ON REPRESENTATIONS OF THE LIE SUPERALGEBRA P (n)

VERA SERGANOVA

Abstract. We introduce a new way to study representation of the Lie superalgebra p (n). Since the center of the universal enveloping algebra U acts trivially on all irreducible representations, we suggest to study the quotient algebra Ū by the radical of U. We show that Ū has a large center which separates irreducible representations. We give a description of Ū factorized by a generic central character. Using this description we obtain character formulae of generic (infinite-dimensional) irreducible representations. We also show that Ū is a quantum deformation of the algebra of functions on some Poisson subvariety in the coadjoint representation.

1. Introduction

In this paper we study representations of the classical Lie superalgebra p (n) introduced in [3]. Let us recall the definitions. Denote by p̃(n) the Lie superalgebra of all endomorphisms of (n|n)-dimensional vector superspace which preserve a fixed odd non-degenerate symmetric form. The Lie superalgebra p (n) is the commutator of p̃(n). It is simple when n ≥ 2.

The main difficulty in representation theory of p̃(n) and p (n) is that the center of the universal enveloping superalgebra of p̃(n) is trivial. The center of the universal enveloping of p (n) is non-trivial see [1], but it acts by the same central character on all irreducible representations. Therefore the powerful technique of central characters seems not applicable. We show that it still can work after some reduction.

The universal enveloping algebra U of p̃(n) has a non-trivial radical I as it was shown in [4]. Let Ū be the quotient algebra U/I. We suggest to study representation theory of Ū instead of U. Using this approach we do not loose any information about irreducible representations but in fact gain some additional information. Our main result is that Ū has a non-trivial center Z, which separates irreducible representation. In Theorem 3.4 below we describe Z precisely. Furthermore we show that for a generic central character χ : Z → ℂ the algebra Ū/Ū Ker χ is isomorphic to the matrix algebra over U (𝔤_{0}) /U (𝔤_{0}) Ker χ_{0} for a certain central character χ_{0} of U (𝔤_{0}). This allows to write character formulae for a generic irreducible weight representation.

The same method is applicable to p (n).

In the last section we suggest some geometric motivation of our construction. We realize the graded algebra Ū as the algebra of function on a Poisson subvariety Q in the coadjoint representation 𝔤^{∗}. This Q is the union of coadjoint orbits of all even elements. It is a specific feature of supergeometry that Q does not coincide with 𝔤^{∗} but is a closed subvariety with singularities. Thus Ū is a quantum deformation of the superalgebra of functions on the Poisson supervariety Q.

Date: October 30, 2001.

I would like to thank M. Gorelik and I. Musson for very useful discussions and remarks.

2. Preliminaries

Let 𝔤 be the Lie superalgebra p̃(n) (n ≥ 2) over ℂ, U be its universal enveloping algebra. Recall that 𝔤_{0} ≅ gl (n), and 𝔤 has a ℤ-grading 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}, where 𝔤_{−1} is isomorphic to Λ^{2}E^{∗} and 𝔤_{1} is isomorphic to S^{2}E as 𝔤_{0}-module, here E stands for the standard module over 𝔤_{0}. Let 𝔤′_{0} = [𝔤_{0}, 𝔤_{0}] ≅ sl (n). The grading of 𝔤 induces the grading of U in the natural way. Thus

U = ⊕_{i=−n(n−1)/2}^{n(n+1)/2}U_{i}.

We fix a Cartan subalgebra 𝔥 in 𝔤, which coincides with Cartan subalgebra of 𝔤_{0} and identify 𝔥 with 𝔥^{∗} by means of the 𝔤_{0}-invariant form (A, B) = tr AB on 𝔤_{0}. If ε_{1}, …, ε_{n} is the standard orthogonal basis in 𝔥, then the roots of 𝔤 are of the following form:

Roots of 𝔤_{0} : ε_{i} − ε_{j}, i ≠ j, 1 ≤ i, j ≤ n

Roots of 𝔤_{−1} : − ε_{i} − ε_{j}, 1 ≤ i < j ≤ n

Roots of 𝔤_{1} : ε_{i} + ε_{j}, 1 ≤ i ≤ j ≤ n

Each root has multiplicity one. For each root α we fix an element X_{α} from the root space such that [X_{α}, X_{−α}] = H_{α} with (H_{α}, H_{α}) = 2. If −α is not a root we choose X_{α} being an arbitrary non-zero element of the root space. For μ ∈ 𝔥^{∗} we denote by μ_{i} its coordinate in the standard basis, i.e. μ = ∑_{i=1}^{n} μ_{i}ε_{i}.

We choose a Borel subalgebra 𝔟 ⊃ 𝔥 such that the positive roots are ε_{i} − ε_{j} for i < j and all roots of 𝔤_{1}. Let 𝔫 = [𝔟.𝔟]. As it was shown in [3], all irreducible finite dimensional 𝔤-modules are modules of highest weight. An irreducible module with highest weight μ is finite-dimensional if and only if μ_{i} − μ_{j} is a non-negative integer for all i < j.

Finally let 𝔤′ = [𝔤, 𝔤], 𝔥′ = 𝔥 ∩ 𝔤′. One can check that 𝔤′ = 𝔤_{−1} ⊕ 𝔤′_{0} ⊕ 𝔤_{1}. Let U′ denote the universal enveloping of 𝔤′.

