A theory of rank for enveloping algebras. I

Victor Protsak Max-Planck-Institut für Mathematik Vivatsgasse 7, 53111, Bonn, Germany

April 17, 2001

Contents

1 Introduction 1

2 General construction of rank ideals 2 2.1 Reductive dual pairs of Lie algebras . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Definition of rank ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Filtered algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Double Commutant Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Relation with reductive dual pair correspondence . . . . . . . . . . . . . . . . 11

3 Rank ideals in U(𝔤𝔩_{n}) 12 3.1 Determinantal varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 The associated graded ideals of rank ideals in U(𝔤𝔩_{n}) . . . . . . . . . . . . . . 14 3.3 Determinants with entries in a noncommutative algebra . . . . . . . . . . . . 15 3.4 Some computations in U(𝔤𝔩_{n}) . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5 Quantum minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.6 Behavior of rank ideals under tensor products . . . . . . . . . . . . . . . . . . 21 3.7 Generators of the rank ideal J_{k} . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.8 A localization property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Rank of highest weight modules over 𝔤𝔩_{n} 26 4.1 Characterization of rank in terms of the highest weight . . . . . . . . . . . . . 27 4.2 Case of finite-dimensional modules . . . . . . . . . . . . . . . . . . . . . . . . 30

1 Introduction

This is the first in a series of papers developing an algebraic analog of the theory of N-rank for unitary representations of classical Lie groups (see [How82, Li89]). Using the theory of reductive dual pairs, we introduce a family of ideals in the enveloping algebra of a complex classical Lie algebra 𝔤. These ideals, called rank ideals, are parametrized by complex classical Lie algebras 𝔤′ forming together with 𝔤 a reductive dual pair (cf. [How79]). More precisely, to each irreducible reductive dual pair of complex Lie algebras (𝔤, 𝔤′) ⊆ 𝔰𝔭(W) we attach a completely prime ideal in U(𝔤) (Definition 2.2.1). Roughly speaking, this ideal is the annihilator in U(𝔤) of the oscillator representation restricted to 𝔤. More generally, we introduce and study a correspondence between ideals in U(𝔤) and U(𝔤′) that is compatible with the reductive dual pair correspondence (see [How89b]). The rank ideal is just the minimal ideal of U(𝔤) that can appear in this algebraic correspondence. These considerations yield new information on the behavior of the annihilators of Harish-Chandra modules of real classical groups under the reductive dual pair correspondence. On the other hand, the algebraic theory of rank that we develop offers a new perspective on primitive, and more generally, prime ideals in enveloping algebras. One application is to the study of Goldie ranks of primitive ideals. Organization of the series is as follows. In the present first part we define the rank ideals and give their detailed description for the case 𝔤 = 𝔤𝔩_{n}. In the second part we establish some properties of rank for primitive ideals in U(𝔤𝔩_{n}) that are very similar to the main results of the theory of N-rank. In the third part we study the general duality correspondence between ideals in U(𝔤) and U(𝔤′). In the projected fourth part we shall give a detailed description of rank ideals in the technically more demanding cases of orthogonal – symplectic reductive dual pairs. Applications of rank theory to the description of the reductive dual pair correspondence are left for a separate paper. Acknowledgements: this paper is based on Chapter 3 of the author’s Yale Ph. D. thesis. I would like to thank Prof. Roger Howe for formulating the problem and for his patient guidance during my studies at Yale, and Dr. Mara Neusel for many constructive comments. I gratefully acknowledge hospitality and financial support of the Max-Planck-Institut für Mathematik, Bonn during the 2000–2001 academic year.

2 General construction of rank ideals

Motivated by problems in the theory of automorphic forms (the classification of singular Siegel modular forms, theta correspondence), R. Howe introduced a notion of N-rank for unitary representations of a symplectic group over a local field, see [How82]. Later J.-S. Li generalized in [Li89] the definition of N-rank and the results of [How82] to other classical groups. He obtained a clean description for the “singular” part of the unitary dual of a classical group G(𝔽) over a local field 𝔽. The main theorem of [Li89] describes a singular unitary representation of G(𝔽) as a quotient of the oscillator representation associated to a reductive dual pair (G(𝔽), G′(𝔽)) in the stable range with the larger member G(𝔽). In the real case the oscillator representation admits a neat algebraic formulation through the theory of Weyl algebra. We use this formulation to introduce a notion of rank for ideals in the enveloping algebra of a complex classical Lie algebra. This notion is an algebraic counterpart of the N-rank, and is also related to the determinantal varieties of matrices of bounded rank.

2.1 Reductive dual pairs of Lie algebras

Let W be a finite-dimensional symplectic complex vector space. Denote by Sp(W) the isometry group of W and by 𝔰𝔭(W) its Lie algebra. We consider an algebraic counterpart of a notion of a reductive dual pair of subgroups of a symplectic group (see [How79]).

Definition 2.1.1. Let 𝔤 and 𝔤′ be Lie subalgebras of the Lie algebra 𝔰𝔭(W). Then (𝔤, 𝔤′) ⊆ 𝔰𝔭(W) is a reductive dual pair if the following two conditions hold:

(1) The Lie subalgebras 𝔤 and 𝔤′ are the mutual centralizers in 𝔰𝔭(W); (2) The actions of 𝔤 and 𝔤′ on the vector space W are reductive (i.e. completely reducible).

In this setting we denote by G and G′ the subgroups of Sp(W) which are the pointwise stabilizers of 𝔤′ and 𝔤, respectively.

It follows immediately from the definitions that Lie G = 𝔤, and Lie G′ = 𝔤′. Thus the groups G and G′ determine the Lie algebras 𝔤 and 𝔤′. As we will presently see, the group G or G′ may be disconnected, namely, an orthogonal group. Thus the Lie algebra 𝔤 alone may not determine the group G. On the other hand, the relationship between the Lie algebra 𝔤 and the group G′ is symmetric, due to the following proposition.

Proposition 2.1.2. The subspace 𝔰𝔭(W)^{G′} of G′-invariants in 𝔰𝔭(W) is precisely the Lie algebra 𝔤. Likewise, 𝔰𝔭(W)^{G} = 𝔤′.

Proof. From the definition of G′ we know that 𝔤 ⊆ 𝔰𝔭(W)^{G′}. On the other hand, since Lie G′ = 𝔤′, we see that 𝔰𝔭(W)^{G′} ⊆ 𝔰𝔭(W)^{𝔤′} = 𝔤, the full centralizer of 𝔤′ in 𝔰𝔭(W) by the definition of a reductive dual pair. Thus 𝔰𝔭(W)^{G′} = 𝔤. Reversing the roles of 𝔤 and 𝔤′ we see that 𝔰𝔭(W)^{G} = 𝔤′.

