}{2}.

As Ω acts on U(−\frac{1}{2}) as a scalar, the representation of the universal enveloping algebra U(L(V) ⋊ sl(2)) on U(−\frac{1}{2}) factors through the quotient of U(L(V) ⋊ sl(2)) by the bilateral ideal I generated by [Ω, A].

The ideal I contains the following elements:

\begin{aligned}[Ω, A] & = (H − \frac{3}{2}) A + 2XB \\ [Ω, B] & = 2Y A − (H + \frac{3}{2}) B, \end{aligned}

hence also

\begin{aligned}a = \frac{1}{2}[A, [Ω, A]] & = −\frac{A^{2}}{2} + X[A, B] \\ b = −\frac{1}{2}[B, [Ω, B]] & = \frac{B^{2}}{2} + Y [A, B] \\ c = − [B, [Ω, A]] & = \frac{AB + BA}{2} + H[A, B] \end{aligned}

and

\begin{aligned}[A, a] & = X[A, [A, B]] \\ [A, b] & = Y [A, [A, B]] − \frac{1}{2}[B, [A, B]] \\ [A, c] & = (H − \frac{1}{2}) [A, [A, B]] \end{aligned} \begin{aligned}[B, a] & = −\frac{1}{2}[A, [A, B]] + X[B, [A, B]] \\ [B, b] & = Y [B, [A, B]] \\ [B, c] & = (H + \frac{1}{2}) [B, [A, B]] \end{aligned}

It follows that

\begin{aligned}−Y [A, a] + X[A, b] − [A, c] + \frac{1}{2}[B, a] & = \frac{1}{4}[A, [A, B]] \\ −Y [B, a] + X[B, b] − [B, c] − \frac{1}{2}[A, b] & = −\frac{1}{4}[B, [A, B]] \end{aligned}

both lie in I, hence [A, B] is a central element of L_{− \frac{1}{2}} . As 2X[A, B] = A^{2} ≠ 0 in End(U(−\frac{1}{2})), the formula defining a implies that [A, B] ≠ 0 in L_{− \frac{1}{2}} and L_{− \frac{1}{2}} = h(2).

2. Representations of sl(n)

(2.1) The Lie algebra sl(n) - standard notation

F is a field of characteristic zero. 𝔤 = sl(n) = sl(n, F) = {X ∈ M_{n}(F) | T r(X) = 0} 𝔤 = 𝔫_{−} ⊕ 𝔥 ⊕ 𝔫_{+}, where 𝔥 = {diagonal matrices} ∩ 𝔤 = a Cartan subalgebra of 𝔤. 𝔫_{+} (resp. 𝔫_{−}) is the set of upper (resp. lower) triangular matrices with zeros on the main diagonal. 𝔟 = 𝔥 ⊕ 𝔫_{+} = a Borel subalgebra of 𝔤. The positive roots of 𝔤 (with respect to 𝔟) are α = e_{i} − e_{j} ∈ 𝔥^{∗} (1 ≤ i < j ≤ n), where e_{i}(diag(d_{1}, …, d_{n})) = d_{i}. The corresponding sl(2)-triple in 𝔤 consists of: X_{α} ∈ 𝔫_{+} with entries (X_{α})_{kl} = δ_{ik}δ_{jl}, Y_{α} = X_{−α} = (X_{α})^{T} ∈ 𝔫_{−} and H_{α} = [X_{α}, Y_{α}] ∈ 𝔥. The simple roots are α_{i} = e_{i} − e_{i+1} (1 ≤ i ≤ n − 1); put X_{i} = X_{α_{i}}, Y_{i} = Y_{α_{i}}, H_{i} = H_{α_{i}}. Each positive root is of the form e_{i} − e_{j} = α_{i} + ⋯ + α_{j−1} (i < j); write X_{i···(j−1)} (resp. Y_{i···(j−1)}) for X_{e_{i}−e_{j}} (resp. Y_{e_{i}−e_{j}}). For example, for n = 3, we have

\begin{matrix}X_{1} = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , X_{2} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) , X_{12} = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , H_{1} = ( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 0 \end{matrix} ) \\ Y_{1} = ( \begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , Y_{2} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ) , Y_{12} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ) , H_{2} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ) \end{matrix}

Commutation relations in 𝔤 are as follows: for every pair α, β of positive roots,

[X_{α}, X_{β}] (resp. [Y_{α}, Y_{β}]) = \{ \begin{matrix}X_{α+β} (resp. − Y_{α+β}) & if α + β is a root \\ 0 & otherwise \end{matrix}
[X_{α}, Y_{α}] = H_{α}, [H_{α}, X_{β}] = β(H_{α})X_{β}, [H_{α}, Y_{β}] = −β(H_{α})Y_{β}

Restriction of the standard scalar product (e_{i}, e_{j}) = δ_{ij} on the span of e_{1}, …, e_{n} defines a scalar product on 𝔥^{∗} satisfying

(α_{i}, α_{j}) = \{ \begin{matrix}2 & if i = j \\ −1 & if i = j ± 1 \\ 0 & otherwise \end{matrix}

Under the induced isomorphism 𝔥^{∗} ⟶^{∼} 𝔥, α corresponds to H_{α}, i.e. (α, β) = β(H_{α}) for all roots α, β. In particular, (α, α_{i}) = α(H_{i}) and

[H_{i}, X_{α}] = (α, α_{i})X_{α}, [H_{i}, Y_{α}] = −(α, α_{i})Y_{α} (1 ≤ i ≤ n − 1).

The induced scalar product on 𝔥 satisfies (H_{α}, H_{β}) = (α, β), in particular (H_{i}, H_{j}) = (α_{i}, α_{j}). The fundamental weights λ_{i} ∈ 𝔥^{∗} (1 ≤ i ≤ n − 1) are highest weights of ⋀^{i} V, where V is the standard n-dimensional representation of 𝔤. They are characterized by

(λ_{i}, α_{j}) = λ_{i}(H_{j}) = δ_{ij}

and satisfy

λ_{1} = (1 − \frac{1}{n}) α_{1} + (1 − \frac{2}{n}) α_{2} + ⋯ + \frac{1}{n}α_{n−1}
λ_{i} = iλ_{1} − (i − 1)α_{1} − (i − 2)α_{2} − ⋯ − α_{i−1} =

= (1 − \frac{i}{n}) (α_{1} + 2α_{2} + ⋯ + iα_{i}) + ((1 − \frac{i + 1}{n}) α_{i+1} + ⋯ + \frac{1}{n}α_{n−1}) .

The corresponding elements of 𝔥 are H_{i}^{∗} ∈ 𝔥 (1 ≤ i ≤ n − 1) such that (H_{i}, H_{j}^{∗}) = δ_{ij} and

H_{i}^{∗} = (1 − \frac{i}{n}) (H_{1} + 2H_{2} + ⋯ + iH_{i}) + ((1 − \frac{i + 1}{n}) H_{i+1} + ⋯ + \frac{1}{n}H_{n−1}) .

The half sum of positive roots is equal to

ρ = \frac{1}{2} ∑_{α>0} α = λ_{1} + ⋯ + λ_{n−1} = ∑_{i=1}^{n−1} \frac{i(n − i)}{2}α_{i} ∈ 𝔥^{∗}.