3. Radical of U

In this section we will show that U has a non-zero Jacobson radical and give some description of this radical. This radical was constructed in [4]. We put d = n (n − 1) /2. Note that d = dim 𝔤_{−1}.

Let M_{μ}^{0} denote Verma module over 𝔤_{0} with highest weight μ and M_{μ} be Verma module over 𝔤. One can verify easily that

M_{μ} = Ind_{𝔤_{0}⊕𝔤_{1}}^{𝔤} M_{μ}^{0}, (3.1)

assuming 𝔤_{1}M_{μ}^{0} = 0.

Let v be a highest vector of M_{μ} and X = Π_{i<j}X_{ε_{i}+ε_{j}}, Y = Π_{i<j}X_{−ε_{i}−ε_{j}}. Finally let Δ (μ) = Π_{i<j} (μ_{i} − μ_{j} + j − i − 1).

Lemma 3.1. XY v = Δ (μ) v.

Proof. We order the roots ε_{i} + ε_{j} in the following way

ε_{i} + ε_{j} < ε_{p} + ε_{q} iff i < p or i = p and j < q.

According to this order we index our roots α_{1}, …, α_{d}.

Then whenever α < β < γ we have

[X_{α}, X_{−β}] ∈ 𝔟_{0},
[[X_{α}, X_{−β}] , X_{−γ}] is either zero or X_{−δ}, with γ < δ.

Now let w_{k} = X_{−α_{k}} … X_{−α_{d}}v, w_{d+1} = v. Using the above relations one verifies that

X_{α_{k}}w_{k} = (μ − α_{k+1} − ⋯ − α_{d}, H_{α_{k}}) w_{k+1}.

If α_{k} = ε_{i} + ε_{j}, then (μ − α_{k+1} − ⋯ − α_{d}, H_{α_{k}}) = μ_{i} − μ_{j} + j − i − 1. Repeating this argument one obtains Xw_{1} = Δ (μ) w_{d+1}.

Let 𝔥_{0}^{∗} = {μ ∈ 𝔥^{∗} | μ_{i} − μ_{j} ∉ ℤ, i ≠ j}. Clearly 𝔥_{0}^{∗} is a Zariski dense set in 𝔥^{∗}.

Lemma 3.2. If μ ∈ 𝔥_{0}^{∗} then M_{μ} is irreducible.

Proof. If we put deg v = 0, then M_{μ} inherits ℤ-grading from U, such that M_{μ} = (M_{μ})_{0} + ⋯ + (M_{μ})_{−d}. Note that (M_{μ})_{−d} = Y M_{μ}^{0} = M_{μ+σ}^{0}, where σ is the sum of all roots of 𝔤_{−1}.The highest vector of M_{μ+σ}^{0} is Y v. By (3.1) M_{μ} is free over U (𝔤_{−1}). Therefore every non-zero submodule N of M_{μ} has a nontrivial intersection with M_{μ+σ}^{0}. The conditions on μ ensure that M_{μ+σ}^{0} is irreducible. Thus, N must contain Y v, and therefore XY v = Δ (μ) v. As μ ∈ 𝔥_{0}^{∗}, Δ (μ) ≠ 0, we have v ∈ N and hence N = M_{μ}.

Lemma 3.3. Let P_{μ} = Ind_{𝔤_{0}}^{𝔤} M_{μ}^{0}.

(a) If μ ∈ 𝔥_{0}^{∗}, then P_{μ} has finite length with irreducible subquotients M_{μ+γ}, where γ runs over sums of non-repeating roots of 𝔤_{1}. Note that the length s of P_{μ} is equal to 2^{n(n+1)/2}.

(b) ⊕_{μ∈𝔥_{0}^{∗}}P_{μ} is a faithful U-module.

Proof. To prove (a) use

P_{μ} = Ind_{𝔤_{0}⊕𝔤_{1}}^{𝔤} (Ind_{𝔤_{0}}^{𝔤_{0}⊕𝔤_{1}} M_{μ}^{0}) .

A filtration of Ind_{𝔤_{0}}^{𝔤_{0}⊕𝔤_{1}} M_{μ}^{0} with 𝔤_{0}-irreducible quotients M_{μ+γ}^{0} induces the filtration on P_{μ} with irreducible quotients M_{μ+γ}.

To show (b) recall that ⊕_{μ∈𝔥_{0}^{∗}}M_{μ}^{0} is a faithful U (𝔤_{0})-module. Therefore ⊕_{μ∈𝔥_{0}^{∗}}P_{μ} = Ind_{𝔤_{0}}^{𝔤} (⊕_{μ∈𝔥_{0}^{∗}}M_{μ}^{0}) is a faithful U-module.

The following theorem is proved in [4]. If M is a module over an associative algebra A, then Ann_{A} M denotes the annihilator of M in A.

Theorem 3.4. The ideal I = ∩_{μ∈𝔥_{μ∈𝔥}} Ann_{U} M_{μ} = ∩_{μ∈𝔥_{0}^{∗}} Ann_{U} M_{μ} coincides with the radical of U.