From now on the notation (𝔤, 𝔤′) or (𝔤, 𝔤′) ⊆ 𝔰𝔭(W) indicates that we are dealing with a reductive dual pair. It turns out that 𝔤 and 𝔤′ are necessarily reductive Lie algebras. Moreover, reductive dual pairs can be completely classified. One proceeds in exactly the same manner as in the case of reductive dual pairs of groups.^{1} Here is an outline of the classification (cf. [How]). First, note that reductive dual pairs are defined intrinsically in terms of the symplectic structure on W. Therefore, for any reductive dual pair (𝔤, 𝔤′) ⊆ 𝔰𝔭(W) and an isometry T : W → W_{1} of symplectic vector spaces, the images of the Lie algebras 𝔤 and 𝔤′ under T also form a reductive dual pair. Such two reductive dual pairs are called isomorphic. Next we introduce a direct sum of reductive dual pairs. Observe that if (𝔤_{1}, 𝔤_{1}′) ⊆ 𝔰𝔭(W_{1}) and (𝔤_{2}, 𝔤_{2}′) ⊆ 𝔰𝔭(W_{2}) are two reductive dual pairs (assume that W_{1} and W_{2} are at least one-dimensional), then letting 𝔤 = 𝔤_{1} ⊕ 𝔤_{2}, 𝔤′ = 𝔤_{1}′ ⊕ 𝔤_{2}′, and W = W_{1} ⊕ W_{2} we obtain a reductive dual pair (𝔤, 𝔤′) ⊆ 𝔰𝔭(W). Such reductive dual pairs are called reducible, the others irreducible. Thus a reductive dual pair is reducible if and only if both 𝔤 and 𝔤′ stabilize a nonzero proper symplectic subspace W_{1} ⊊ W, because then they also stabilize its orthogonal complement W_{2} = W_{1}^{⊥}, and (𝔤, 𝔤′) is a direct sum. Clearly, (ir)reducibility is preserved under isomorphisms. A basic result in the theory of reductive dual pairs is that every reductive dual pair is isomorphic to a direct sum of irreducible reductive dual pairs, with the summands uniquely determined. We describe three families of irreducible reductive dual pairs. Let U and V be vector spaces, G = GL(U) and G′ = GL(V) the groups of linear automorphisms of U and V, and 𝔤 = 𝔤𝔩(U) and 𝔤′ = 𝔤𝔩(V) the corresponding Lie algebras. We let W = hom(U, V) ⊕ hom(V, U) and endow it with the symplectic form

⟨(S_{1}, T_{1}), (S_{2}, T_{2})⟩ = tr(S_{1}T_{2} − S_{2}T_{1}), S_{i} ∈ hom(U, V), T_{i} ∈ hom(V, U), i = 1, 2. (2.1.3)

The groups GL(U) and GL(V) act on W by pre- and post-multiplication, preserving the symplectic form. The differentials of these actions are infinitesimally symplectic actions of 𝔤𝔩(U) and 𝔤𝔩(V) on W. Then (𝔤, 𝔤′) is an irreducible reductive dual pair in 𝔰𝔭(W). This is the general linear case, characterized by the condition that 𝔤 and 𝔤′ preserve a lagrangian subspace of W (such reductive dual pairs are called “type II”, cf. [How79]). Up to an isomorphism, this reductive dual pair is specified by the dimensions of U and V. Let U and V be two vector spaces endowed with nondegenerate bilinear forms, denoted (·, ·) and (·, ·)′. We require either that U is a symplectic vector space and V is an orthogonal vector space (the symplectic–orthogonal case), or that U is an orthogonal vector space and V is a symplectic vector space (the orthogonal–symplectic case). Denote by G and G′ the isometry groups of the forms on U and V, respectively, and by 𝔤 and 𝔤′ their Lie algebras. We let W = hom(U, V). Define the conjugate ∗ : hom(U, V) → hom(V, U) by the condition (Su, v)′ = (u, S^{∗}v) for any u ∈ U, v ∈ V. Endow the vector space W with a symplectic structure by the formula

1It is not a priori clear that the groups G and G′ are mutual centralizers, thus forming a reductive dual pair in the usual sense.

⟨S_{1}, S_{2}⟩ = tr(S_{1}S_{2}^{∗}), S_{1}, S_{2} ∈ hom(U, V).  (2.1.4)

As in the general linear case, the groups G and G′ act on W by pre- and post-multiplication and preserve the symplectic form. Differentials of these actions define embeddings of 𝔤 and 𝔤′ into the symplectic Lie algebra 𝔰𝔭(W). Then (𝔤, 𝔤′) is an irreducible reductive dual pair in 𝔰𝔭(W). In contrast with the general linear case above, there is no lagrangian subspace of W invariant under both 𝔤 and 𝔤′ (such reductive dual pairs are called “type I”, cf. [How79]). Note that symplectic vector spaces are classified up to an isomorphism by their dimension, a positive even integer. Since the field ℂ is algebraically closed, orthogonal vector spaces are also classified by their dimension, which may be any positive integer. The three families of reductive dual pairs just described admit explicit realizations in terms of matrices.

Theorem 2.1.5. Let (𝔤, 𝔤′) ⊆ 𝔰𝔭(W) be a reductive dual pair.

(1) The reductive dual pair (𝔤, 𝔤′) is isomorphic to a direct sum of irreducible reductive dual pairs. Moreover, the irreducible summands and their multiplicities are uniquely determined.

(2) Suppose that (𝔤, 𝔤′) is irreducible. Then it is isomorphic to one of reductive dual pairs described above.

2.2 Definition of rank ideals

Our next goal is to define rank ideals associated with reductive dual pairs of Lie algebras. Let W be the Weyl algebra of the symplectic vector space W, namely, W is the associative algebra with a unit generated by the vector space W with the defining relations w_{1} · w_{2} − w_{2} · w_{1} = ⟨w_{1}, w_{2}⟩·1 for any w_{1}, w_{2} ∈ W (see [How95], Appendix 2 for a detailed discussion). The action of the group Sp(W) on the vector space W extends canonically to an action on W by algebra automorphisms, the so-called conjugation action. Taking commutators, we can consider Weyl algebra as a Lie algebra. Then there is a canonical Sp(W)-equivariant embedding ψ of the Lie algebra 𝔰𝔭(W) into W compatible with the brackets (i.e., a Lie algebra homomorphism). Finally, the differential of the conjugation action of Sp(W) on W is the map from 𝔰𝔭(W) to End W which is the composition of ψ : 𝔰𝔭(W) → W with the adjoint action of W. Restricting the map ψ : 𝔰𝔭(W) → W to the Lie subalgebras 𝔤 ⊆ 𝔰𝔭(W) and 𝔤′ ⊆ 𝔰𝔭(W), we obtain two maps from Lie algebras 𝔤 and 𝔤′ to the Weyl algebra that are compatible with the brackets. By the universal property of enveloping algebras, these maps uniquely extend to homomorphisms of the enveloping algebras of 𝔤 and 𝔤′ to the Weyl algebra W. We denote these homomorphisms by

R : U(𝔤) → W and L : U(𝔤′) → W.

Equivalently, one may first extend ψ to an associative algebra homomorphism from the enveloping algebra U(𝔰𝔭(W)) to the Weyl algebra, still denoted ψ. Restriction of the resulting map to subalgebras U(𝔤) and U(𝔤′) of U(𝔰𝔭(W)) yields the same maps R and L. Now we are ready to give our basic definition.

Definition 2.2.1. Let (𝔤, 𝔤′) be an irreducible reductive dual pair. The corresponding rank ideal is an ideal in U(𝔤) which is the kernel of the homomorphism R : U(𝔤) → W.