The following scalar products will appear later on:

(λ_{i}, λ_{j}) = (1 − \frac{max(i, j)}{n}) min(i, j)

(λ, 2ρ) = i(n − i) = n(λ_{i}, λ_{i})

The Casimir operator

Ω = ∑_{α>0}(X_{α}Y_{α} + Y_{α}X_{α}) + ∑_{i=1}^{n−1} H_{i}^{∗}H_{i}

lies in the centre Z(𝔤) of the universal enveloping algebra U(𝔤) of 𝔤. We have

\frac{1}{2} ∑_{α>0} H_{α} = ∑_{i=1}^{n−1} \frac{i(n − i)}{2}H_{i},

hence

Ω = 2 ∑_{α>0} X_{α}Y_{α} + ∑_{i=1}^{n−1} H_{i}(H_{i}^{∗} − i(n − i)) = 2 ∑_{α>0} Y_{α}X_{α} + ∑_{i=1}^{n−1} H_{i}(H_{i}^{∗} + i(n − i)).

The Weyl group W ⊂ Aut(𝔥^{∗}) is generated by reflections s_{i} : x ↦ (α_{i}, x)α_{i} − x (1 ≤ i ≤ n − 1); it is isomorphic to S_{n} generated by transpositions s_{i} : i ↔ i + 1. The Coxeter element c = s_{1} ⋯ s_{n−1} ∈ W has order n. Conjugation by

w_{0} = ( \begin{matrix}0 & ⋯ & 0 & 1 \\ 0 & ⋯ & 1 & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ 1 & ⋯ & 0 & 0 \end{matrix} )

is the longest element in the Weyl group.

(2.2) Lowest weight Verma modules

For μ ∈ 𝔥^{∗}, denote by V (μ) the lowest weight Verma module of U(𝔤) (not necessarily irreducible) with lowest weight μ + ρ. It is free of rank one over U(𝔫_{+}) with a basis vector (= a lowest weight vector) v satisfying

U(𝔫_{−})v = 0, H_{i}v = (μ_{i} + 1)v (μ_{i} = μ(H_{i}), 1 ≤ i ≤ n − 1).

The Casimir operator Ω acts on V (μ) by the scalar

∑_{i=1}^{n−1}(μ + ρ)(H_{i})((μ + ρ)(H_{i}^{∗}) − i(n − i)) = (μ + ρ, μ − ρ) = (μ, μ) − (ρ, ρ).

Conjugation by w_{0} interchanges 𝔫_{−} with 𝔫_{+} and transforms V (μ) into M(w_{0} · μ) (the usual Verma module with highest weight w_{0} · μ − ρ).

The module

U(μ) = V (μ) ⊕ V (μ + λ_{1}) ⊕ ⋯ ⊕ V (μ + λ_{n−1})

is an analogue of V (μ) ⊕ V (μ + 1) from 1.2 for sl(n).

(2.3) Action of L(V) on U(μ)

Denote by V = ⊕_{i=1}^{n}F · A_{i} the standard n-dimensional representation of sl(n), where A_{1} is a highest weight vector (of weight λ_{1}) and Y_{i} · A_{i} = A_{i+1} (1 ≤ i ≤ n − 1). Let L(V) be the free Lie algebra on V. We want to extend the action of sl(n) on U(μ) (for every μ ∈ 𝔥^{∗}) to an action of the semi-direct product L(V) ⋊ sl(n). All we have to do is to construct operators

A_{1} : V (μ + λ_{k}) ⟶ V (μ + λ_{k+1}) (k ∈ ℤ/nℤ)

(where λ_{0} = 0 and λ_{n+i} = λ_{i}) satisfying

(2.3.1)  \begin{aligned}[X_{i}, A_{1}] & = 0 \\ [H_{i}, A_{1}] & = δ_{i1}A_{1}  (1 ≤ i ≤ n − 1) \\ (ad(Y_{i}))^{1+δ_{i1}}A_{1} & = 0 \end{aligned}

and define inductively

A_{j+1} = [Y_{j}, A_{j}] : V (μ + λ_{k}) ⟶ V (μ + λ_{k+1}) (k ∈ ℤ/nℤ, 1 ≤ j ≤ n − 1).

For each k = 0, …, n − 1 fix a lowest weight vector u_{k} ∈ V (μ + λ_{k}). If the operators A_{j} exist they must be of the form

A_{j+1} : u_{k} ↦ P_{j,k} · u_{k+1}, (0 ≤ j, k ≤ n − 1)

where P_{j,k} = 0 for j > k and P_{j,k} ∈ U(𝔫_{+}) is a homogeneous element of degree

deg(P_{j,k}) = (λ_{1} − α_{1} − ⋯ − α_{j}) + (μ + λ_{k}) − (μ + λ_{k+1}) = α_{j+1} + ⋯ + α_{k}

for j ≤ k. Here U(𝔫_{+}) is graded by the root lattice ⊕ℤα_{i} and deg(X_{α}) = α. This implies that, for every 1 ≤ j ≤ n − 1 and 0 ≤ k ≤ n − 1,

P_{j,k} · u_{k+1} = [Y_{j}, A_{j}](u_{k}) = Y_{j}A_{j}(u_{k}) − A_{j}Y_{j}(u_{k}) = Y_{j}A_{j}(u_{k}) = Y_{j}P_{j−1,k} · u_{k+1} =
= (Y_{j}P_{j−1,k} − P_{j−1,k}Y_{j}) · u_{k+1} = [Y_{j}, P_{j−1,k}] · u_{k+1}.

The commutator [Y_{j}, P_{j−1,k}] does not necessarily lie in U(𝔫_{+}); it may contain H_{j}. As A_{1} commutes with all X_{i}, we have

A_{1} : P · u_{k} ↦ P P_{0,k} · u_{k+1} (k ∈ ℤ/nℤ)

for all P ∈ U(𝔫_{+}).

(2.4) Proposition. For every μ ∈ 𝔥^{∗} there are P_{j,k} ∈ U(𝔫_{+}) of degree α_{j+1} + ⋯ + α_{k} (0 ≤ j ≤ k ≤ n − 1), unique up to a scalar multiple P_{j,k} ↦ t_{k}P_{j,k} depending on k, such that operators

A_{j+1} : u_{k} ↦ \{ \begin{matrix}P_{j,k} · u_{k+1} & if j ≤ k \\ 0 & if j > k \end{matrix}

(for 0 ≤ j, k ≤ n − 1) satisfy (2.3.1) and define an action of L(V) ⋊ sl(n) on U(μ).

(2.5) Remark. For the first few values of j, k we have

\begin{aligned}P_{0,0} & = a_{0} · 1 \\ P_{0,1} & = a_{1} · X_{1} \\ P_{0,2} & = a_{2} · (X_{1}X_{2} − (1 + μ_{2})X_{12}) \\ P_{0,3} & = a_{3} · (X_{1}X_{2}X_{3} − (1 + μ_{2} + μ_{3})X_{12}X_{3} − (1 + μ_{3})X_{1}X_{23} + (1 + μ_{2} + μ_{3})(1 + μ_{3})X_{123}) \\ P_{1,1} & = −a_{1} · (1 + μ_{1}) \\ P_{1,2} & = −a_{2} · (1 + μ_{1} + μ_{2})X_{2} \\ P_{1,3} & = −a_{3} · (1 + μ_{1} + μ_{2} + μ_{3})(X_{2}X_{3} − (1 + μ_{3})X_{23}) \\ P_{2,2} & = a_{2} · (1 + μ_{2})(1 + μ_{1} + μ_{2}) \\ P_{2,3} & = a_{3} · (1 + μ_{2} + μ_{3})(1 + μ_{1} + μ_{2} + μ_{3})X_{3} \\ P_{3,3} & = −a_{3} · (1 + μ_{3})(1 + μ_{2} + μ_{3})(1 + μ_{1} + μ_{2} + μ_{3}) \end{aligned}