Proof. The assertion ∩_{μ∈𝔥^{∗}} Ann_{U} M_{μ} = ∩_{μ∈𝔥_{0}^{∗}} Ann_{U} M_{μ} follows from Zariski density of 𝔥_{0}^{∗}. By Lemma 3.2 M_{μ} is irreducible if μ ∈ 𝔥_{0}^{∗}, and therefore rad U ⊂ I. To show that I ⊂ rad U it suffices to prove that I is nilpotent. Note that I^{s} (P_{μ}) = 0 for all μ ∈ 𝔥_{0}^{∗}. Since ⊕_{μ∈𝔥_{0}^{∗}}P_{μ} is faithful we obtain I^{s} = 0.

Corollary 3.5. Let Ū = U/I. Then the ℤ-grading on U induces the ℤ-grading on Ū such that Ū = Ū_{−d} ⊕ ⋯ ⊕ Ū_{d}. The natural maps U (𝔤_{0} ⊕ 𝔤_{−1}) → Ū and 𝔤 → Ū are injective. Finally ∩_{μ∈𝔥_{0}^{∗}} Ann _{\overline{U}} M_{μ} = \{0\}.

The following theorem can be proven in the same manner as Theorem 3.4.

Theorem 3.6. The ideal I′ = U′ ∩ (∩_{μ∈(𝔥′)^{∗}} Ann_{U′} M_{μ}) coincides with the radical of U′.

4. The center of Ū

In this section we will describe the center Z of Ū. As follows from Corollary 3.5 U (𝔥) is a subalgebra of Ū. Therefore the standard Harish-Chandra projection h : U → U (𝔥) with the kernel 𝔫^{−}U + U𝔫^{+} induces the map Ū → S (𝔥). Thus, one can define the Harish-Chandra homomorphism h : Z → U (𝔥) = S (𝔥). We identify S (𝔥) with polynomial algebra on 𝔥^{∗}. As in the usual case for any z ∈ Z

z|_{M_{μ}} = h (z) (μ) Id . (4.1)

Therefore the last statement of Corollary 3.5 implies that h is injective. We will describe h (Z).

Let W be the Weyl group of 𝔤_{0}, note that W is isomorphic to S_{n}. Define a W - action on 𝔥^{∗} by μ^{w} = w (μ + ρ_{0}) − ρ_{0}, where ρ_{0} is the half sum of even positive roots. Denote by S (𝔥)^{W} the subring of W -invariant polynomials on 𝔥^{∗}. Let Θ (μ) = Π_{i≠j} ((μ + ρ_{0}, ε_{i} − ε_{j}) − 1). Note that Θ ∈ S (𝔥)^{W}.

Lemma 4.1. h (Z) ⊂ S (𝔥)^{W}.

Proof. Let 𝔡 = {μ ∈ 𝔥^{∗} | μ_{i} − μ_{j} ∈ ℤ_{≥0}, i < j}. Then 𝔡 is a Zariski dense set in 𝔥^{∗}. Furthermore for any w ∈ W

Hom_{𝔤} (M_{μ^{w}}, M_{μ}) = Hom_{𝔤_{0}} (M_{μ^{w}}^{0}, M_{μ}^{0}) = ℂ.

Therefore every element z ∈ Z acts by the same scalar on M_{μ^{w}} and on M_{μ}. By (4.1) every p ∈ h (Z) satisfies p (μ^{w}) = p (μ) for all μ ∈ 𝔡. Since 𝔡 is Zariski dense, this implies that p (μ^{w}) = p (μ) for all μ ∈ 𝔥^{∗}.

Lemma 4.2. Let μ ∈ 𝔥^{∗} be such that μ_{n−1} = μ_{n} and p ∈ h (Z). Then p (μ + t (ε_{n−1} + ε_{n})) = p (μ) for any t ∈ ℂ.

Proof. Let v be a highest vector of Verma module M_{μ}. An easy calculation shows that w = X_{−ε_{n}−ε_{n−1}}v is 𝔟-invariant, and therefore Hom_{𝔤} (M_{μ−ε_{n−1}−ε_{n}}, M_{μ}) ≠ 0. By (4.1) this implies p (μ) = p (μ − ε_{n−1} − ε_{n}). Applying this relation several times we obtain p (μ) = p (μ + t (ε_{n−1} + ε_{n})) for all t ∈ ℤ, which by Zariski density of ℤ in ℂ implies the same relation for all t ∈ ℂ.

Corollary 4.3. Let (μ + ρ_{0}, ε_{i} − ε_{j}) = 1 and p ∈ h (Z). Then p (μ) = p (μ + t (ε_{i} + ε_{j})) for all t ∈ ℂ.

Lemma 4.4. h (Z) ⊂ ΘS (𝔥)^{W} + ℂ.

Proof. Let

l_{i,j} = {μ ∈ 𝔥^{∗}| (μ + ρ_{0}, ε_{i} − ε_{j}) = 1} .