Note that the image of the homomorphism R is contained in the subalgebra W^{G′} of G′- invariants in the Weyl algebra. This follows from the definition of the group G′ and the Sp(W)-equivariance of the map ψ. We will see below that the homomorphism R is onto W^{G′} (Theorem 2.4.5). Note also that the map R is equivariant with respect to the natural actions of G on its enveloping algebra U(𝔤) and on the Weyl algebra. The equivariance of the map ψ has another consequence, which is essential in applications of rank ideals to description of the reductive dual pair correspondence.

Proposition 2.2.2. The rank ideals associated to two isomorphic reductive dual pairs coincide.

We are now making explicit the definition of rank ideals in the general linear case.

Definition 2.2.3. For any p, q ≥ 1, denote by M_{p,q} the complex vector space of p×q matrices. The trace form is the bilinear pairing M_{p,q} ⊕ M_{q,p} → ℂ given by the formula

(A, B) ↦ tr(AB),

where A ∈ M_{p,q}, B ∈ M_{q,p}.

The trace form is a nondegenerate bilinear form, and we use it to identify the dual space M_{p,q}^{∗} with M_{q,p}. Let k, n ≥ 1, V = M_{k,n}, V^{∗} = M_{n,k}, and W = V ⊕V^{∗} = M_{k,n} ⊕M_{n,k}. Then W is a symplectic vector space, where the symplectic form is given by

⟨(X_{1}, Y_{1}), (X_{2}, Y_{2})⟩ = tr(X_{1}Y_{2} − X_{2}Y_{1}), X_{1}, X_{2} ∈ M_{k,n}, Y_{1}, Y_{2} ∈ M_{n,k}.

The vector spaces V = M_{k,n} and V^{∗} = M_{n,k} are lagrangian subspaces of W. In this situation the Weyl algebra of W may be realized as the algebra of polynomial coefficient differential operators on V, denoted PD(V). Let P(M_{k,n}) be the algebra of polynomial functions on the vector space V = M_{k,n}. Polynomial coefficient differential operators act on the polynomial functions. Thus the vector space P(M_{k,n}) acquires a structure of a module over the algebra PD(M_{k,n}). Consider the following actions of the Lie algebras 𝔤 = 𝔤𝔩_{n} = M_{n,n} and 𝔤′ = 𝔤𝔩_{k} = M_{k,k} on W:

E · (X, Y) = (−XE, EY), E′ · (X, Y) = (−E′^{t}X, Y E′^{t}),  (2.2.4)

where E ∈ M_{n,n}, E′ ∈ M_{k,k}, X ∈ M_{k,n}, Y ∈ M_{n,k}. We claim that (𝔤, 𝔤′) is an irreducible reductive dual pair.^{2} In fact, 𝔤 and 𝔤′ stabilize the lagrangian subspaces V = M_{k,n} and V^{∗} = M_{n,k}, and 𝔤 and 𝔤′ sit inside the subalgebra 𝔤𝔩(V) ⊂ 𝔰𝔭(W). This yields an explicit formula for the images of the standard generators of 𝔤𝔩_{n} under the map R. Denote by E_{ij} ∈ 𝔤𝔩_{n} the n × n matrix with the unit at the (i, j)th entry and zeros elsewhere. Using the normalized symbol map, we get the following expression:

R(E_{ij}) = R^{k}(E_{ij}) = ∑_{t=1}^{k} \frac{1}{2}(x_{ti}∂_{tj} + ∂_{tj}x_{ti}) = ∑_{t=1}^{k} x_{ti}∂_{tj} + \frac{k}{2} δ_{ij}.  (2.2.5)

A similar formula holds for L^{n}(E′_{rs}). The upper index k reflects the fact that in the definition of the map R we used the reductive dual pair (𝔤𝔩_{n}, 𝔤𝔩_{k}) with 𝔤′ = 𝔤𝔩_{k}.

Definition 2.2.6. The kth rank ideal in U(𝔤𝔩_{n}), denoted J_{k}, is the kernel of the homomorphism R^{k} : U(𝔤𝔩_{n}) → PD(M_{k,n}) defined by Equation 2.2.5.

It follows from Theorem 3.2.4 below that for k ≥ n, the rank ideal J_{k} is zero. The definition of rank ideals can be naturally extended to the case k = 0. Let P(M_{k,n}) be the trivial one-dimensional module over 𝔤𝔩_{n} on which all generators of 𝔤𝔩_{n} act by 0, and let the rank ideal J_{0} be its annihilator. The ideal J_{0} is the so-called the augmentation ideal of U(𝔤𝔩_{n}).

Example 2.2.7. Let us consider the case k = 1. Denote the standard generators of PD(ℂ^{n}) by x_{1}, …, x_{n}, ∂_{1}, …, ∂_{n}. The commutation relations in PD(ℂ^{n}) are

[x_{i}, ∂_{j}] = δ_{ij}, [x_{i}, x_{l}] = [∂_{l}, ∂_{m}] = 0.

The map R on the generators of 𝔤𝔩_{n} is given by

R(E_{ij}) = \frac{1}{2}(x_{i}∂_{j} + ∂_{j}x_{i}) = x_{i}∂_{j} + \frac{1}{2}δ_{ij},  (2.2.8)

and the generator E′_{11} of 𝔤𝔩_{1} is mapped into L(E′_{11}) = ∑_{i=1}^{n} x_{i}∂_{i} + \frac{n}{2} = deg +\frac{n}{2}, where deg is the degree operator, giving the homogeneity degree of homogeneous polynomials in P(ℂ^{n}).

Remark 2.2.9. The action of 𝔤𝔩_{n} defined by Equation 2.2.5 is a twist of the differential of the action of the group GL_{n} on the vector space of k × n matrices by right multiplication. Suppose that k ≤ n, then the latter action has an open dense orbit consisting of matrices of maximal rank. Using the results of Borho and Brylinski ([BB82], Corollary 3.6), one can get a description of the kth rank ideal in U(𝔤𝔩_{n}) as an induced ideal.

We conclude this section by an elementary observation on the behavior of rank ideals in U(𝔤𝔩_{n}) under restriction. Let m ≤ n. Consider a natural embedding of Lie algebras 𝔤𝔩_{m} → 𝔤𝔩_{n} given by adding to a m × m matrix n − m zero columns on the right and n − m zero rows on the bottom.

Proposition 2.2.10. The intersection of the kth rank ideal in the algebra U(𝔤𝔩_{n}) with the subalgebra U(𝔤𝔩_{m}) is the kth rank ideal in the enveloping algebra U(𝔤𝔩_{m}).

2The action of 𝔤𝔩_{k} defined here differs from the action described by Equation 2.1.3 by an automorphism E′ ↦ −E′^{t}. This has an effect of producing more natural formulas for infinitesimal character correspondence.

Proof. If in Equation 2.2.5 we restrict the range of indices to 1 ≤ i, j ≤ m, we see that the composition of the embedding of 𝔤𝔩_{m} to 𝔤𝔩_{n} with the map R^{k} for 𝔤𝔩_{n} is the map R^{k} for 𝔤𝔩_{m}. The same remains true on the level of enveloping algebras:

U(𝔤𝔩_{m}) → U(𝔤𝔩_{n}) →^{R^{k}} PD(M_{k,n}).