Here μ_{i} = μ(H_{i}) and a_{0}, …, a_{n−1} ∈ F are arbitrary constants. (2.6) Proof of Proposition. We begin by fixing a notation. For integers 1 ≤ a ≤ b ≤ n − 1 put I = [a, b] = {a, a + 1, …, b} and X_{I} = X_{a···b} (= (adX_{a})(adX_{a+1}) ⋯ (adX_{b−1})X_{b}). If I_{1} = [a_{1}, b_{1}] and I_{2} = [a_{2}, b_{2}] with a_{2} = b_{1} + 1, we write (I_{1})(I_{2}) for the partition of [a_{1}, b_{2}] into I_{1} and I_{2} (and similarly for (I_{1}) ⋯ (I_{r})). We allow a = b + 1, in which case I = ∅ and X_{∅} = 1. Proof of uniqueness. Assume that P_{j,k} with the required properties exist. For 0 ≤ j ≤ k ≤ n − 1 a basis of {P ∈ U(𝔫_{+}) | deg(P) = α_{j+1} + ⋯ + α_{k}} is given by X_{I_{1}} ⋯ X_{I_{ℓ}}, for all partitions [j + 1, k] = (I_{1}) ⋯ (I_{ℓ}). This means that

P_{j,k} = ∑_{[j+1,k]=(I_{1})···(I_{ℓ})} c_{j,k}(I_{1}| ⋯ |I_{ℓ})X_{I_{1}} ⋯ X_{I_{ℓ}}

for some c_{j,k}(I_{1}| ⋯ |I_{ℓ}) ∈ F.

Consider first the case j = 0. Fix a partition [1, k] = (I_{1}) ⋯ (I_{ℓ}) with I_{r} = [a_{r}, b_{r}], a_{1} = 1, b_{ℓ} = k, b_{r} + 1 = a_{r+1}. It follows from [Y_{i}, X_{i}] = −H_{i} and

\begin{aligned}[Y_{i}, X_{i···j}] & = X_{(i+1)···j}  (i < j) \\ [Y_{i}, X_{k···i}] & = −X_{k···(i−1)}  (i > k) \end{aligned}

that

[Y_{i}, X_{I_{1}} ⋯ X_{I_{ℓ}}] = \{ \begin{matrix}0 & if (∀r) i ≠ a_{r}, b_{r} \\ X_{I_{1}} ⋯ X_{I_{r}\\{i\}} ⋯ X_{I_{ℓ}} & if i = a_{r} ≠ b_{r} \\ −X_{I_{1}} ⋯ X_{I_{r}\\{i\}} ⋯ X_{I_{ℓ}} & if i = b_{r} ≠ a_{r} \\ X_{I_{1}} ⋯ X_{I_{r−1}}X_{I_{r+1}} ⋯ X_{I_{ℓ}}(1 − H_{i}) & if i = a_{r} = b_{r}, r < ℓ \\ X_{I_{1}} ⋯ X_{I_{ℓ−1}}(−H_{i}) & if i = a_{r} = b_{r}, r = ℓ \end{matrix}

Using

H_{i}(u_{k+1}) = (μ + ρ + λ_{k+1})(H_{i})u_{k+1} = (μ_{i} + 1 + δ_{i,k+1})u_{k+1},

we obtain

[Y_{i}, A_{1}](u_{k}) = [Y_{i}, P_{0,k}] · u_{k+1} = 0 (i > k)

and

[Y_{i}, A_{1}](u_{k}) = ∑ [(−δ_{ik} − μ_{i})c_{0,k}(J_{1}| ⋯ |J_{a}|i|K_{1}| ⋯ |K_{b}) + (1 − δ_{ik})c_{0,k}(J_{1}| ⋯ |J_{a}|iK_{1}| ⋯ |K_{b})−
−(1 − δ_{i1})c_{0,k}(J_{1}| ⋯ |J_{a}i|K_{1}| ⋯ |K_{b})] X_{J_{1}} ⋯ X_{J_{a}}X_{K_{1}} ⋯ X_{K_{b}} · u_{k+1} (i ≤ k),

where the sum is taken over all partitions [1, i−1] = (J_{1}) ⋯ (J_{a}) and [i+1, k] = (K_{1}) ⋯ (K_{b}). The conditions [Y_{i}, A_{1}] = 0 (2 ≤ i ≤ n − 1) are then equivalent to

(2.6.1)

\begin{matrix}−(1 + μ_{k})c_{0,k}(J_{1}| ⋯ |J_{a}|k) & = c_{0,k}(J_{1}| ⋯ |J_{a}k) (1 < i = k) \\ −μ_{i}c_{0,k}(J_{1}| ⋯ |J_{a}|i|K_{1}| ⋯ |K_{b}) + c_{0,k}(J_{1}| ⋯ |J_{a}| & iK_{1}| ⋯ |K_{b}) = c_{0,k}(J_{1}| ⋯ |J_{a}i|K_{1}| ⋯ |K_{b}) (1 < i < k) \end{matrix}

for all partitions [1, i − 1] = (J_{1}) ⋯ (J_{a}), [i + 1, k] = (K_{1}) ⋯ (K_{b}). These equations determine recursively all coefficients c_{0,k}(I_{1}| ⋯ |I_{ℓ}) as unique multiples of a_{k} = c_{0,k}(1|2| ⋯ |k). This proves the uniqueness of P_{0,k} (up to scalar multiples P_{0,k} ↦ t_{k}P_{0,k}). The formulas [Y_{j}, A_{j}] = A_{j+1} then determine all P_{j,k} (again up to scalar multiples P_{j,k} ↦ t_{k}P_{j,k}). Proof of existence. One can verify by direct calculation that

c_{0,k}(I_{1}| ⋯ |I_{ℓ}) = a_{k} ∏_{{i=1 \atop i,i+1∈same Ir}}^{k−1} (−(1 + μ_{i+1} + ⋯ + μ_{k}))

(for all partitions (I_{1}) ⋯ (I_{ℓ}) = [1, k]) are solutions of the equations (2.6.1). This implies that the corresponding operators

A_{1} : u_{k} ↦ P_{0,k} · u_{k+1}

satisfy [Y_{i}, A_{1}] = 0 for all i = 2, …, n − 1. We compute

A_{2}(u_{k}) = [Y_{1}, A_{1}](u_{k}) =
= ∑_{[2,k]=(I_{1})···(I_{ℓ})} ((−μ_{1})c_{0,k}(1|I_{1}| ⋯ |I_{ℓ}) + c_{0,k}(1I_{1}| ⋯ |I_{ℓ})) X_{I_{1}} ⋯ X_{I_{ℓ}} · u_{k+1},  (k ≥ 1)

hence

c_{1,k}(I_{1}| ⋯ |I_{ℓ}) = −a_{k}(1 + μ_{1} + ⋯ + μ_{k}) ∏_{{i=2 \atop i,i+1∈same Ir}}^{k−1} (−(1 + μ_{i+1} + ⋯ + μ_{k})). (k ≥ 1)

Continuing this process, we obtain by induction (for k ≥ j)

(2.6.2) P_{j,k} = a_{k} ∏_{i=1}^{j}(−(1 + μ_{i} + ⋯ + μ_{k})) ∑_{[j+1,k]=(I_{1})···(I_{ℓ})} X_{I_{1}} ⋯ X_{I_{ℓ}} ∏_{{i=j+1 \atop i,i+1∈same Ir}}^{k−1} (−(1 + μ_{i+1} + ⋯ + μ_{k})).

As P_{1,k} is a polynomial in X_{2}, …, X_{n−1} but not in X_{1}, we have (ad(Y_{1}))^{2}A_{1} = [Y_{1}, A_{2}] = 0. This means that A_{1} satisfies (2.3.1). Proposition is proved.