We want to show that if p ∈ h (Z), then p is constant on ∪_{i≠j}l_{i,j}. Since p is W - invariant, it suffices to show that p is constant on one hyperplane l_{i,j}, for example on l_{1,2}. Let μ ∈ l_{1,2}, i.e. μ_{1} = μ_{2}. Put λ = μ + (μ_{3} − μ_{1}) (ε_{1} + ε_{2}). By Corollary 4.3 p (μ) = p (λ). Note that λ ∈ l_{2,3}. Put ν = λ + (μ_{1} − μ_{3} + 1) (ε_{2} + ε_{3}). Then p (μ) = p (λ) = p (ν). Let w = (132) ∈ W. Then ν^{w} = μ + 2ε_{3}. Thus we obtain the relation p (μ) = p (μ + 2ε_{3}). In the same way as in the proof of lemma 5 we can get that p (μ) = p (μ + tε_{3}) for all t ∈ ℂ. Using W -invariance of p we obtain that p (μ) = p (μ + tε_{i}) for all t ∈ ℂ and i = 3, …, n. This implies that p is constant on l_{1,2} and therefore p is constant on ∪_{i≠j}l_{i,j}.

For any x ∈ U denote by x_{0} the image of x under the projection onto U (𝔤_{0}) with the kernel 𝔤_{1}U +U𝔤_{−1}. Recall that U = U (𝔤_{1})⊗U (𝔤_{0})⊗U (𝔤_{−1}) and U (𝔤_{1}) = S (𝔤_{1}) ⊂ U.

Lemma 4.5. (Y S^{d} (𝔤_{1}))_{0} ≠ 0.

Proof. We will show that (Y X)_{0} ≠ 0. Consider the Verma module M_{μ} with a highest vector v. Then (Y X)_{0} Y v = Y XY v = Δ (μ) Y v by Lemma 3.1. Therefore (Y X)_{0} ≠ 0.

Let U^{k} be the k-th term of the natural filtration of U. Lemma 4.5 enables us to define a non-trivial ad 𝔤′_{0}-invariant map

φ : S^{d} (𝔤_{1}) → U^{2d} (𝔤_{0})

by

φ (x) = (Y x)_{0} .

Let {X_{i}} be a basis in some ad 𝔤_{0}-invariant subspace complementary to Ker φ. Then {(Y X_{i})_{0}} is a linearly independent set in U^{2d} (𝔤_{0}) and could be extended to some basis of U^{2d} (𝔤_{0}). Recall that U^{2d} (𝔤_{0}) ≅ S^{≤2d} (𝔤_{0}) as ad 𝔤_{0} module and it admits a non-degenerate ad 𝔤_{0}-invariant bilinear symmetric form B. Use the notation (Y X_{i})′_{0} for the vectors of the dual basis with respect to B. Let

T = ΣX_{i} (Y X_{i})′_{0} .

Then T is ad 𝔤′_{0} invariant and

(Y T)_{0} = Σ (Y X_{i} (Y X_{i})′_{0})_{0} = Σ (Y X_{i})_{0} (Y X_{i})′_{0} ≠ 0. (4.2)

Lemma 4.6. (a) Let Y^{ad} = Π_{i<j} ad X_{−ε_{i}−ε_{j}} and S = Y^{ad} (T), and S̄ be its image in Ū. Then S̄ ∈ Z and h \binom{}{S̄} is proportional to Θ.

(b) Let z ∈ Z (U (𝔤_{0})). Put S_{z} = Y^{ad} (zT), and let S̄_{z} be its image in Ū. Then

S̄_{z} ∈ Z and h \binom{}{S̄_{z}} = h \binom{}{S̄} h (z).

Proof. (a) First of all S̄ ≠ 0. Indeed, if v is a highest vector of M (μ) then

S̄Y v = SY v = Y^{ad} (T) Y v = Y T Y v = (Y T)_{0} Y v.

Since (Y T)_{0} ≠ 0, (Y T)_{0} Y v ≠ 0 for some μ, and therefore S̄ ≠ 0.

Note that T̄ ∈ Ū_{d}, hence 𝔤_{1}T̄ = T̄𝔤_{1} = 0, and ad 𝔤′_{0}T̄ = 0. Thus, T̄ is ad 𝔟 invariant vector of weight −σ. Let V be ad 𝔤-submodule of Ū generated by T̄. Since S̄ ≠ 0, V ≅ Ind_{𝔤_{0}⊕𝔤_{1}}^{𝔤} ℂT̄ and S̄ generates the trivial ad 𝔤-submodule of V. Thus, S̄ ∈ Z.

To find h \binom{}{S̄} note that S̄ acts by zero on the trivial 𝔤-module, hence h \binom{}{S̄} (0) = 0. On the other hand the degree of the polynomial h \binom{}{S̄} is not greater than 2d as S̄ ∈ Ū^{2d}. Therefore, by Lemma 4.4, h \binom{}{S̄} must be proportional to Θ.

(b) The arguments similar to (a) show that S̄_{z} ∈ Z. Finally, if v is a highest vector of M_{μ}, then

h \binom{}{S̄_{z}} (μ) v = S̄_{z}v = Y^{ad} (zT) v = zT Y v = zY^{ad} (T) v = zSv. (4.3)

Note that zSv = (h (z)) (μ) Sv. Therefore h \binom{}{S̄_{z}} = h \binom{}{S̄} h (z).

Corollary 4.7. ℂ + ΘS (𝔥)^{W} ⊂ h (Z).