The map R^{k} for U(𝔤𝔩_{m}) is the composition of the two arrows, and the first map is injective. Therefore, the intersection of J_{k} = Ker R^{k} ⊆ U(𝔤𝔩_{n}) with the embedded U(𝔤𝔩_{m}) is the kth rank ideal in U(𝔤𝔩_{m}).

2.3 Filtered algebra

An important role in the study of enveloping algebras, their ideals, and their quotients is played by methods of filtered algebra. Here we collect the relevant definitions and a few useful facts about filtered vector spaces, algebras, and maps (cf. [NvO82, Bou72]).

Definition 2.3.1. A filtered vector space X is a vector space endowed with a nonnegative increasing filtration {F^{j}X : j ≥ 0} by finite-dimensional linear subspaces, satisfying X = ⋃_{j≥0} F^{j}X. A filtered algebra A is a a filtered vector space which is an associative algebra, and the two structures are compatible in the following sense:

F^{0}A = ℂ · 1, and F^{i}A · F^{j}A ⊆ F^{i+j}A.

Subspaces and quotients of a filtered vector space become filtered vector spaces in a natural way. Let grX = ⊕_{j≥0}F^{j}X/F^{j−1}X be the associated graded space of a filtered vector space X. For any nonzero x ∈ X, its degree is the smallest d such that x ∈ F^{d}X, and its principal symbol is the image of x in gr^{d}X = F^{d}X/F^{d−1}X. Recall that a linear map ϕ : X → Y between two filtered vector spaces (in particular, filtered algebras) is a filtered map if for all j ≥ 0 we have ϕ(F^{j}X) ⊆ F^{j}Y. If moreover for all j ≥ 0 we have Im ϕ ∩ F^{j}Y = ϕ(F^{j}X), then ϕ is called a strict map. Denote by grϕ : grX → grY the associated graded map of a filtered map ϕ. Here is the key property of strict maps (see [NvO82], Theorem D.III.3).

Proposition 2.3.2. Suppose that ϕ : X → Y is a filtered map of filtered vector spaces. The associated graded map grϕ is surjective if and only if ϕ is surjective and strict, which is to say ϕ(F^{i}U) = F^{i}V for any i ≥ 0. If this is the case, then Ker grϕ = gr Ker ϕ.

Let X and Y be two filtered algebras and ϕ a filtered homomorphism, i.e. a filtered map preserving the algebra structure. Then grϕ : grX → grY is also a homomorphism. Suppose that grϕ is surjective. Then Proposition 2.3.2 yields surjectivity of ϕ and allows one to compute the associated graded of the ideal Ker ϕ ⊆ X (which is an ideal in the algebra grX). As another application of Proposition 2.3.2, we derive a well-known criterion for generation of an ideal in a filtered algebra.

Proposition 2.3.3. Let A be a filtered algebra, I a (left, right, or two-sided) ideal in A. Suppose that t_{1}, …, t_{N} ∈ I, and grt_{1}, …, grt_{N} generate the (left, right, or two-sided) ideal grI in grA. Then t_{1}, …, t_{N} generate I.

Proof. The proof is identical in all three cases. For example, consider the case of a left ideal I ⊂ A. Let ϕ be the natural map At_{1} ⊕ … ⊕ At_{N} → I coming from multiplication. Since grt_{1}, …, grt_{N} generate grI, the map grϕ is surjective. Then by Proposition 2.3.2, ϕ is surjective, meaning that t_{1}, …, t_{N} generate I.

Let 𝔤 be a finite-dimensional Lie algebra. Its enveloping algebra U(𝔤) has a natural structure of a filtered algebra, where F^{0}U(𝔤) = ℂ, F^{1}U(𝔤) = ℂ ⊕ 𝔤, and F^{j}U(𝔤) is the linear span of i-fold products for i ≤ j of elements of 𝔤. The algebra U(𝔤) is generated by the linear subspace F^{1}U(𝔤) of elements of degree one, and F^{j}U(𝔤) is the linear span of j-fold products of elements of F^{1}U(𝔤). By the Poincaré–Birkhoff–Witt theorem, the associated graded algebra of U(𝔤) is the commutative algebra S(𝔤), the symmetric algebra of 𝔤.

Definition 2.3.4. An almost commutative algebra is a filtered algebra A such that grA is commutative and A is generated by its subspace F^{1}A.

This notion was introduced by M. Duflo, who proved that the class of almost commutative algebras coincides with the class of quotients of enveloping algebras of finite-dimensional Lie algebras.

2.4 Double Commutant Theorem

As a first step towards an explicit description of rank ideals, we shall compute their associated graded ideals. In this computation and in the construction of a duality correspondence between ideals in Part III, a crucial role is played by a variant of Howe’s Double Commutant Theorem (Theorem 7 from [How89a]). To state and prove it, we have to use a few more properties of the Weyl algebra listed below. These properties are quite easy to verify (again, see [How95] for details).

(1) The Weyl algebra W is a filtered algebra. Here F^{1}W = ℂ ⊕ W, and W^{j} is spanned by i-fold products for i ≤ j of elements of W ⊂ W.

(2) The associated graded algebra grW is the graded commutative algebra S(W) = ⊕_{j≥0}S^{j}(W), the symmetric algebra of the vector space W.

(3) The conjugation action of Sp(W) preserves the filtration on W, and under this action W decomposes into a direct sum of mutually non-isomorphic algebraic Sp(W)-modules W^{j} satisfying W^{j} ⊆ F^{j}W. Here W^{0} = ℂ, W^{1} = W, and W^{2} = ψ(𝔰𝔭(W)). In general, W^{j} is the linear span of w^{j} for w ∈ W.

(4) Let us choose a lagrangian subspace of W with a basis {x_{1}, …, x_{n}}. The stabilizer of the flag determined by this basis is a Borel subgroup B ⊂ Sp(W). Then x_{1}^{j} is a highest weight vector in W^{j} with highest weight jω_{1}. Here ω_{1} is the first fundamental weight of Sp(W), corresponding to the defining module W for Sp(W).

(5) Restriction of the principal symbol map gr : W → S(W) to the subspace W^{j} is an Sp(W)- equivariant linear isomorphism from W^{j} onto S^{j}(W) (which is an irreducible Sp(W)- module with highest weight jω_{1}). Using the direct sum decomposition W = ⊕_{j≥0}W^{j}, these isomorphisms combine into a linear Sp(W)-equivariant isomorphism from W to S(W), called the normalized symbol map.

The center of the symplectic group Sp(W) is the cyclic group of order 2 generated by the transformation σ : w ↦ −w, w ∈ W. This group fixes pointwise the subspaces W^{j} with j even, and acts by the sign character on the subspaces W^{j} with j odd.

Definition 2.4.1. The even Weyl algebra A is the subalgebra W^{σ} = ⊕_{i≥0}W^{2i} ⊂ W. We endow it with the filtration F^{j}A = ⊕_{0≤i≤j}W^{2i}.

Due to the properties above, the even Weyl algebra A is a filtered algebra. Note that the filtration on A that we have just defined is different from the filtration induced by the filtration on W: if a ∈ A ⊂ W, the A-degree of a is one half of the W-degree of a. One concludes that the associated graded algebra gr_{A}A of A is the even symmetric algebra S(W)^{σ} = ⊕_{i≥0}S^{2i}(W). (To avoid confusion, we denote the principal symbol map with respect to the filtration on A by gr_{A}.) In particular, gr_{A}A is a commutative algebra.