(2.7) Assume that all a_{k} ≠ 0. One can reparametrize u_{k} ↦ t_{k}u_{k} and A_{1} ↦ t_{n}A_{1} with t_{i} ∈ F^{∗}; this changes (a_{0}, a_{1}, …, a_{n−1}) to (a_{0}t_{n}t_{0}^{−1}t_{1}, a_{1}t_{n}t_{1}^{−1}t_{2}, …, a_{n−1}t_{n}t_{n−1}^{−1}t_{0}). This transformation leaves the expression (a_{0} ⋯ a_{n−1}) (mod F^{∗n}) ∈ F^{∗}/F^{∗n} invariant; replacing F by F ( ^{n} \sqrt{a_{0} ⋯ a_{n−1}}) and choosing suitable t_{i} we can (and will) assume that a_{0} = ⋯ = a_{n−1} = 1. One can write P_{j,k} in a more compact form as follows. We have

(2.7.1)  \begin{matrix}[X_{i}, X_{i+1}^{a}] & = aX_{i+1}^{a−1}X_{i(i+1)} \\ [X_{i}^{a}, X_{i+1}] & = aX_{i}^{a−1}X_{i(i+1)} \end{matrix}  (a = 0, 1, 2, ⋯)

We shall formally use these formulas for all a ∈ F, in the form

X_{i+1}^{1−a}X_{i}X_{i+1}^{a} = X_{i}^{a}X_{i+1}X_{i}^{1−a} = X_{i+1}X_{i} + aX_{i(i+1)}.

We claim that, with the normalization a_{0} = ⋯ = a_{n−1} = 1, we have

(2.7.2)  \begin{aligned}P_{0,k} & = X_{k}^{1+μ_{k}}X_{k−1}^{1+μ_{k−1}+μ_{k}} ⋯ X_{2}^{1+μ_{2}+···+μ_{k}}X_{1}X_{2}^{−μ_{2}−···−μ_{k}} ⋯ X_{k−1}^{−μ_{k−1}−μ_{k}}X_{k}^{−μ_{k}} \\ P_{j,k} & = ∏_{i=1}^{j}(−(1 + μ_{i} + ⋯ + μ_{k}))X_{k}^{1+μ_{k}} ⋯ X_{j+2}^{1+μ_{j+2}+···+μ_{k}}X_{j+1}X_{j+2}^{−μ_{j+2}−···−μ_{k}} ⋯ X_{k}^{−μ_{k}} \end{aligned}

Indeed, induction based on (2.7.1) shows that all P_{j,k} defined by (2.7.2) are in fact elements of U(𝔫_{+}). As [Y_{i}, P_{j,k}] = 0 for i ≤ j and

[Y_{j+1}, P_{j,k}] = P_{j+1,k} \frac{(−H_{j+1} − μ_{j+2} − ⋯ − μ_{k})}{(−1 − μ_{j+1} − ⋯ − μ_{k})},

we have indeed [Y_{j+1}, A_{j+1}] = A_{j+2}. This shows that P_{j,k} satisfy (2.3.1) and gives an alternative proof of the existence statement of Proposition 2.4.

(2.8) Proposition. For μ ∈ 𝔥^{∗}, the following conditions are equivalent:

(i) Every z ∈ Z(𝔤) acts on U(μ) as a scalar. (ii) Ω acts on U(μ) as a scalar. (iii) μ = −\frac{ρ}{n}. (iv) μ, μ + λ_{1}, …, μ + λ_{n−1} lie in one W -orbit.

Proof. (i) ⟹ (ii) is automatic.

(iv) ⟺ (i) follows from the fact that, for every w ∈ W, z acts on V (μ) and V (w · μ) by the same scalar (Harish-Chandra’s theorem). (ii) ⟺ (iii): the condition (ii) is equivalent to

(∀i = 1, …, n − 1) (μ, μ) = (μ + λ_{i}, μ + λ_{i}) ⟺ (∀i) − 2(μ, λ_{i}) = (λ_{i}, λ_{i}) = (\frac{2ρ}{n}, λ_{i}) ⟺

⟺ (∀i) (2μ + \frac{2ρ}{n}, λ_{i}) = 0 ⟺ μ = −\frac{ρ}{n}.

(iii) ⟹ (iv): the Coxeter element c = s_{1} ⋯ s_{n−1} ∈ W satisfies

c^{i} (−\frac{ρ}{n}) = −\frac{ρ}{n} + λ_{i} (1 ≤ i ≤ n − 1).

Proposition is proved.

(2.9) This important property of the weight −\frac{ρ}{n} was discovered by Bernstein (unpublished), who also posed the question of determining the structure of the Lie algebra L(n) defined in 3.1 below. Another approach can be found in an unpublished part of the author’s thesis [Ne].

The weight −\frac{ρ}{n} also corresponds to “exceptional characters” ω introduced by Kazhdan-Patterson [KP, p. 71] (the apparent discrepancy of signs is caused by our convention of using lowest weight, rather than highest weight, vectors). The significance of exceptional characters (for n-fold coverings of GL(n)) is explained by Cor. I.3.6 and Thm. II.2.5(b) of [KP], where it is shown that local components V_{0}(ω) of the corresponding metaplectic representations have unique Whittaker models.

(2.10) It seems appropriate to call U(−\frac{ρ}{n}) the Weil representation of sl(n). For this value of μ = −\frac{ρ}{n} we have μ_{i} = −\frac{1}{n} for all i, hence

P_{0,k} = X_{k}^{1− \frac{1}{n}} X_{k−1}^{1− \frac{2}{n}} ⋯ X_{2}^{1− \frac{k−1}{n}} X_{1}X2^{\frac{k−1}{n}} ⋯ Xk−1^{\frac{2}{n}}Xk^{\frac{1}{n}}

P_{j,k} = (∏_{i=1}^{j}(\frac{k − i + 1}{n} − 1)) X_{k}^{1− \frac{1}{n}} X_{k−1}^{1− \frac{2}{n}} ⋯ X_{j+2}^{1− \frac{k−1−j}{n}} X_{j+1}Xj+2^{\frac{k−1−j}{n}} ⋯ Xk−1^{\frac{2}{n}}Xk^{\frac{1}{n}} .

3. Higher Heisenberg Lie algebras

(3.1) The Lie algebra h(n)

From now on, we restrict our attention to the space U(−\frac{ρ}{n}) as a representation of L(V) ⋊ sl(n). Denote by L(n) the image of L(V) in End(U(−\frac{ρ}{n})). We know from 1.1 that L(2) = h(2) is the Heisenberg Lie algebra.

Definition. The Heisenberg Lie algebra h(n) of level n is the quotient of L(V) by the Lie ideal generated by (ad(x))^{n}y, for all x, y ∈ V.

For n = 2 this gives the usual Heisenberg Lie algebra: if A, B is a basis of V, then we are dividing L(V) by relations [A, [A, B]] = [B, [A, B]] = 0, i.e. we are forcing [A, B] to be a central element of h(2).

(3.2) Theorem. The canonical surjection L(V) ⟶ L(n) factors through L(V) ⟶ h(n) ⟶ L(n), i.e. the operators A_{1}, …, A_{n} define a representation of h(n) on U(−\frac{ρ}{n}).