Lemma 4.4 and Corollary 4.7 immediately imply

Theorem 4.8. h (Z) = ℂ + ΘS (𝔥)^{W}.

Let Ū′ = U′/I′, Z′ be the center of Ū′ and h: Z′ → S (𝔥′)^{∗} be the Harish-Chandra homomorphism. One can prove the following theorem by repeating the arguments of this section.

Theorem 4.9. h (Z′) = ℂ + ΘS (𝔥′)^{W}.

5. The category of Ū-modules

Let μ ∈ 𝔥^{∗}. Define χ_{μ} : Z → ℂ by the formula χ_{μ} (z) = h (z) (μ) for all z ∈ Z. We call μ typical if Δ (μ) ≠ 0. Otherwise we call μ atypical. The following statement follows immediately from Theorem 4.8.

Lemma 5.1. Let μ, ν ∈ 𝔥^{∗}. Then χ_{μ} = χ_{ν} iff either both μ and ν are atypical or μ and ν are both typical and μ = ν^{w} for some w ∈ W

Define Ū_{μ} = Ū/Ū Ker χ_{μ}. Let U_{μ}^{0} = U (𝔤_{0}) /U (𝔤_{0}) Ker χ_{μ}^{0}, where χ_{μ}^{0} : Z (U (𝔤_{0})) → ℂ is defined using the Harish-Chandra projection restriction to Z (U (𝔤_{0})). By Ū_{μ} − mod (respectively U_{μ}^{0} − mod ) we denote the category of all left Ū_{μ}-modules (resp. U_{μ}^{0}-modules). In the similar way one defines Ū′_{μ} for μ ∈ (𝔥′)^{∗}.

We will prove the following

Theorem 5.2. If μ ∈ 𝔥^{∗} is typical then the categories Ū_{μ} − mod and U_{μ}^{0} − mod are equivalent.

Let us introduce the following notations. Put z_{0} = diag (1_{n}, −1_{n}) ∈ Z (𝔤_{0}). We assume now that μ is typical and fixed. If M is a 𝔤-module we put

M^{𝔤_{1}} = {m ∈ M | 𝔤_{1}m = 0}
M_{0} = {m ∈ M | z_{0}m = μ (z_{0}) m} .

Lemma 5.3. If M is an Ū_{μ}-module then (a) M^{𝔤_{1}} is an U_{μ}^{0}-module;

(b) M^{𝔤_{1}} = M_{0};

(c) The functor Inv : Ū_{μ} − mod → U_{μ}^{0} − mod , defined by Inv M = M_{0} = M^{𝔤_{1}} is faithful and exact.

Proof. The first statement follows from the formula S̄_{z}m = zS̄m for any m ∈ M^{𝔤_{1}}, z ∈ Z (U (𝔤_{0})). Since S̄m = χ_{μ} \binom{}{S̄} m ≠ 0, we get

zm = \frac{S̄_{z}m}{χ_{μ} \binom{}{S̄}} = χ_{μ}^{0} (z) m.

By (a) z_{0}m = μ (z_{0}) m for any m ∈ M^{𝔤_{1}}, and therefore M^{𝔤_{1}} ⊆ M_{0}. Let m ∈ M_{0}. Then M′ = U (𝔤_{1}) m is a z_{0}-invariant finite-dimensional subspace of M with z_{0} weight decomposition M′ = ⊕_{i≥0}M′_{μ(z_{0})+i}. In particular M′_{μ(z_{0})} = ℂm. Since (M′)^{𝔤_{1}} ≠ 0, and (M′)^{𝔤_{1}} ⊆ M_{0}, we obtain m ∈ (M′)^{𝔤_{1}} and M′ = ℂm. Therefore M^{𝔤_{1}} = M_{0}.

Faithfulness of Inv follows from the fact that U (𝔤_{1}) is isomorphic to finite-dimensional Grassman algebra. Exactness is a direct corollary of the identity Inv M = M_{0}.

Let Ū^{+} denote the image of U (𝔤_{0} ⊕ 𝔤_{1}) in Ū under the natural projection. Since U (𝔤_{0}) is a subalgebra of Ū it is possible to define a Ū^{+}-module structure on every 𝔤_{0}-module by putting 𝔤_{1}N = 0.

Let Ind N = Ū ⊗Ū^{+} N.

Lemma 5.4. Let N be a U_{μ}^{0}-module. Then (a) Ind N is Ū_{μ}-module.

(b) Ind N ≅ U (𝔤_{−1}) ⊗_{ℂ} N as 𝔤_{0}-module.

(c) The functor Ind: U_{μ}^{0} − mod → Ū_{μ} − mod is exact and faithful.

(d) Inv (Ind N) ≅ N.

Proof. Since Ind N = ŪN, it suffices to check that zv = χ_{μ} (z) v for every z ∈ Z and v ∈ N. Define the projection h_{0} : Ū_{0} → U (𝔤_{0}) with the kernel Ū_{0} ∩ Ū𝔤_{1}. One can easily verify that h_{0} (Z) ⊂ Z (U (𝔤_{0})), zv = h_{0} (z) v and χ_{μ} (z) = χ_{μ}^{0} (h_{0} (z)). Now (a) follows immediately.

