Star-products and the eigenvalue of the Casimir

William Graham^{1}

1 Introduction

These are informal notes of a talk on joint work with David Vogan. What it is about is in essence a calculation. A consequence of this calculation is the non-existence of invariant star-products on certain nilpotent orbits.

2 The eigenvalue of the Casimir

Let G be a complex simple Lie group with Lie algebra 𝔤, and let J ⊂ U(𝔤) be a primitive ideal, i.e. the annihilator of an irreducible U(𝔤)-module. Let Ω be the Casimir element of U(𝔤) (with respect to a nondegenerate G-invariant bilinear form B on G); so Ω = ∑ x_{i}x_{i}^{∗}, where {x_{i}} and {x_{i}^{∗}} are dual bases of 𝔤 with respect to B. Under the natural map U(𝔤) → U(𝔤)/J, the element Ω maps to a scalar c_{J}. This scalar I will call the eigenvalue of the Casimir.

One would like to construct primitive ideals attached to certain nilpotent orbits in 𝔤^{∗} – one might call this quantizing the orbit. There is a kind of quantization in symplectic geometry called star-products, which is formally very similar to the construction of primitive ideals. Since nilpotent orbits are symplectic, it is natural to try to construct primitive ideals using star-products.

In fact what I am going to explain is that if we have a star-product satisfying some conditions that are natural from the viewpoint of symplectic geometry, then we do get a primitive ideal J_{star}, and moreover we can actually calculate c_{J_{star}}.

This has the satisfying feature that we can calculate a number. This is a little surprising because the theorems about the existence of star-products in symplectic geometry are not very explicit. On the other hand, it has the unsatisfying consequence that in some cases the number c_{J_{star}} is not compatible with known results about primitive ideals, showing that in these cases such star-products do not exist. We are then calculating numbers attached to non-existent star-products. I don’t know if they have any significance.

1Department of Mathematics, University of Georgia, Boyd Graduate Research Center, Athens, GA 30602

3 Star-products in symplectic geometry

The notion of star-product was introduced as a way to formulate quantum mechanics (see [BFFLS]). In classical mechanics, the observables are C^{∞}(M), where M is a symplectic manifold. In quantum mechanics, the observables are some noncommutative algebra related to C^{∞}(M). This algebra is given, in the star-product formulation, as a deformation of the ring R = C^{∞}(M). Formally, a star-product is an associative algebra structure ∗_{t} on the ring R[[t]] which is t-linear and for r, s ∈ R takes the form

r ∗_{t} s = ∑ t^{i}m_{i}(r, s),

where m_{i} : R ⊗ R → R are bilinear maps satisfying certain conditions. Associativity imposes strong conditions on the m_{i}; in addition we require:

(1) m_{0} is the usual product, and m_{i}(constant, r) = m_{i}(r, constant) = 0 for i > 0 (this implies that 1 ∈ R is the multiplicative identity under ∗_{t}).

(2) m_{1} = \frac{1}{2}\{, \}, where \{, \} is the Poisson bracket.^{1}

(3) m_{i} is symmetric (resp. anti-symmetric) if i is even (resp. odd). These conditions are not so serious. More serious is:

(4) The m_{i} are bidifferential operators, i.e., in local coordinates we have

m_{i}(r, s) = ∑ a_{IJ} \frac{∂^{I}r}{∂x^{I}} \frac{∂^{J}s}{∂x^{J}} .

Finally, there is a result of Lichnerowicz [L] which says that given a star-product (i.e. ∗_{t} satisfying these conditions), we can modify it so that:

(5) Each m_{i} has (bi)-order at most (i, i).

It is difficult to write down star-products explicitly, but for example in ℝ^{2}, with coordinates x, y and Poisson bracket

\{, \} = \frac{∂}{∂x} ⊗ \frac{∂}{∂y} − \frac{∂}{∂y} ⊗ \frac{∂}{∂x},

we can define a star-product by the rule

m_{i} = \frac{1}{2^{i}i!} (\frac{∂}{∂x} ⊗ \frac{∂}{∂y} − \frac{∂}{∂y} ⊗ \frac{∂}{∂x})^{i} .  (3.1)

This defines a star-product on ℝ^{2}; a similar formula works for ℝ^{2n}.