Proof. We prove a slightly stronger statement, with L(n) replaced by another Lie algebra L̃(n), intermediate between L(V) and L(n). Let χ : Z(sl(n)) ⟶ F be the ring homomorphism given by the action on U(−\frac{ρ}{n}) (using Proposition 2.8) and denote by J_{χ} the bilateral ideal of U(L(V)⋊sl(n)) generated by z −χ(z)1 (for all z ∈ Z(sl(n))). Let L̃(n) be the image of L(V) in U(L(V) ⋊ sl(n))/J_{χ}; then L(V) ⟶ L(n) factors through L(V) ⟶ L̃(n) ⟶ L(n). In order to prove the theorem, it is enough to show that (ad(A_{1}))^{n}A_{2} ∈ J_{χ}.

As a first step, note that [z, x] = [z −χ(z)1, x] lies in J_{χ} for all z ∈ Z(sl(n)) and x ∈ L(V). In particular, [Ω, v] ∈ J_{χ} for all v ∈ V. The commutators are computed in U(L(V) ⋊ sl(n)); one must be careful to distinguish between X ∗ v (the action of X ∈ sl(n) on v ∈ V) and Xv (multiplication in U(L(V) ⋊ sl(n))). The basic commutation rule is [X, v] = Xv − vX = X ∗ v (X ∈ sl(n), v ∈ V). With this notation we have

[Ω, A_{1}] = ∑_{α>0}(X_{α}(Y_{α} ∗A_{1})+(X_{α} ∗A_{1})Y_{α} +Y_{α}(X_{α} ∗A_{1})+(Y_{α} ∗A_{1})X_{α})+ ∑_{i=1}^{n−1}(λ_{1}(H_{i})A_{1}H_{i}^{∗} +λ_{1}(H_{i}^{∗})H_{i}A_{1}) =

= ∑_{i=1}^{n−1}(X_{1···i}A_{i+1} + A_{i+1}X_{1···i}) + A_{1}H_{1}^{∗} + ∑_{i=1}^{n−1} (1 − \frac{i}{n}) H_{i}A_{1} = ∑_{i=1}^{n−1}(2X_{1···i}A_{i+1} − A_{1}) + (2H_{1}^{∗} − λ_{1}(H_{1}^{∗}))A_{1} =

= 2 ((H_{1}^{∗} − \frac{n^{2} − 1}{2n})A_{1} + ∑_{i=1}^{n−1} X_{1···i}A_{i+1}) .

Applying successively ad(Y_{1}), …, ad(Y_{n−1}) we obtain formulas for all [Ω, A_{i}]. They can be written in matrix form as

\frac{1}{2}[Ω, A] = (M − cI)A,

where c = \frac{n^{2}−1}{2n}, A = (A_{1}, …, A_{n})^{T} and

(3.2.1)  M = ( \begin{matrix}H_{1}^{∗} & X_{1} & X_{12} & ⋯ & X_{1···(n−1)} \\ Y_{1} & H_{2}^{∗} − H_{1}^{∗} & X_{2} & ⋯ & X_{2···(n−1)} \\ Y_{12} & Y_{2} & H_{3}^{∗} − H_{2}^{∗} & ⋯ & X_{3···(n−1)} \\ ⋮ & ⋮ & ... & ... & ⋮ \\ Y_{1···(n−1)} & Y_{2···(n−1)} & ⋯ & Y_{n−1} & −H_{n−1}^{∗} \end{matrix} ) ∈ M_{n}(U(sl(n))).

Note that H_{i+1}^{∗} − H_{i}^{∗} = H_{1}^{∗} − H_{1} − ⋯ − H_{i} (H_{0}^{∗} = H_{n}^{∗} = 0). Taking commutators with A_{1}, we obtain equalities in M_{n}(U(L(V) ⋊ sl(n)))

[A_{1}, M] = \frac{A_{1}}{n} I + ( \begin{matrix}−A_{1} & 0 & ⋯ & 0 \\ −A_{2} & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ −A_{n} & 0 & ⋯ & 0 \end{matrix} )

(ad(A_{1}))^{i}M = ( \begin{matrix}0 & 0 & ⋯ & 0 \\ −(ad(A_{1}))^{i−1}A_{2} & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ −(ad(A_{1}))^{i−1}A_{n} & 0 & ⋯ & 0 \end{matrix} ) (i > 1)

It follows from

(ad(A_{1}))^{j}A = ( 0 ; (ad(A_{1}))^{j}A_{2} ; ⋮ ; (ad(A_{1}))^{j}A_{n} )  (j > 0)

that

((ad(A_{1}))^{i}M)((ad(A_{1}))^{j}A) = 0 (i > 1, j > 0).

This implies that

\frac{1}{2}(ad(A_{1}))^{n}[Ω, A] = (ad(A_{1}))^{n}((M − cI)A) = ∑_{i+j=n} \binom{n}{i}((ad(A_{1}))^{i}(M − cI))((ad(A_{1}))^{j}A) =
= ((ad(A_{1}))^{n}M)A + n[A_{1}, M](ad(A_{1}))^{n−1}A + (M − cI)(ad(A_{1}))^{n}A =

( 0 ; −(ad(A_{1}))^{n−1}A_{2}A_{1} ; ⋮ ; −(ad(A_{1}))^{n−1}A_{n}A_{1} ) + ( 0 ; A_{1}(ad(A_{1}))^{n−1}A_{2} ; ⋮ ; A_{1}(ad(A_{1}))^{n−1}A_{n} ) + (M − cI)(ad(A_{1}))^{n}A = (M − (c − 1)I)(ad(A_{1}))^{n}A.

It will be shown in Corollary 3.12 below that M − λI is invertible in M_{n}(U(L(V) ⋊ sl(n))/J_{χ}) for all λ ∉ \{c, c − \frac{1}{n}, …, c − \frac{n−1}{n}\}. As [Ω, A_{i}] ∈ J_{χ}, it follows that

(ad(A_{1}))^{n}A = (M − (c − 1)I)^{−1}(M − (c − 1)I)(ad(A_{1}))^{n}A = 0

in (U(L(V)⋊sl(n))/J_{χ})^{n}, hence (ad(A_{1}))^{n}A_{i} ∈ J_{χ} for all i. This implies that L(V) ⟶ L̃(n) factors through L(V) ⟶ h(n) ⟶ L̃(n) as claimed.

(3.3) About the matrix M

The matrix M ∈ M_{n}(U(sl(n))) from (3.2.1) has very interesting properties. In the simplest case n = 2,

M = ( \begin{matrix}\frac{H}{2} & X \\ Y & −\frac{H}{2} \end{matrix} ) , M^{2} = ( \begin{matrix}\frac{H^{2}}{4} + XY & X \\ Y & \frac{H^{2}}{4} + Y X \end{matrix} ) ,

hence

M^{2} − M − \frac{Ω}{2}I = 0 in M_{2}(U(sl(2))).

For every ring homomorphism φ : Z(sl(2)) ⟶ F put

U(sl(2))_{φ} = U(sl(2)) ⊗_{Z(sl(2)),φ} F.

If φ = χ_{μ} is given by the action on V (μ), then φ(Ω) = \frac{μ^{2}−1}{2} and

(M − \frac{1 + μ}{2}) (M − \frac{1 − μ}{2}) = M^{2} − M − \frac{μ^{2} − 1}{4}I = 0 in M_{2}(U(sl(2))_{χ_{μ}}).

In particular, for μ = −\frac{1}{2},

(M − \frac{3}{4}) (M − \frac{1}{4}) = 0 in M_{2}(U(L(V) ⋊ sl(2))/J_{χ}).