Proposition 2.4.2. The algebra A is generated by its subspace W^{2}. Moreover, the algebra grA is generated by its subspace grW^{2}.

Proof. Due to Proposition 2.3.2, the second statement implies the first. Nevertheless, to better reveal what is going on, we will establish both statements simultaneously. The subspace W^{2} ⊂ W is invariant under the conjugation action of the symplectic group Sp(W). Therefore, the subalgebra A′ of the Weyl algebra generated by W^{2} is Sp(W)- invariant. On the other hand, x_{1}^{2i} = (x_{1}^{2})^{i} ∈ A′ is a highest weight vector in W^{2i} for the conjugation action. Therefore, A′ ⊃ W^{2i} for any i ≥ 0. It follows that A′ ⊇ ⊕_{i≥0}W^{2i}, hence A′ = A. The same argument works for the associated graded situation and shows that the algebra generated by grW^{2} = S^{2}(W) is the full even symmetric algebra grA.

Remark 2.4.3. By similar methods, one can show that A does not have a non-trivial Sp(W)- invariant multiplicatively closed linear subspace. Since the adjoint action of the subspace W^{2} on the Weyl algebra coincides with the differential of the conjugation action, any ideal of A would be automatically Sp(W)-invariant, and we obtain that A is a simple algebra. We see that the filtered algebra A is an almost commutative algebra (Definition 2.3.4). One can be more precise: the algebra A is a quotient of the enveloping algebra of the Lie algebra W^{2} = 𝔰𝔭(W).

Proposition 2.4.4. The even Weyl algebra A = W^{σ} is precisely the image of the homomorphism ψ : U(𝔰𝔭(W)) → W. Moreover, the filtration on A coincides with the image under the map ψ of the canonical filtration on U(𝔰𝔭(W)), and ψ is a strict map.

Proof. The enveloping algebra U(𝔰𝔭(W)) is generated by its subspace 𝔰𝔭(W). By the properties above, the space 𝔰𝔭(W) is mapped by ψ onto W^{2}, which by Proposition 2.4.2 generates the whole algebra A. The filtration on A was set up exactly so as to make it the image under ψ of the natural filtration on U(𝔰𝔭(W)). Moreover, we know that grψ is onto grA, because grA is generated by grW^{2} = grψ(𝔰𝔭(W)). By Proposition 2.3.2 we conclude that ψ is strict.

More generally, it turns out that for an arbitrary irreducible reductive dual pair, the algebra of G′-invariants on the even Weyl algebra is an almost commutative algebra, more precisely, the image under the map R of the enveloping algebra U(𝔤). (The situation of the proposition corresponds to a reductive dual pair 𝔤 = 𝔰𝔭(W) and 𝔤′ = 0, then G = Sp(W) and G′ = {1, σ} is the center of Sp(W).)

Theorem 2.4.5. Let (𝔤, 𝔤′) ⊆ 𝔰𝔭(W) be an irreducible reductive dual pair. Then the map R : U(𝔤) → A is a strict homomorphism of filtered algebras. The image of R is precisely the subalgebra A^{G′} of the G′-invariants in A. It also coincides with with the algebra W^{G′} of G′-invariants in the Weyl algebra, and with the algebra A^{𝔤′} of 𝔤′-invariants in A.

Proof. The map R is the restriction of the filtered homomorphism ψ to the subalgebra U(𝔤) ⊆ U(𝔰𝔭(W)) with the induced filtration, hence R itself is a filtered map. This allows one to speak about the associated graded map grR. As in the proof of Proposition 2.4.4, we will show that R is strict and that the algebra A^{G′} is generated by its subspace W^{2} ∩ A^{G′} by establishing the corresponding generation property on the level of associated graded spaces. In the place of Proposition 2.4.2 we shall use the following fact from Classical Invariant Theory.

Proposition 2.4.6. The algebra S(W)^{G′} of G′-invariants on the symmetric algebra S(W) is generated by its subspace S^{2}(W)^{G′}.

This statement may be checked for each of the three families of irreducible reductive dual pairs described above using the First Fundamental Theorem of Classical Invariant Theory (see [How89a, How95]). We will discuss the general linear case below. The image of the subspace 𝔤 ⊂ S(𝔤) = grU(𝔤) under the associated graded map grR : S(𝔤) → gr_{A}A = S(W)^{σ} is gr_{A}R(𝔤) = gr_{A}(W^{2})^{G′} = S^{2}(W)^{G′}, where the first equality is by Proposition 2.1.2. Since the subspace 𝔤 generates the symmetric algebra S(𝔤), the image of the algebra S(𝔤) = grU(𝔤) under the homomorphism grR is the algebra S(W)^{G′} = gr_{A}A^{G′}. By the criterion of Proposition 2.3.2, the homomorphism R is strict onto A^{G′}. The situation is captured by the following commutative diagram:

U(𝔤) ⟶^{R} A^{G′} ⟶ A ⟶ W
↓  ↓^{s}  ↓^{s}  ↓^{s}
S(𝔤) ⟶^{grR} gr_{A}A^{G′} ⟶ S(W)^{σ} ⟶ S(W).

(2.4.7)

Here we denoted by s the normalized symbol map for the Weyl algebra and its restrictions to A and A^{G′}. Since A = W^{σ} and σ ∈ G′, we see that W^{G′} = A^{G′}. Slightly modifying the argument, one gets that the image of the map R also equals A^{𝔤′}. Instead of Proposition 2.4.6 one uses the fact that S(W)^{G′} = (S(W)^{σ})^{𝔤′} is generated by S^{2}(W)^{𝔤′} = 𝔤.

The argument that we gave proceeds along the lines of Howe’s original proof. The paper [How89a] treats a more general case involving a reductive dual pair of Lie superalgebras, and the proof above easily extends to that case. An important consequence of the theorem is the following Double Commutant Theorem.

Theorem 2.4.8. The subalgebras R(U(𝔤)) and L(U(𝔤′)) are the mutual commutants in the algebra A.

Proof. The algebra L(U(𝔤′)) is generated by its subspace L(𝔤′). Hence the commutant of L(U(𝔤′)) in A consists of the elements of A commuting with L(𝔤′). Due to the equivariance of the map ψ, this commutant is A^{𝔤′}, which equals R(U(𝔤)) by Theorem 2.4.5. Likewise, the commutant of R(U(𝔤)) in A is A^{𝔤′} = L(U(𝔤′)).