In fact, I could use the same formulas to define (in this case) a star-product on R[[t]] where R is the polynomial ring ℝ[x, y] (rather than C^{∞}(M)). In this case, I can define a new product ∗_{1} on R by

r ∗_{1} s = r ∗_{t} s|_{t=1}

(the sum converges). The algebra (R, ∗_{1}) is isomorphic to the noncommutative algebra R[x, \frac{∂}{∂x}].

The following theorem was proved by deWilde-Lecomte and independently and later, using a different method, by Fedosov (see [DL], [F], [D]).

Theorem 3.1 Any symplectic manifold admits a star-product, unique up to (some kind of) equivalence.

1The factor of \frac{1}{2} is usually omitted; it can be eliminated by rescaling the Poisson bracket.

4 Nilpotent orbits

Nilpotent coadjoint orbits in real Lie groups are symplectic manifolds, so it is reasonable to consider star-products on them. We are interested in simple Lie algebras; it is convenient to work with adjoint orbits, which are identified (using a nondegenerate invariant bilinear form) with coadjoint orbits. Let G_{ℝ} be a real reductive group with Lie algebra 𝔤_{ℝ} and let O_{ℝ} ⊂ 𝔤_{ℝ} be a nilpotent orbit. Then O_{ℝ} is a cone (i.e. invariant under dilation), so it makes sense to say a function r on O_{ℝ} is homogeneous of degree k: it transforms appropriately under dilation.

If m_{0} is the ordinary product on R = C^{∞}(O_{ℝ}), and m_{1} the Poisson bracket, and if r_{1} and r_{2} are homogeneous of degrees k_{1} and k_{2}, then m_{0}(r_{1}, r_{2}) (resp. m_{1}(r_{1}, r_{2})) is homogeneous of degree k_{1} + k_{2} (resp. k_{1} + k_{2} − 1).

Definition 4.1 A star-product on O_{ℝ} is called graded if for r_{1}, r_{2} as above, each m_{i}(r_{1}, r_{2}) is homogeneous of degree k_{1} + k_{2} − i.

Lemma 4.2 If O_{ℝ} admits a G_{ℝ}-invariant star-product, then it admits a graded G_{ℝ}- invariant star-product.

5 Complex nilpotent orbits

Most of the rest of this talk will deal with complex nilpotent orbits (i.e. nilpotent G-orbits in 𝔤, with G and 𝔤 as above). These have G-invariant holomorphic symplectic forms. By definition, a holomorphic symplectic form is a nondegenerate 2-form on the holomorphic tangent spaces, which is holomorphic as a section of the bundle ⋀^{2} T^{∗}X. For example, on ℂ^{2} there is the holomorphic symplectic form dz_{1} ∧ dz_{2}. This is not a real symplectic form on ℂ^{2}, but its restriction to ℝ^{2} ⊂ ℂ^{2} is dx_{1} ∧ dx_{2} − dy_{1} ∧ dy_{2}, which is a real symplectic form.

On a complex manifold with a holomorphic symplectic form, there is a notion of holomorphic star-product (defined by the obvious analogues of (1)-(5) above).

Now suppose 𝔤_{ℝ} ⊂ 𝔤_{ℝ} ⊗ ℂ, G_{ℝ} ⊂ G, O_{ℝ} = G_{ℝ} · x ⊂ O = G · x.

Lemma 5.1 O_{ℝ} admits a graded G_{ℝ}-invariant star-product if and only if O admits a graded G-invariant holomorphic star-product.

6 Deformations and primitive ideals

Most of the results of this section are due to Vogan [V], who observed that the problem of finding certain primitive ideals in U(𝔤) can be formulated in terms of deformations of algebras.

Suppose J ⊂ U(𝔤) is an ideal. Then J inherits a filtration J_{0} ⊂ J_{1} ⊂ ⋯ from U(𝔤), and gr J = ⊕J_{i}/J_{i_{1}} = I is a graded ideal in gr U(𝔤) = S(𝔤). The zero set V (I) ⊂ 𝔤^{∗} ≃ 𝔤 is a cone. The following result is due to Joseph and Borho-Brylinski.