We are going to prove similar identities for 𝔤 = sl(n). It will be convenient to consider the Casimir element Ω as a 𝔤-invariant element of 𝔤 ⊗ 𝔤:

Ω = ∑_{α>0}(X_{α} ⊗ Y_{α} + Y_{α} ⊗ X_{α}) + ∑_{i=1}^{n−1} \frac{H_{i} ⊗ H_{i}^{∗} + H_{i}^{∗} ⊗ H_{i}}{2} ∈ S^{2}(𝔤)^{𝔤} ⊂ (𝔤 ⊗ 𝔤)^{𝔤}.

For every pair of representations ρ_{V_{i}} : 𝔤 ⟶ End_{F} (V_{i}) (i = 1, 2) consider elements

(1 ⊗ ρ_{V_{2}})(Ω) ∈ (𝔤 ⊗_{F} End_{F} (V_{2}))^{𝔤}

(ρ_{V_{1}} ⊗ ρ_{V_{2}})(Ω) ∈ (End_{F} (V_{1}) ⊗_{F} End_{F} (V_{2}))^{𝔤} ⟶^{∼} End_{F} (V_{1} ⊗_{F} V_{2})^{𝔤}.

If V_{2} = V, then End_{F} (V) = M_{n}(F) and

(1 ⊗ ρ_{V})(Ω) = ( \begin{matrix}H_{1}^{∗} & Y_{1} & Y_{12} & ⋯ & Y_{1···(n−1)} \\ X_{1} & H_{2}^{∗} − H_{1}^{∗} & Y_{2} & ⋯ & Y_{2···(n−1)} \\ X_{12} & X_{2} & H_{3}^{∗} − H_{2}^{∗} & ⋯ & Y_{3···(n−1)} \\ ⋮ & ⋮ & ... & ... & ⋮ \\ X_{1···(n−1)} & X_{2···(n−1)} & ⋯ & X_{n−1} & −H_{n−1}^{∗} \end{matrix} ) = M^{T}

(as an element of 𝔤⊗_{F} M_{n}(F) ⊂ U(𝔤)⊗_{F} M_{n}(F)). We want to show that M^{T} satisfies a polynomial equation of degree n with coefficients in Z(𝔤) · I, by considering the action on V (μ) for variable μ ∈ 𝔥^{∗}.

(3.4) Lemma. Assume that V_{i} (i = 0, 1, 2) are 𝔤-modules such that V_{0} ⊂ V_{1} ⊗ V_{2} and Ω acts on each V_{i} by a scalar d_{i}. Then the restriction of (ρ_{V_{1}} ⊗ ρ_{V_{2}})(Ω) to End(V_{0})^{𝔤} is equal to \frac{1}{2}(d_{0} − d_{1} − d_{2}) · I.

Proof. As explained to us by A. Wassermann, the formula

(X ⊗ 1 + 1 ⊗ X)(Y ⊗ 1 + 1 ⊗ Y) + (X ⊗ 1 + 1 ⊗ X)(Y ⊗ 1 + 1 ⊗ Y)−
−(XY + Y X) ⊗ 1 − 1 ⊗ (XY + Y X) = 2(X ⊗ Y + Y ⊗ X)

implies

Ω_{V_{1}⊗V_{2}} − Ω_{V_{1}} ⊗ I_{V_{2}} − I_{V_{1}} ⊗ Ω_{V_{2}} = 2(ρ_{V_{1}} ⊗ ρ_{V_{2}})(Ω);

restricting to V_{0} we get the required statement.

(3.5) Corollary. For every V (ν) ⊂ V (μ)⊗V (λ), the restriction of (ρ_{V (μ)} ⊗ρ_{V (λ)})(Ω) to End(V (ν)) is equal to

\frac{1}{2} ((ν, ν) − (μ, μ) − (λ, λ) + (ρ, ρ)) · I.

(3.6) For every μ ∈ 𝔥^{∗} we have

V (μ) ⊗ V = ⨁_{i=1}^{n} V (μ − (λ_{i} − λ_{i−1})) (λ_{0} = λ_{n} = 0)

(note that this formula can be used to give a conceptual proof of Proposition 2.4). Applying Corollary 3.5 to V (μ) and V ⊂ V (−λ_{n−1} − ρ), we obtain

(ρ_{V (μ)} ⊗ ρ_{V})(Ω) = ⨁_{i=1}^{n} C_{i} · I_{V (μ−(λ_{i}−λ_{i−1}))}

with

C_{i} = C_{i}(μ) = \frac{1}{2} ((μ − (λ_{i} − λ_{i−1}), μ − (λ_{i} − λ_{i−1})) − (μ, μ) − (−λ_{n−1} − ρ, −λ_{n−1} − ρ) + (ρ, ρ)) =

= −(μ, λ_{i} − λ_{i−1}) − (λ_{n−1}, ρ) = −(μ, λ_{i} − λ_{i−1}) − \frac{n − 1}{2}.

The coefficients q_{i} of the polynomial

∑_{i=0}^{n} q_{i}X^{i} = ∏_{i=1}^{n} (X − C_{i}(μ)) = ∏_{i=1}^{n} (X + (μ, λ_{i} − λ_{i−1}) + \frac{n − 1}{2})

are polynomial functions of μ ∈ 𝔥^{∗}, i.e. q_{i} ∈ S(𝔥). The set of weights {λ_{i} − λ_{i−1}} forms a W -orbit, hence all q_{i} ∈ S(𝔥)^{W} are W -invariant.

Recall the definition of the Harish-Chandra isomorphism, slightly modified as we work with lowest weight Verma modules: the algebra U(𝔤) is graded by the root lattice ⊕ℤα_{i} and its component of degree zero U(𝔤)_{0} is equal to the centralizer of U(𝔥) = S(𝔥). The subspace L = U(𝔤)𝔫_{−} ∩U(𝔤)_{0} = 𝔫_{+}U(𝔤)∩U(𝔤)_{0} is a bilateral ideal in U(𝔤)_{0} satisfying U(𝔤)_{0} = L ⊕ S(𝔥). The Harish-Chandra isomorphism

ω : Z(𝔤) ⟶^{∼} S(𝔥)^{W}

is given by the composition

Z(𝔤) ↪ U(𝔤)_{0} = L ⊕ S(𝔥) ⟶^{pr} S(𝔥) ⟶^{γ} S(𝔥),

where  γ(λ ↦ P (λ)) = (λ ↦ P (λ + ρ))  (λ ∈ 𝔥^{∗}). For μ ∈ 𝔥^{∗}, any z ∈ Z(𝔤) acts on V (μ) by the

scalar ω(z)(μ).

Define

Q(X) = ∑_{i=0}^{n} ω^{−1}(q_{i})X^{i} ∈ Z(sl(n))[X].

For n = 2 the scalar product is normalized by (1, 1) = \frac{1}{2}, λ_{1} = 1, C_{1} = −\frac{μ+1}{2}, C_{2} = \frac{μ−1}{2},

(X − C_{1})(X − C_{2}) = X^{2} + X + \frac{1 − μ^{2}}{4} , Q(X) = X^{2} + X − \frac{Ω}{2}.

(3.7) Proposition. The matrix M^{T} satisfies Q(M^{T}) = 0 in M_{n}(U(sl(n))) = U(sl(n)) ⊗_{F} M_{n}(F).

Proof. An element x ∈ U(sl(n)) is equal to zero if and only if its image in End(V (μ)) is zero for all μ ∈ 𝔥^{∗}. This means that it is enough to check that

(ρ_{V (μ)} ⊗ 1)(Q(M^{T})) = Q((ρ_{V (μ)} ⊗ 1)(M^{T})) = Q((ρ_{V (μ)} ⊗ 1)(1 ⊗ ρ_{V})Ω)) =

= Q((ρ_{V (μ)} ⊗ ρ_{V})(Ω)) =^{?} 0,

which follows from the definition of Q.