To prove (b) recall that U (𝔤_{−1}) is a subalgebra of Ū and Ind N is free over U (𝔤_{−1}).

(c) follows (b).

To prove (d) use that Ind N = N ⊕ 𝔤_{−1} Ind N. Let

N′ := Inv (Ind N) ∩ (𝔤_{−1} Ind N) .

It suffices to show that N′ = {0}. Indeed, let M′ ⊂ Ind N be a submodule generated by N′. Then M′ ⊆ 𝔤_{−1} Ind N, and therefore T M′ = 0. But then S̄M′ = 0. Since μ is typical, S̄ acts on M′ by a non-zero constant. Hence M′ = {0} and N′ = {0}.

Lemma 5.5. The functors Inv: U_{μ}^{0}− mod → Ū_{μ}− mod and Ind: Ū_{μ}− mod → U_{μ}^{0} − mod establish equivalence of categories.

Proof. By Lemma 5.4 (d) Inv (Ind N) ≅ N. Therefore it suffices to show that Ind (Inv M) ≅ M. Since Inv (Ind (Inv M)) ≅ Inv M and Inv is faithful and exact, we obtain Ind (Inv M) ≅ M. In fact we use only the left exactness of Inv, which is always true.

Note that Lemma 5.5 implies Theorem 5.2.

Remark 5.6. Note that in the same way one can prove that the category of right Ū_{μ}-modules and the category of right U_{μ+σ}^{0}-modules are equivalent. The shift by σ comes from rewriting formula (4.3) for right 𝔤-modules.

Theorem 5.7. For a typical μ ∈ 𝔥^{∗} the algebra Ū_{μ} is isomorphic to the matrix algebra U_{μ}^{0} ⊗_{ℂ} End_{ℂ} (U (𝔤_{−1})).

Proof. We are using the following classical result (see [2]). Let A and B be 𝔤_{0}- modules. Denote by F (A, B) the subspace of Hom_{ℂ} (A, B) on which the adjoint action of 𝔤_{0} is locally finite. The natural homomorphism ρ_{0} : U_{μ}^{0} → F (M_{μ}^{0}, M_{μ}^{0}) is an isomorphism. We will show that the natural homomorphism ρ : Ū_{μ} → F (M_{μ}, M_{μ}) is also an isomorphism. First of all since M_{μ} = Ind M_{μ}^{0} and M_{μ}^{0} is faithful, Morita equivalence of U_{μ}^{0} and Ū_{μ} implies that M_{μ} must be faithful. Thus ρ is injective. To show the surjectivity of ρ consider F (M_{μ}, M_{μ}) as a left Ū_{μ}-module. Then by Theorem 5.2

F (M_{μ}, M_{μ}) = Ind (Inv F (M_{μ}, M_{μ})) = Ind F (M_{μ}, M_{μ}^{0}) .

Note that F (M_{μ}, M_{μ}^{0}) is a right Ū_{μ}-module and therefore by Remark 5.6

F (M_{μ}, M_{μ}^{0}) = Ind (Inv F (M_{μ}, M_{μ}^{0})) = Ind F (M_{μ+σ}^{0}, M_{μ}^{0}) .

Thus, it suffices to check that F (M_{μ+σ}^{0}, M_{μ}^{0}) ⊆ Im ρ. Indeed, it follows from

F (M_{μ+σ}^{0}, M_{μ}^{0}) = F (M_{μ}^{0}, M_{μ}^{0}) ρ (T) = ρ (U (𝔤_{0}) T) .

Finally, the isomorphism of 𝔤_{0}-modules M_{μ} ≅ M_{μ}^{0} ⊗_{ℂ} U (𝔤_{−1}) implies

F (M_{μ}, M_{μ}) ≅ F (M_{μ}^{0}, M_{μ}^{0}) ⊗_{ℂ} End_{ℂ} (U (𝔤_{−1})) ,

and that completes the proof of the theorem.

Corollary 5.8. Let M_{0} be an irreducible U_{μ}^{0}-module for some typical μ ∈ 𝔥^{∗}. Then M = Ind_{𝔤_{0}⊕𝔤_{1}}^{𝔤} M_{0} is an irreducible 𝔤-module. Moreover, if M_{0} is a semi-simple 𝔥module with finite weight multiplicities, then M is also 𝔥-semi-simple with finite weight multiplicities and ch M = ch M_{0}Π_{i≠j} (1 + εe^{−ε_{i}−ε_{j}}).

Theorem 5.9. Let ν ∈ (𝔥′)^{∗} and μ ∈ 𝔥^{∗} be such that μ_{|𝔥′} = ν. Then the algebra Ū′_{ν} is isomorphic to Ū_{μ}.

Proof. Let (U_{ν}^{0})′ = U (𝔤′_{0}) /U (𝔤′_{0}) Ker χ_{ν}^{0}. One can prove repeating the above arguments that Ū′_{ν} is isomorphic to (U_{ν}^{0})′ ⊗ End_{ℂ} (U (𝔤_{−1})). Now theorem follows from the obvious isomorphism (U_{ν}^{0})′ ≅ U_{μ}^{0}.