Theorem 6.1 If J is a primitive ideal then V (gr J) = \overline{O} ⊂ 𝔤 is a nilpotent orbit closure.

What this means is that \sqrt{gr J} is the ideal I(\overline{O}) of functions vanishing on \overline{O}. It is natural to ask for ideals J with gr J = I(\overline{O}). These would be the largest ideals attached to an orbit, and therefore should be related to the smallest representations attached to an orbit.

Given such an ideal, the ring A = U(𝔤)/J has a natural filtration such that gr A = R(\overline{O}) = S(𝔤)/I(\overline{O}), (here R(Z) denotes the ring of regular functions on the algebraic variety Z). In fact (writing simply \overline{R} for R(\overline{O})) the ring A is isomorphic to (\overline{R}, ∗_{1}). Here ∗_{t} = ∑ t^{i}m_{i}, where the m_{i} are G-invariant maps \overline{R} ⊗ \overline{R} → \overline{R}, satisfying conditions (1)-(3)

above^{2} and moreover a condition related to the grading of \overline{R}:

(G) m_{i} has degree −i as an operator, i.e., if r, s are homogeneous elements of \overline{R} of degrees a, b, then m_{i}(r, s) is homogeneous of degrees a + b − i.

I will not prove that from an ideal J one can get ∗_{t} on \overline{R}. However, it is easy to go the other way, i.e., to construct J from \overline{R} and ∗_{t}. Indeed, because \overline{R} has no terms of negative degree, ∗_{1} = ∗_{t}|_{t=1} defines an algebra structure on \overline{R}. There is an algebra map

Φ : U(𝔤) → (\overline{R}, ∗_{1}),

defined as follows. For X ∈ 𝔤, Φ(X) = X viewed as a function on O ⊂ 𝔤^{∗} ≃ 𝔤. The conditions on the m_{i} imply that

Φ(X) ∗_{1} Φ(Y) − Φ(Y) ∗_{1} Φ(X) = Φ([X, Y])

so Φ extends to a surjective algebra map U(𝔤) → (\overline{R}, ∗_{1}). The kernel of this map will be a primitive ideal J with gr J = I(\overline{O}).

6.1 Normality of orbit closures

There is one technical point connected with normality of the orbit closure \overline{O}. This point can be ignored if the orbits we are dealing with have normal closures, as is the case for the examples considered at the end of the paper. Therefore this section is not essential for the rest of the talk.

The technical point is this. If \overline{O} is normal, then the rings \overline{R} = R(\overline{O}) and R = R(O) coincide; in general R is the normalization of \overline{R}. Instead of considering a product ∗_{t} on \overline{R} we could consider a product ∗_{t} on R, again satisfying conditions (1)-(3) and (G). Since R is in general a proper subset of \overline{R}, these notions are different. However, the kernel J of a map U(𝔤) → (R, ∗_{1}) will again be a primitive ideal. For this reason, even in the non-normal case it is worth considering ∗_{t} on R, as well as on \overline{R}.

7 The eigenvalue of the Casimir

The results of the previous section motivate studying G-invariant ∗_{t} on R = R(O) satisfying the associativity constraints, plus conditions (1)-(3) and (G). It is natural therefore to ask in addition for (4) and (5) (in the context of algebraic, or holomorphic, bidifferential operators).

2Actually, for condition (3), we need J^{t} = J, where the superscript is the transpose anti-involution of 𝔤. This holds if J is the annihilator of a unitary representation.

Recall that we have identified 𝔤 with 𝔤^{∗} via a nondegenerate form B. Pick f ∈ O ⊂ 𝔤, and put f into a standard triple {h, e, f}. Under the action of the 𝔰𝔩_{2} they span, decompose 𝔤 into irreducibles as 𝔤 = ⊕_{n}V_{n}^{⊕a_{n}}, where V_{n} is the irreducible 𝔰𝔩_{2}-module of highest weight n. In other words, a_{n} is the multiplicity of V_{n} in 𝔤.