(3.8) In order to pass from M^{T} to M we need some notation. If R is a ring, denote the opposite multiplication on R by a{· \atop op}b = ba; the ring R with opposite multiplication will be denoted by R^{op}. With this notation, three matrices A, B, C ∈ M_{n}(R) satisfy

(3.8.1)  C = AB in M_{n}(R) ⟺ C^{T} = B^{T} A^{T} in M_{n}(R^{op}).

As (−X){· \atop op}(−Y) − (−Y){· \atop op}(−X) = −[X, Y] in U(𝔤) for all X, Y ∈ 𝔤, the map X ↦ −X extends to an isomorphism of rings

θ : U(𝔤) ⟶ U(𝔤)^{op}, θ(X_{1} ⋯ X_{k}) = (−1)^{k}X_{k} ⋯ X_{1}.

The Harish-Chandra isomorphism satisfies ω(θ(μ)) = ω(−μ).

After all these preliminaries, put

∑_{i=0}^{n} p_{i}X^{i} = ∏_{i=1}^{n} (X + (μ, λ_{i} − λ_{i−1}) − \frac{n − 1}{2}) ∈ S(𝔥)^{W} [X]

P (X) = ∑_{i=0}^{n} ω^{−1}(p_{i})X^{i} ∈ Z(sl(n))[X].

For n = 2, ∑_{i=0}^{2} p_{i}X^{i} = (X + \frac{μ−1}{2}) (X − \frac{μ+1}{2}) = X^{2} − X + \frac{1−μ^{2}}{4} and P (X) = X^{2} − X − \frac{Ω}{2}.

(3.9) Proposition. The matrix M satisfies P (M) = 0 in M_{n}(U(sl(n))) = U(sl(n)) ⊗_{F} M_{n}(F). Proof. We have Q(M^{T}) = 0 in M_{n}(U(sl(n))) by Proposition 3.7. It follows from (3.8.1) that Q(M) = 0 in M_{n}(U(sl(n))^{op}), hence (θ(Q))(θ(M)) = 0 in M_{n}(U(sl(n))). As θ(M) = −M, this implies that P (M) = 0 for

ω(P)(X) = ∑_{i=0}^{n} ω(θ ∘ ω^{−1}(q_{i}))(−1)^{n−i}X^{i} = (−1)^{n} ∏_{i=1}^{n}(−X − C_{i}(−μ)) = ∏_{i=1}^{n}(X + C_{i}(−μ)) =

= ∏_{i=1}^{n} (X + (μ, λ_{i} − λ_{i−1}) − \frac{n − 1}{2}) .

(3.10) Corollary. For every μ ∈ 𝔥^{∗},

∏_{i=1}^{n} (M + ((μ, λ_{i} − λ_{i−1}) − \frac{n − 1}{2}) I) = 0 in M_{n}(U(sl(n))_{χ_{μ}}).

(3.11) Proposition. The matrix M satisfies the equation

∏_{j=0}^{n−1} (M − (c − \frac{j}{n}) I) = 0 (c = \frac{n^{2} − 1}{2n})

in M_{n}(U(L(V) ⋊ sl(n))/J_{χ}). Proof. For μ = −\frac{ρ}{n} and j = 0, …, n − 1 we have

−C_{j+1}(−μ) = (\frac{ρ}{n}, λ_{j+1} − λ_{j}) + \frac{n − 1}{2} = \frac{n − 1 − 2j}{2n} + \frac{n − 1}{2} = \frac{n^{2} − 1}{2n} − \frac{j}{n} = c − \frac{j}{n},

hence

ω(P) (−\frac{ρ}{n}) = ∏_{j=0}^{n−1} (X − (c − \frac{j}{n})) .

(3.12) Corollary. M − λI is invertible in M_{n}(U(L(V) ⋊ sl(n))/J_{χ}) for every λ ≠ c, c − \frac{1}{n}, ⋯, c − \frac{n−1}{n}.

4. Further properties of U(−\frac{ρ}{n}) and h(n)

(4.1) The operators Δ, Δ̃ Let μ ∈ 𝔥^{∗} be arbitrary. We are interested in the action of operators

Δ = ∑_{σ∈S_{n}} sgn(σ)A_{σ(1)} ⋯ A_{σ(n)},  Δ̃ = ∑_{σ∈S_{n}} sgn(σ)A_{σ(n)} ⋯ A_{σ(1)} = (−1)^{n(n−1)/2}Δ

on U(μ). We have

Δ(u_{k}) = d_{k}(μ)u_{k}, Δ̃(u_{k}) = d̃_{k}(μ)u_{k} (0 ≤ k ≤ n − 1)

for some constants d̃_{k}(μ) = (−1)^{n(n−1)/2}d_{k}(μ) ∈ F. As [X, Δ] = 0 for every X ∈ 𝔤, it follows that Δ(P · u_{k}) = d_{k}(μ)P · u_{k} P ∈ U(𝔫_{+}).

(4.2) Lemma. For every μ ∈ 𝔥^{∗},

d̃_{0}(μ) = ∏_{1≤i≤j≤n−1} (−(1 + μ_{i} + ⋯ + μ_{j})) = ∏_{α>0} (−(1 + (μ, α)))

d_{0}(μ) = ∏_{1≤i≤j≤n−1} (1 + μ_{i} + ⋯ + μ_{j}) = ∏_{α>0} (1 + (μ, α)) .

Proof. For σ ≠ 1, A_{σ(n)} ⋯ A_{σ(1)}(u_{0}) = 0, hence Δ̃(u_{0}) = A_{n} ⋯ A_{1}(u_{0}). The result follows from the formulas (2.6.2), which give

A_{j+1}(u_{j}) = ∏_{i=1}^{j}(−(1 + μ_{i} + ⋯ + μ_{j}))u_{j+1}.

(4.3) For μ = −\frac{ρ}{n}, μ_{i} = −\frac{1}{n} for all i = 1, …, n − 1, hence

(4.3.1)  d_{0} (−\frac{ρ}{n}) = ∏_{j=1}^{n−1} (\frac{j}{n})^{j} .

For n = 3, an easy calculation shows that

\begin{aligned}d̃_{0}(μ) & = −(1 + μ_{1})(1 + μ_{2})(1 + μ_{1} + μ_{2}) \\ d̃_{1}(μ) & = −μ_{1}(1 + μ_{2})(μ_{1} + μ_{2}) \\ d̃_{2}(μ) & = −(1 + μ_{1})μ_{2}(μ_{1} + μ_{2}) \end{aligned}  ⟹  Δ = \frac{4}{27} · I on U(−\frac{ρ}{3}).

For bigger n explicit computations become very complicated. (4.4) Conjecture. For every k = 0, …, n − 1,

d̃_{k}(μ) = (−1)^{k(n−k)}d̃_{0}(μ − λ_{k}) = (−1)^{k(n−k)} ∏_{α>0} (−(1 + (μ − λ_{k}, α))) .

(4.5) A short calculation shows that

d̃_{0} (−\frac{ρ}{n} − λ_{k}) = (−1)^{k(n−k)}d̃_{0} (−\frac{ρ}{n}) .