6. Geometric realization of Gr Ū

As in classical case Gr U ≅ S (𝔤) is a supercommutative algebra with Poisson bracket. It can be realized as the algebra of functions on 𝔤^{∗}. Since Gr I is a Poisson ideal, the graded algebra Gr Ū is also an algebra with Poisson bracket. We realize it as an algebra of functions on some Poisson supervariety. The reader can find all necessary details on supergeometry in [5] and [6]. For an affine supervariety V we define by O (V) the ring of regular functions on V.

Consider 𝔤^{∗} as a supervariety with the natural action of the supergroup G. Its underlying variety is 𝔤_{0}^{∗} ⊂ 𝔤^{∗}. We define Q as G𝔤_{0}^{∗}. We will show that Q is a closed supervariety with singularities.

Recall that one can identify 𝔤_{0} with 𝔤_{0}^{∗}, and we denote by R_{0} the open subset of all regular elements of 𝔤_{0}^{∗}. By R we denote the virtual supervariety GR_{0}.

Lemma 6.1. Let x ∈ 𝔤_{0}^{∗} and 𝔤_{x} be the Lie superalgebra of the stabilizer of x. Then dim (𝔤_{x} ∩ 𝔤_{1}) ≥ n. If x ∈ R_{0} then dim (𝔤_{x} ∩ 𝔤_{1}) = n and 𝔤_{x} ∩ 𝔤_{1} = {0}. If x ∉ R_{0}, then dim (𝔤_{x} ∩ 𝔤_{1}) > n and dim (𝔤_{x} ∩ 𝔤_{−1}) > 0.

Proof. We realize x ∈ 𝔤_{0}^{∗} by an n × n matrix, a ∈ 𝔤_{−1} by a skewsymmetric matrix and b ∈ 𝔤_{1} by a symmetric matrix. Then

𝔤_{x} ∩ 𝔤_{−1} = {a ∈ 𝔤_{−1} | ax − x^{t}a = 0}

and in the same way

𝔤_{x} ∩ 𝔤_{1} = {b ∈ 𝔤_{1} | xb − bx^{t} = 0} .

Since b is symmetric and xb − bx^{t} is skewsymmetric,

dim 𝔤_{x} ∩ 𝔤_{1} ≥ dim 𝔤_{1} − dim 𝔤_{−1} = n.

That proves the first statement.

To check two other statements note that dim 𝔤_{x} ∩ 𝔤_{−1} and dim 𝔤_{x} ∩ 𝔤_{1} are constant on the conjugacy class of x. Therefore it is sufficient to prove the statements for a diagonal x and for a nilpotent x which can be easily done by direct calculations.

By Lemma 6.1 one should expect that dim Q = (n^{2}, n^{2} − n) and that the singularities of Q are at all non-regular elements. We will construct a resolution of Q.

Let 𝔭 = 𝔤_{0} + 𝔤_{1}, P be the subgroup with the Lie superalgebra 𝔭, and K = G/P. Then K is an algebraic supervariety of dimension (0|d). Therefore K is isomorphic to 𝔤_{−1} and O (K) ≅ S (𝔤_{−1}^{∗}). Let V be the linear bundle on K defined as V = G ×_{P} (𝔤/ [𝔭, 𝔭])^{∗}. Then the cotangent bundle T^{∗} ≅ G ×_{P} (𝔤/𝔭)^{∗} is a subbundle of V. The underlying variety V_{0} is isomorphic to 𝔤_{0}^{∗}, and since it is affine, the supervariety V splits. The embeddings T^{∗} → V and 𝔤_{0}^{∗} ≅ V_{0} → V induce the isomorphism

V ≅ T^{∗} × 𝔤_{0}^{∗}. (6.1)

Now we define the map r : V → 𝔤^{∗} in the obvious way. A point of V is a pair (𝔭_{y}, ϕ) where 𝔭_{y} is a parabolic subalgebra conjugate to 𝔭 and ϕ is a linear functional on 𝔤 annihilating [𝔭_{y}, 𝔭_{y}]. We put r (𝔭_{y}, ϕ) = ϕ.

Lemma 6.2. (a) r is a G-equivariant map; (b) The image of r coincides with Q;

(c) r is a rational isomorphism between r^{−1} (R) and R.

Proof. (a) is obvious from construction of r. It is also obvious that r induces the identity map 𝔤_{0}^{∗} → 𝔤_{0}^{∗} of the underlying subvarieties. Therefore (b) follows from (a). To check (c) one calculates easily that d_{x}r has a maximal rank for any regular point x ∈ R_{0}.

Corollary 6.3. V is a resolution of Q.

The last corollary allows one to describe the structure sheaf O_{Q} as a subsheaf of O_{V}. Indeed by (6.1)

O_{V} = S (𝔤_{−1}^{∗}) ⊗ S (𝔤_{−1}) ⊗ O_{𝔤_{0}^{∗}}.

Identify K with 𝔤_{−1}, T^{∗} with 𝔤_{−1} × 𝔤_{−1}^{∗}, and V with 𝔤_{−1} × 𝔤_{−1}^{∗} × 𝔤_{0}^{∗}. Then for any x ∈ 𝔤_{0}^{∗}

r^{−1} (x) = (𝔤_{−1} ∩ 𝔤_{x}) × {0} × {x} .