2.5 Relation with reductive dual pair correspondence

The definition of rank ideals originates in the theory of reductive dual pairs and local Howe duality over ℝ. We would like to explain how rank ideals arise in this context. The theory of reductive dual pairs is centered around the study of certain correspondences between representations of classical groups. These correspondences, called dual pair correspondences, are constructed by restricting the oscillator representation of a symplectic group to two groups forming a reductive dual pair. The theory makes sense for any local or finite field 𝔽, but we consider the case 𝔽 = ℝ, the field of real numbers. Let W_{ℝ} be a real symplectic vector space. Denote by Sp(W_{ℝ}) the group of invertible linear transformations of W_{ℝ} preserving the symplectic form. The oscillator representation ω is a projective representation of Sp(W_{ℝ}) that becomes a unitary linear representation of a suitable two-fold cover Mp(W_{ℝ}) of the symplectic group, the metaplectic group. The oscillator representation can be described algebraically using the notion of the Weyl algebra of a symplectic vector space as follows. Denote by W the complexification of W_{ℝ}, it is a symplectic vector space over ℂ. Suppose that W = V ⊕ V^{∗} is a lagrangian polarization of W, then the Weyl algebra W is isomorphic to the algebra PD(V) of the polynomial coefficient differential operators on V. Note that by definition, PD(V) is an algebra of operators on the vector space P = P(V) of polynomial functions on V, so that P is a module over the Weyl algebra. The complexification of the Lie algebra of the group Sp(W_{ℝ}) is the symplectic Lie algebra 𝔰𝔭(W). The embedding ψ : 𝔰𝔭(W) → W considered above is the infinitesimal version of the canonical homomorphism from Sp(W_{ℝ}) to the algebra of linear operators in the space of the oscillator representation. Since P is a module over W, by restriction it becomes a module over the Lie algebra 𝔰𝔭(W). A choice of a polarization W = V ⊕ V^{∗} satisfying a certain positivity condition corresponds to a choice of a maximal compact subgroup K in Sp(W_{ℝ}) (see [How]). When these choices are made and W is identified with PD(V), the vector space P = P(V) becomes the so-called Fock model of ω. The complexified Lie algebra of K is the Lie algebra 𝔤𝔩(V), which is the stabilizer in 𝔰𝔭(W) of the lagrangian subspaces V and V^{∗} in the direct sum decomposition W = V ⊕ V^{∗}. Let (G_{ℝ}, G_{ℝ}′) be a reductive dual pair in a symplectic group Sp(W_{ℝ}) (cf. Definition 2.1.1). Denote by 𝔤 and 𝔤′ the complexified Lie algebras of the groups G_{ℝ} and G_{ℝ}′. The Lie algebras 𝔤 and 𝔤′ form a reductive dual pair in 𝔰𝔭(W). The groups G_{ℝ} and G_{ℝ}′ commute, and we may form their product G_{ℝ} · G_{ℝ}′, which is a subgroup of Sp(W_{ℝ}). The theory of reductive dual pairs considers the restriction of ω to G_{ℝ} · G_{ℝ}′ and asks to describe the decomposition of this restriction into simple G_{ℝ} · G_{ℝ}′-modules, in a sense which has to be made precise. Actually, one is forced to consider two-fold covers of the groups G_{ℝ} and G_{ℝ}′. In the algebraic approach to the oscillator representation outlined above, one restricts the 𝔰𝔭(W)-module P = P(V) to 𝔤 and 𝔤′ and inquires which 𝔤 ⊕ 𝔤′-modules may appear as quotients P/N for some 𝔤 ⊕ 𝔤′- invariant subspace N. A full translation of the decomposition problem into algebraic language involves Harish-Chandra modules, see [How89b]. We will neglect a part of the structure and consider just modules over the Lie algebras 𝔤 and 𝔤′. Let us consider the quotient P/N as a U(𝔤)-module. Clearly, its annihilator Ann_{U(𝔤)}(P/N) contains the ideal J = Ann_{U(𝔤)} P ⊆ U(𝔤). Tracing back the definitions, we see that J is what we have called the rank ideal associated to a reductive dual pair (𝔤, 𝔤′). Thus J is a two-sided ideal in U(𝔤) which provides a lower bound for annihilators of modules over U(𝔤) occurring in the Howe duality for the reductive dual pair (G_{ℝ}, G_{ℝ}′). Although a priori the ideal J may have depended on the reductive dual pair (G_{ℝ}, G_{ℝ}′) of real groups, in fact J is entirely determined by the reductive dual pair of complex Lie algebras (𝔤, 𝔤′) (cf. Proposition 2.2.2). More generally, the formulas

I ↦ I′ = I · A ∩ L(U(𝔤′)) and I′ ↦ I = I′ · A ∩ R(U(𝔤))  (2.5.1)

define a pair of correspondences between ideals I ⊆ R(U(𝔤)) and I′ ⊆ L(U(𝔤′)) with a similar lower bound property. These questions are treated in detail in Part III.

3 Rank ideals in U(𝔤𝔩_{n})

The main result of this section, Theorem 3.7.1, describes a natural system of generators of the kth rank ideal J_{k} ⊆ U(𝔤𝔩_{n}). These generators, which we call quantum minors, are certain noncommutative determinants in the standard generators of the Lie algebra 𝔤𝔩_{n}.

3.1 Determinantal varieties

Our purpose here is to review the determinantal varieties, paying particular attention to a description of a natural system of generators for their defining ideals. Denote by M_{n,n} the vector space of n × n matrices. Consider the action of the group H = GL_{n} × GL_{n} on M_{n,n} by left and right multiplication:

A ↦ hAg^{−1}, g, h ∈ GL_{n}, A ∈ M_{n,n}.  (3.1.1)

This gives rise to the rational representation of GL_{n}×GL_{n} on the space P(M_{n,n}) of polynomial functions on M_{n,n} given by the formula

(h, g) · f(A) = f(h^{−1}Ag), g, h ∈ GL_{n}, A ∈ M_{n,n}, f ∈ P(M_{n,n}).

For any 1 ≤ i, j ≤ n, denote by x_{ij} the linear functional on the vector space M_{n,n} which is the (i, j)-entry of an n × n matrix. Each x_{ij} may be viewed as a polynomial function on M_{n,n}. We arrange these functions into an n × n matrix X over the algebra P(M_{n,n}) so that for all 1 ≤ i, j ≤ n the (i, j) entry X_{ij} of X is the polynomial function x_{ij} ∈ P(M_{n,n}). Let A be an n × n matrix and I = (i_{0}, i_{1} …, i_{l}) and J = (j_{0}, j_{1}, …, j_{m}) two sequences of indices between 1 and n. Then we denote by A_{IJ} the (l+1)×(m+1) matrix whose (r, s)-entry is A_{i_{r}j_{s}} (0 ≤ r ≤ l, 0 ≤ s ≤ m). If I and J are strictly increasing sequences then A_{IJ} is a submatrix of A consisting of the entries of A in the rows i_{0} < i_{1} < … < i_{l} and in the columns j_{0} < j_{1} < … < j_{m}. For arbitrary sequences I and J rows and columns of A_{IJ} may repeat or occur in a different order than in A. Now suppose that I and J are two sequences of indices which have the same length k + 1, not necessarily increasing. Denote by δ_{IJ} the function det X_{IJ} ∈ P(M_{n,n}). Extending the usual terminology, we call any δ_{IJ} a minor of order k + 1 of the matrix X (with rows in I and columns in J). Let {ϵ_{1}, …, ϵ_{n}} be the standard basis in ℂ^{n}. Denote by {ϵ_{1}^{∗}, …, ϵ_{n}^{∗}} the dual basis in ℂ^{n}^{∗}. Define

v_{I}^{∗} = ϵ_{i_{0}}^{∗} ∧ … ∧ ϵ_{i_{k}}^{∗}, v_{J} = ϵ_{j_{0}} ∧ … ∧ ϵ_{j_{k}}.