Conjecture 4.4 would then imply that Δ acts on U(−\frac{ρ}{n}) by the scalar (4.3.1). (4.6) The Lie algebra L(3) = G_{2} For n = 3 and μ = −\frac{ρ}{3} we have

\begin{aligned}A_{1} & : u_{0} ↦ u_{1} \\ A_{1} & : u_{1} ↦ X_{1}u_{2} \\ A_{1} & : u_{2} ↦ (X_{1}X_{2} − \frac{2}{3}X_{12})u_{0} \end{aligned} \begin{aligned}A_{2} & : u_{0} ↦ 0 \\ A_{2} & : u_{1} ↦ −\frac{2}{3}u_{2} \\ A_{2} & : u_{2} ↦ −\frac{1}{3}X_{2}u_{0} \end{aligned} \begin{aligned}A_{3} & : u_{0} ↦ 0 \\ A_{3} & : u_{1} ↦ 0 \\ A_{3} & : u_{2} ↦ \frac{2}{9}u_{0} \end{aligned}

This implies that

\begin{aligned}[A_{1}, A_{2}] & : u_{0} ↦ \frac{2}{3}u_{2} \\ [A_{1}, A_{2}] & : u_{1} ↦ \frac{2}{3}(X_{1}X_{2} − \frac{1}{3}X_{12})u_{0} \\ [A_{1}, A_{2}] & : u_{2} ↦ \frac{2}{3}X_{2}u_{1} \end{aligned}  \begin{aligned}[A_{1}, [A_{1}, A_{2}]] & : u_{0} ↦ −\frac{2}{9}X_{12}u_{0} \\ [A_{1}, [A_{1}, A_{2}]] & : u_{1} ↦ −\frac{2}{9}X_{12}u_{1} \\ [A_{1}, [A_{1}, A_{2}]] & : u_{2} ↦ −\frac{2}{9}X_{12}u_{2} \end{aligned}

Both [A_{1}, [A_{1}, A_{2}]] and X_{12} are highest weight vectors of 8-dimensional irreducible representations of sl(3). This implies that we have equalities in End(U(−\frac{ρ}{3}))

\begin{aligned}[A_{1}, [A_{1}, A_{2}]] & = −\frac{2}{9}X_{12} \\ [A_{2}, [A_{1}, A_{2}]] & = −\frac{2}{9}X_{2} \\ [A_{1}, [A_{3}, A_{1}]] & = −\frac{2}{9}X_{1} \\ [A_{1}, [A_{2}, A_{3}]] & = −\frac{2}{9}H_{1}^{∗} \end{aligned}  \begin{aligned}[A_{2}, [A_{3}, A_{2}]] & = −\frac{2}{9}(H_{2}^{∗} − H_{1}^{∗}) \\ [A_{2}, [A_{2}, A_{3}]] & = −\frac{2}{9}Y_{1} \\ [A_{3}, [A_{3}, A_{1}]] & = −\frac{2}{9}Y_{2} \\ [A_{3}, [A_{2}, A_{3}]] & = −\frac{2}{9}Y_{12} \end{aligned}

It follows that L(3) ⊂ End(U(−\frac{ρ}{3})) is spanned by A_{1}, A_{2}, A_{3}, [A_{1}, A_{2}], [A_{1}, A_{3}], [A_{2}, A_{3}] and the eight double commutators listed above. This is a 14-dimensional simple Lie algebra of type G_{2}. Moreover, L(3) contains the image of sl(3) in End(U(−\frac{ρ}{3})); this seems to be a specific feature of the case n = 3.

This representation of G_{2} on U(−\frac{ρ}{3})) appeared in the work of S. Gelfand [Ge] and also in [Ne]. On the group level, the corresponding representation of a 3-fold covering of G_{2} was studied by Savin [Sa].

(4.7) Proposition. dim h(n) = ∞ for n > 2.

Proof. We argue by induction. Write V_{n} for V and V_{n−1} for the subspace spanned by A_{i} with 1 ≤ i ≤ n − 1. By definition, I(n) := Ker(L(V_{n}) ⟶ h(n)) is the Lie ideal generated by (ad(x))^{n}y (x, y ∈ V_{n}); this implies that L(V_{n−1}) ∩ I(n) is equal to the Lie ideal generated by (ad(x))^{n}y (x, y ∈ V_{n−1}). It follows that L(V_{n−1}) ∩ I(n) ⊂ I(n − 1), hence h(n − 1) is a subquotient of h(n) and dim h(n) ≥ dim h(n − 1). It remains to show that dim h(3) = ∞. Put a = [A_{1}, [A_{2}, A_{3}]], b = [A_{1}, [A_{1}, A_{3}]] ∈ L(V_{3}). It is enough to show that the images of (ad(a))^{k}b (k = 1, 2, …) are linearly independent in h(3). Suppose that ∑_{k≥1} c_{k}(ad(a))^{k}b ∈ I(3) for some c_{k} ∈ F. As I(n) is a homogeneous ideal of L(V_{n}) for every n (with respect to the grading induced from the tensor algebra T (V_{n})), it follows that, for all k ≥ 1, c_{k}(ad(a))^{k}b ∈ I(3), hence c_{k}(ad(a))^{k}b = 0 in h(3). However, the image of (ad(a))^{k}b in End(U(−\frac{ρ}{3})) is non-zero for all k ≥ 1, as (ad(H_{1}^{∗}))^{k}X_{1} = X_{1} in sl(3). This shows that all c_{k} = 0 as claimed.

5. Open questions

(5.1) Can h(n) be related to some known class of infinite-dimensional Lie algebras? (5.2) Can the representation U(−\frac{ρ}{n}) of h(n) be characterized in a manner similar to the Stone-von Neumann theorem? (5.3) The Kazhdan-Patterson space V_{0}(ω) mentioned in 2.9 is a representation space of the n-fold covering of SL(n, K) analogous to V (−\frac{ρ}{n}) (here K is a local field containing μ_{n}). Is there a group-version of h(n) defined over K and acting on the space obtained from V_{0}(ω) by adding n − 1 representations analogous to V (−\frac{ρ}{n} + λ_{i})? (5.4) The representation of h(n)⋊sl(n) on U(−\frac{ρ}{n}) depends on a parameter a ∈ F^{∗}/F^{∗n} (equal to a_{0} ⋯ a_{n−1} (mod F^{∗n})), i.e. on the form x ↦ ax^{n} of degree n on a one-dimensional vector space over F. It would be interesting to construct natural representations of sl(n, F) corresponding to other forms of degree n, such as norm forms for extensions E/F of degree n or reduced norms for division algebras of degree n^{2} over F (in the well-known case n = 2 everything works even on the level of groups; for the case n = 3 see [Ka]). (5.5) Is there an infinite-dimensional analogue of Howe’s dual reductive pairs in the context of 5.4?

References

[Ge] S.I. Gelfand, The Weil representation of a Lie algebra of type G_{2} and related representations of SL_{3}, Funktsional. Anal. i Prilozhen. 14:1 (1980), 51–52. (Russian)

[Ka] D.A. Kazhdan, The minimal representation of D_{4}, in: Operator algebras, unitary representations, enveloping algebras, and invariant theory (ed. A. Connes, M. Duflo, A. Joseph, R. Rentschler), Progress in Math. 92, Birkhäuser Boston, Boston, MA, 1990. pp. 125–158.

[KP] D.A. Kazhdan, S.J. Patterson, Metaplectic forms, Publ. Math. de l’I.H.E.S., 59 (1984), 35–142. (Corrections: ibid. 62 (1985), 419.)

[Ne] J. Nekovář, Jacobi forms, preprint 1988.

[Sa] G. Savin, An analogue of the Weil representation for G_{2}, J. Reine Angew. Math. 434 (1993), 115–126.

[We] A. Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211.

Jan Nekovář Department of Pure Mathematics and Mathematical Statistics University of Cambridge 16 Mill Lane Cambridge CB2 1SB UK

nekovar@dpmms.cam.ac.uk
http://www.dpmms.cam.ac.uk/~nekovar/