Theorem 7.1 Suppose ∗_{t} is a G-invariant star-product satisfying conditions (1)-(5) and (G). Then the eigenvalue of the Casimir, denoted Φ(Ω), is

Φ(Ω) = \frac{dim 𝔤}{B(e, f)} ∑_{n} a_{n}C_{n}.  (7.2)

Here the numbers C_{n} depend only on n, and can be explicitly calculated.

I will not give the explicit formula for C_{n}, but simply note that C_{0} = C_{1} = 0, C_{2} = −1/8.

8 Outline of proof

To begin, I will make more explicit the identification between 𝔤 and 𝔤^{∗}. Elements of S(𝔤) are functions on 𝔤^{∗}; hence since O ⊂ 𝔤^{∗}, they define functions on O. Since 𝔤 = S^{1}(𝔤), elements of 𝔤 define degree 1 functions on O. In particular this is true of h, e, f. Under the identification of 𝔤 with 𝔤^{∗} by the form B, for x ∈ 𝔤, the value of x at f ∈ O, x(f), is x(f) = B(x, f).

8.1 Ingredients of the proof

We can divide the proof into 4 steps:

1. Reduce to understanding m_{2}.

2. Reduce further to understanding the scalar m_{2}(e, f).

3. Define a new map m : R ⊗ R → ℂ by m(r ⊗ s) = m_{2}(r, s)(f). Then we need properties of m following from:

(a) Associativity of ∗_{t};

(b) G-invariance of m_{2};

(c) The order of m_{2} is (2, 2);

(d) m_{2} vanishes on constants.

4. Vanishing property of elements of 𝔤^{f} (viewed as elements of R) at f ∈ O. Here 𝔤^{f} ⊂ 𝔤 is the stabilizer subalgebra of f.

9 Some details

Step 1 consists of the following lemma.

Lemma 9.1 Choose an orthonormal basis {x_{i}} (with respect to the form B). Then Φ(Ω) = ∑_{i} m_{2}(x_{i}, x_{i}).

Proof. Since x_{i} has degree 1 in R, our assumptions on the m_{k} imply that deg m_{k}(x_{i}, x_{i}) = 2 − k. In particular, m_{k}(x_{i}, x_{i}) = 0 for k ≥ 3. Therefore,

Φ(Ω) = ∑_{i} x_{i}^{2} + \frac{1}{2}[x_{i}, x_{i}] + m_{2}(x_{i}, x_{i}).

But ∑_{i} x_{i}^{2} is an invariant positive degree element in S(𝔤), and therefore vanishes on O. That is, ∑_{i} x_{i}^{2} is 0 as an element of R. Since [x_{i}, x_{i}] = 0, the result follows. □

For Step 2, we can simplify matters a little by rescaling the form B so that B(e, f) = 1; this is harmless since both sides of (7.2) transform in the same way under rescaling. Now, 𝔤 ⊂ R, so

m_{2}|_{𝔤⊗𝔤} : 𝔤 ⊗ 𝔤 → ℂ

defines a symmetric bilinear form on 𝔤. Since 𝔤 is simple, this form is a multiple λ of B, and the scalar λ is given by m_{2}(e, f). Thus,

Φ(Ω) = ∑ m_{2}(x_{i}, x_{i}) = ∑ λB(x_{i}, x_{i}) = dim 𝔤 · m_{2}(e, f).

By our assumptions on m_{2}, m_{2}(e, f) is a degree 0 element of R, i.e. a scalar. To evaluate it, it suffices to evaluate it at any point of O, in particular, at the special point f ∈ O.

We have defined m : R ⊗ R → ℂ by m(r, s) = m_{2}(r, s)(f). By the above remarks, Theorem 7.1 will follow from:

Theorem 9.2 m(e, f) = ∑ a_{n}C_{n}, where a_{n} and C_{n} are as above.