The set {v_{I}^{∗}|1 ≤ i_{0} < i_{1} < … < i_{k} ≤ n} is a basis in ∧^{k+1}ℂ^{n∗} and the set {v_{J}|1 ≤ j_{0} < j_{1} < … < j_{k} ≤ n} is a basis in ∧^{k+1}ℂ^{n}. The module ∧^{k+1}ℂ^{n} is an irreducible module over GL_{n}. Its highest weight is a fundamental weight for GL_{n}, and is denoted ω_{k+1}.

Proposition 3.1.2. The assignment

v_{I}^{∗} ⊗ v_{J} ↦ δ_{IJ}  (3.1.3)

extends to a GL_{n} × GL_{n}-equivariant linear isomorphism between the irreducible GL_{n} × GL_{n}- module ∧^{k+1}ℂ^{n∗} ⊗ ∧^{k+1}ℂ^{n} and the linear span of the minors δ_{IJ} of order k + 1. The highest weight vector in the linear span of the minors is the lower left minor of size k + 1, which we denote by δ^{k+1}.

Proof. Since determinant is skew-symmetric in rows and columns, minor δ_{IJ} is skew-symmetric in row indices I and column indices J. The determinant is a linear function of the rows of a matrix, and it is also a linear function of the columns. Identify the columns with elements of the standard basis {ϵ_{1}, …, ϵ_{n}} in ℂ^{n} and the rows with the elements of the dual basis in {ϵ_{1}^{∗}, …, ϵ_{n}^{∗}} in ℂ^{n∗}. The left and right GL_{n}-actions on M_{n,n} are described, respectively, as elementary row and column transformations of n × n matrices, and their effect on a minor δ_{IJ} of order k + 1 is the same as on the corresponding v_{I}^{∗} ⊗ v_{J} ∈ ∧^{k+1}ℂ^{n∗} ⊗ ∧^{k+1}ℂ^{n}. It follows that the linear span of the minors of order k + 1 is stable under the GL_{n} × GL_{n}-action and moreover, that the assignment (3.1.3) extends to a GL_{n} × GL_{n}-equivariant linear map. This map is surjective by definition, and injective because its kernel is a submodule of an irreducible GL_{n} × GL_{n}-module, hence trivial. The upper Borel subgroup of the left copy of GL_{n}, GL_{n} × 1 ⊂ GL_{n} × GL_{n} acting on M_{n,n} by Equation 3.1.1 preserves the span of the last k + 1 rows of the matrix A. Similarly, the upper Borel subgroup of the right copy of GL_{n}, 1 × GL_{n} ⊂ GL_{n} × GL_{n} preserves the span of the first k + 1 columns. This implies that the minor δ^{k+1} is a common eigenvector of the two Borel subgroups. It is easy to verify directly that the weight of δ^{k+1} is (ω_{k+1}^{∗}, ω_{k+1}).

Let us view M_{n,n} as an affine algebraic variety with Zariski topology. By the reduction theory for matrices, the orbits of the GL_{n} × GL_{n}-action on M_{n,n} are locally closed subvarieties of matrices of a fixed rank k, 0 ≤ k ≤ n. Since the group GL_{n} × GL_{n} is connected, its orbits are irreducible algebraic varieties. Their closures are irreducible subvarieties of M_{n,n}. Let R_{k} be the closure of the orbit consisting of matrices of rank k, the kth determinantal variety. For example, R_{0} = {0}, corresponding to a unique closed orbit, and R_{n} = M_{n,n}, corresponding to the open dense orbit GL_{n} ⊂ M_{n,n} of matrices of rank n. Denote by I_{k} the defining ideal of R_{k}.

Theorem 3.1.4. Each determinantal variety R_{k} is the union of orbits of rank at most k. Let I′_{k} be the ideal in P(M_{n,n}) generated by the minors δ_{IJ} of order k + 1. Then I′_{k} = I_{k}.

Proof. From elementary linear algebra, R_{k} is the closed subvariety of M_{n,n} on which all minors δ_{IJ} of size k + 1 vanish. The orbit of matrices of rank k consists of matrices in R_{k} for which there is a non-vanishing minor of size k. These facts imply the closure relation on the orbits and show that R_{k} consists of matrices of rank at most k. Moreover, we conclude that the closed subvariety defined by the ideal I′_{k} is precisely R_{k}. By Hilbert’s Nullstellensatz, the ideal I_{k} is the radical of I′_{k}, and to prove the second statement of the theorem it remains to show that I′_{k} is prime. The primeness of I′_{k} is best understood in the context of the theory of multiplicity-free actions. We sketch the argument, since we will need parts of it later in the discussion of inclusion relation between rank ideals (cf. [How95], Section 2.6 and [HU91], Sections 12–13). The action on M_{n,n} of the product of the upper Borel subgroups of GL_{n} has an open dense orbit, consisting of matrices X for which δ^{l}(X) ≠ 0 for 1 ≤ l ≤ n (again, use the reduction theory for matrices). This implies that the action of GL_{n} ×GL_{n} on M_{n,n} is multiplicity-free and imposes a strong restriction on the form of nonzero GL_{n} × GL_{n}-invariant prime ideals in P(M_{n,n}). Namely, every such ideal must be generated by an irreducible GL_{n} × GL_{n}-submodule of P(M_{n,n}) with highest weight (ω^{∗}, ω), where ω is a fundamental weight for 𝔤𝔩_{n}, and ω^{∗} is its contragredient. By Proposition 3.1.2 the ideal I′_{k} satisfies this condition with ω = ω_{k+1}, and I′_{k} for 0 ≤ k < n, in fact, exhaust all ideals satisfying the condition. Thus every GL_{n} × GL_{n}- invariant prime ideal of P(M_{n,n}) is contained among {I′_{k} : 0 ≤ k < n}. Expanding a minor of order k + 1 in a column, we see that I′_{k} ⊇ I′_{k+1}. On the other hand, ideals I_{k} are prime and GL_{n} × GL_{n}-invariant, and I′_{k} is contained in I_{k} for any 0 ≤ k < n. From this we conclude that I′_{k} = I_{k} for any 0 ≤ k < n.

3.2 The associated graded ideals of rank ideals in U(𝔤𝔩_{n})

Here we prove that the associated graded ideal of the kth rank ideal in U(𝔤𝔩_{n}) is the kth determinantal ideal. There is a chain of GL_{k} × GL_{n}-equivariant identifications

PD(M_{k,n}) = P(M_{k,n} ⊕ M_{k,n}^{∗}) = P(M_{k,n} ⊕ M_{n,k}) = P(M_{k,n}) ⊗ P(M_{n,k}).  (3.2.1)

Here the first equality is the normalized symbol map, and the second equality is given by the trace form. For the generators of P(M_{k,n}) we get that

x_{ti} ↦ x_{ti} ⊗ 1,  ∂_{uj} ↦ 1 ⊗ x_{ju}.  (3.2.2)

By the Poincaré–Birkhoff–Witt Theorem, the associated graded algebra of U(𝔤𝔩_{n}) is the symmetric algebra of M_{n,n}. Using the trace form, we will identify grU(𝔤𝔩_{n}) = S(M_{n,n}) with the algebra P(M_{n,n}) of polynomial functions on M_{n,n}. On the elements of degree 1 the identification is given by

grE_{ij} = x_{ji}.  (3.2.3)

Theorem 3.2.4. For k ≤ n the associated graded grJ_{k} ⊆ S(M_{n,n}) of the kth rank ideal is identified via Equation 3.2.3 with the kth determinantal ideal of P(M_{n,n}). For k ≥ n the ideal J_{k} is zero.