For any x ∈ 𝔤_{0} define

ζ_{x} = {ϕ ∈ 𝔤_{−1}^{∗} | ϕ (𝔤_{−1} ∩ 𝔤_{x}) = 0} .

By Lemma 6.1 ζ_{x} = 𝔤_{−1}^{∗} if and only if x ∈ R_{0}. All functions lifted from Q must be constant on the fibers of r^{−1} (x). Therefore

O_{Q} = {f ∈ O_{V} | f (x) ∈ S (𝔤_{−1}) ⊕ S (𝔤_{−1}^{∗}) ζ_{x} ⊗ S (𝔤_{−1}) for all x ∈ 𝔤_{0}} .

In particular, O_{V} (R_{0}) ≅ O_{Q} (R_{0}).

Now we are ready to prove the following theorem.

Theorem 6.4. The Poisson superalgebras Gr Ū and O (Q) are isomorphic.

Proof. The restriction homomorphism ξ: Gr U → O (Q) is surjective. To calculate Ker ξ we are going to construct a homomorphism γ : Gr U → O (V) such that γ = r^{∗} ∘ ξ.

Let D be the algebra of differential operators on K and D̄ = D ⊗ U (𝔤_{0}). Since O (K) = S (𝔤_{−1}^{∗}), D ≅ End (S (𝔤_{−1}^{∗})) and D̄ ≅ End (S (𝔤_{−1}^{∗})) ⊗ U (𝔤_{0}). Define the following U-module

F = Hom_{U(𝔤_{0}+𝔤_{1})} (U, U (𝔤_{0})) ,

where U (𝔤_{0}) acts on U (𝔤_{0}) by left multiplications and 𝔤_{1} acts on U (𝔤_{0}) trivially. Note that F ≅ U (𝔤_{−1}^{∗}) ⊗ U (𝔤_{0}) ≅ S (𝔤_{−1}^{∗}) ⊗ U (𝔤_{0}). Therefore one can define the natural injective homomorphism i: D̄ → End (F). Let j: U → End (F) be the homomorphism induced by the representation of U. Then j (U) ⊆ i \binom{}{D̄}, and therefore the homomorphism γ̄ : U → D̄ given by γ̄ = i^{−1} ∘ j is well defined.

Lemma 6.5. Ker γ̄ = I.

Proof. Let F′ = HomŪ^{+} (Ū, U (𝔤_{0})). Generally speaking F′ is a submodule of F. But since U (𝔤_{0}) and U (𝔤_{−1}) are both subalgebras of Ū, we obtain

F′ ≅ U (𝔤_{−1}^{∗}) ⊗ U (𝔤_{0}) ≅ S (𝔤_{−1}^{∗}) ⊗ U (𝔤_{0}) ≅ F.

Therefore F is well defined Ū-module which implies that I ⊆ Ann_{U} F.

On the other hand, F ≅ Ind U (𝔤_{0}) ⊗ C_{−σ}, where C_{−σ} is the one-dimensional 𝔤module of weight −σ. Hence F is a faithful Ū-module, which implies Ann_{U} F = I. Since Ker γ̄ = Ann_{U} F, we obtain the lemma.

The algebra D̄ has the filtration induced by the natural filtration of U (𝔤_{0}) and the natural filtration of D. By (6.1) Gr D̄ = O (V). It is clear that γ̄ preserves the filtration. Therefore γ̄ induces the homomorphism γ : Gr U → O (V) of graded algebras.

Lemma 6.6. γ = r^{∗} ∘ ξ.

Proof. The homomorphism γ induces the morphism θ : V → 𝔤^{∗} of affine supervarieties such that θ^{∗} = γ. We have to check that θ = r. Let V_{0} ⊂ V be the underlying subvariety canonically embedded in V. Recall that V_{0} is isomorphic to 𝔤_{0}^{∗}. One can see from the definitions of r and θ that r (V_{0}) = θ (V_{0}) = 𝔤_{0}^{∗} ⊂ 𝔤^{∗}. Furthermore, r and θ coincide on V_{0}. Note that θ is G-equivariant as well as r. Since V = GV_{0}, we obtain that θ = r.

Since r^{∗} is injective, Ker ξ = Ker γ = Gr I. Therefore Gr Ū is isomorphic to O (Q).

References

1. M. Gorelik, The center of a simple p-type lie superalgebra, Journal of Algebra.

2. A. Joseph, Kostant’s problem, Goldie rank and the Gelfand-Kirillov conjecture, Inventiones Mathematicae 56 (1980), no. 3, 191–213.

3. V. G. Kac, Lie superalgebras, Adv. Math. 26 (1977), 8–96.

4. E. Kirkman and J. Kuzmanovich, Minimal prime ideals in enveloping algebras of lie superalgebras, Proceedings of American Mathematical Society 124 (1996), 1693–1702.

5. Yu. Manin, Gauge field theory and complex geometry, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 289, Springer-Verlag, Berlin, 1988, Translated from the Russian by N. Koblitz and J. R. King.

6. I. Penkov, Borel-Weil-Bott theory for classical lie supergroups, Contemporary Problems in Mathematics, vol. 32, VINITI, Moscow, 1988.

Dept. of Mathematics, University of California at Berkeley, Berkeley, CA 94720

  E-mail address: serganov@math.berkeley.edu