To establish this theorem, we need to carry out Step 3, which gives some properties of the map m. The properties are (the reasons they hold are sketched in Step 3 above):

(a) Suppose a, b, c ∈ R. Then

a(f) · m(b, c) − m(ab, c) + m(a, bc) − m(a, b) · c(f) = N(a, b, c),

where we define

N(a, b, c) = \frac{1}{4}\{\{a, c\}, b\}(f).

If a, b, c ∈ 𝔤 (recall that 𝔤 ⊂ R) then the right hand side is

\frac{1}{4}B([a, c], [b, f]).  (9.3)

(b) For α ∈ R ⊗ R, we have m((ad f)α) = 0.

(c) If a ∈ R vanishes to order ≥ 3 at f, then for all b ∈ R, we have m(a, b) = 0.

(d) m(a + constant, b) = m(a, b).

Finally, Step 4 says: If x is in the stabilizer algebra 𝔤^{f}, then x, viewed as an element of R, vanishes to order 2 at f.

10 Completion of the proof for 𝔰𝔩_{2}

We will use the facts from the previous section to complete the proof in the special case where 𝔤 = 𝔰𝔩_{2}, and O is the non-trivial nilpotent orbit. We must show that

m(e, f) = C_{2} = −\frac{1}{8}.

Let I_{f,k} ⊂ R denote the ideal of functions vanishing to order k at f ∈ O.

Lemma 10.1 m(e, f) = −\frac{1}{4}m(e, h^{2}).

Proof. Since h^{2}+4ef is invariant, it vanishes on O, so as elements of R we have ef = −\frac{1}{4}h^{2}. As e(f) = 1, we have f = −\frac{1}{4}h^{2} modulo I_{f,3}. Since m_{2} has order (2, 2), we conclude that m(e, f) = −\frac{1}{4}m(e, h^{2}). □

Lemma 10.2 m(e, h^{2}) = N(e, e, f).

Proof. Again using the equation h^{2} + 4ef = 0 in R, we have

e ⊗ h^{2} = 2e ⊗ ef − f ⊗ e^{2} + (ad f)α

for some α ∈ R ⊗ R. Therefore,

m(e, h^{2}) = 2m(e, ef) − m(e^{2}, f) = 2m(e, f) − m(e^{2}, f),

since ef = f modulo I_{f,3} (as f vanishes to order 2 at f). By Step 3 above,

\begin{aligned}N(e, e, f) & = & e(f)m(e, f) − m(e^{2}, f) + m(e, ef) − m(e, e)f(f) \\ & = & 2m(e, f) − m(e^{2}, f), \end{aligned}

proving the lemma. □

We conclude that m(e, f) = −\frac{1}{4}N(e, e, f). But we can calculate N(e, e, f) from (9.3); this gives m(e, f) = −\frac{1}{8}, completing the proof.

11 Nonexistence of star-products

Suppose that for some orbit O, we know the scalars c_{J} for the primitive ideals J with gr J = I(\overline{O}). We can compare these scalars with the scalar c_{J_{star}} of Theorem 7.1. If c_{J_{star}} does not equal one of the possible c_{J}, we conclude that O does not admit a G-invariant star-product with properties (1)-(5) and (G).

In particular, this is the case for the nilpotent orbits O_{p} in 𝔰𝔭_{2n} corresponding to the partitions (2^{p}, 1^{2n−2p}). These orbits have normal closures [H]. For most p, there is a unique completely prime primitive ideal J_{p} in U(𝔤) whose associated variety is the orbit closure \overline{O_{p}} (the annihilator of a holomorphic discrete series representation studied by Rossi-Vergne and others). This implies that J_{p} is the only possible ideal with gr J_{p} = \overline{O}_{p}. The infinitesimal characters of the ideals J_{p} are known, so the scalars c_{J_{p}} can be calculated; and for p > 1 they disagree with the scalar produced by Theorem 7.1. We conclude that these orbits have no G-invariant star-products with properties (1)-(5) and (G). For p = 1 the orbit admits a double covering by ℂ^{2n} − {0}; there is a star-product on ℂ^{2n} − {0} defined by (3.1), and this descends to a star-product on the orbit having all of the above properties.

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