Proof. This follows from Theorem 2.4.5 in the case of the reductive dual pair (𝔤𝔩_{n}, 𝔤𝔩_{k}). Since R is a strict map, by Proposition 2.3.2 we have that gr Ker R = Ker grR. Let us compute the right hand side explicitly. In the present case the commutative diagram (2.4.7) yields

U(𝔤𝔩_{n}) ⟶^{R} PD(M_{k,n})^{GL_{k}} ⟶  PD(M_{k,n})
↓  ↓^{s}  ↓^{s}
P(M_{n,n}) ⟶^{μ^{∗}} P(M_{k,n} ⊕ M_{n,k})^{GL_{k}} ⟶ P(M_{k,n}) ⊗ P(M_{n,k}).

(3.2.5)

Equation 2.2.5 for R^{k}(E_{ij}) along with the identifications (3.2.2) and (3.2.3), results in the following explicit formula for the map μ^{∗}:

μ^{∗}(x_{ji}) = s(R(E_{ij})) = ∑_{t=1}^{k} x_{ti}∂_{tj} = ∑_{t=1}^{k} x_{ti} ⊗ x_{jt}.

Thus the homomorphism μ^{∗} of commutative algebras is dual to the morphism of affine algebraic varieties μ : M_{n,k} ⊕ M_{k,n} → M_{n,n} given by multiplication of matrices. The surjectivity of μ^{∗} that made its appearance in the proof of Theorem 2.4.5 and followed from Proposition 2.4.6 is synonymous to the geometric form of the First Fundamental Theorem of Classical Invariant Theory for the action of GL_{k} on M_{n,k} ⊕ M_{k,n}, namely, that the algebra of polynomial invariants coincides with the pull-back of P(M_{n,n}) with respect to μ (cf. [How95], Equation (2.2.3.7)). The image of the morphism μ is the set of n×n matrices of rank at most min(k, n), which is M_{n,n} when k ≥ n, and R_{k}, the kth determinantal variety, for k < n. Since Ker μ^{∗} ⊆ P(M_{n,n}) is the defining ideal of Im μ ⊆ M_{n,n}, Ker μ^{∗} equals the kth determinantal ideal I_{k} for k < n and zero for k ≥ n.

A parallel argument for the map L : U(𝔤𝔩_{k}) → PD(M_{k,n}) shows that L is injective for k ≤ n and gr Ker L equals the nth determinantal ideal in P(M_{k,k}) for k > n.

3.3 Determinants with entries in a noncommutative algebra

Here we establish several properties of a noncommutative generalization of the determinant of square matrices. These properties will be used in the proof of Theorem 3.7.1, describing generators of the rank ideals in U(𝔤𝔩_{n}). Consider an associative algebra A. Since algebra A need not be commutative, there are several feasible analogs of the determinant. We define the determinant of a matrix by means of the full expansion, arranging the entries in each product in the order of their columns.

Definition 3.3.1. Let A = (A_{ij}) be a k by k square matrix with entries in A. The determinant (more precisely, column determinant) of A is the element of A defined by the formula:

det A = ∑_{σ∈S_{k}} sgn(σ)\overrightarrow{∏}_{1≤s≤k}A_{σ(s)s}.  (3.3.2)

Here S_{k} is the permutation group on k symbols, sgn(σ) is the sign of a permutation σ, and the ordered product \overrightarrow{∏}_{s} is the product in which the factors are arranged from left to right in the order of increasing s.

For example,

det | \begin{matrix}a & b \\ c & d \end{matrix} | = ad − cb.  (3.3.3)

Ordered products enjoy many properties of the ordinary products. For example, they can be split into several subproducts, provided that the order of indices is kept. Another useful property is the Leibniz Rule: for any a, a_{1}, …, a_{k} ∈ A we have

[a, \overrightarrow{∏}_{1≤s≤k}a_{s}] = ∑_{r=1}^{k} (\overrightarrow{∏}_{s<r}a_{s}) [a, a_{r}]\overrightarrow{∏}_{s>r}a_{s}.

In this formula and below, the ordered product over the empty set of indices is set to 1. In the next three propositions we collect some properties of the column determinant. Throughout A is a k by k matrix with entries in A.

Proposition 3.3.4 (Linearity).

(1) The determinant is a multilinear function of the columns.

(2) The determinant is a skew-symmetric multilinear function of the rows.

Proposition 3.3.5 (Expansion rules).

(1) Decompose the sequence (1, …, k) into q consecutive segments J_{1} ∐ … ∐ J_{q}. Split A accordingly into q vertical strips, so that the strip number t consists of consecutive columns of A with numbers {s|s ∈ J_{t}}. Denote by S_{J} = S_{J_{1}} ×…×S_{J_{q}} the subgroup of S_{k} stabilizing each J_{t}. Let S^{J} be the canonical system of left coset representatives of minimal length for the subgroup S_{J} ⊆ S_{k}. It consists of permutations in S_{k} preserving the order of elements in each J_{t}. Then

det A = ∑_{σ∈S^{J}} sgn(σ)\overrightarrow{∏}_{1≤t≤q} det(A_{σ(J_{t})J_{t}}).  (3.3.6)

In particular, the “expansion in a column” holds for the first or last column.

(2) Suppose that all entries of A in column r commute with all entries of the next column r+1. Denote by A′ be the matrix A with these two columns switched. Then det A′ = − det A.

(3) Suppose that all 2 × 2 minors in two consecutive columns r and r + 1 in A vanish. Then det A = 0.

Finally, we see how the determinant behaves under maps of algebras. If ϕ : A → B is a map of associative algebras and A is a square matrix with entries in A, apply ϕ to A entrywise and denote the resulting matrix with entries in B by ϕ(A).

Proposition 3.3.7. Suppose that ϕ : A → B is a homomorphism of associative algebras. Then ϕ(det A) = det ϕ(A).

Proposition 3.3.8. Let A be a filtered algebra and gr : A → grA be the principal symbol map. Suppose that all entries of a matrix A have the same degree, and det grA ≠ 0. Then gr det A = det grA.

Proof. Most of these facts are direct analogs of the corresponding properties of determinants over a commutative ring. They are proved in the usual way, interpreting all products as ordered products with the ordering determined by the columns. Thus Proposition 3.3.4 follows directly from the definition of the determinant (see Equation 3.3.2). In order to prove Proposition 3.3.5, let us decompose S_{k} into the union of its left S_{J}-cosets. Completely expanding each factor in (3.3.6) and replacing each iterated ordered product \overrightarrow{∏}_{1≤t≤q} (\overrightarrow{∏}_{s∈J_{t}}) with \overrightarrow{∏}_{1≤s≤q}, we obtain the full expansion (3.3.2) of det A. Parts two and three of the proposition follow from the more general first statement. Indeed, let q = 3, J_{1} = [1, …, r − 1], J_{2} = [r, r + 1] and J_{3} = [r + 2, …, k] (J_{1} or J_{3} may be empty). Expand det A as in Equation 3.3.6. The assumption of the second part implies that 2 × 2 minors in the middle term of each triple product in the expansion are de facto commutative, hence skew-symmetric in the columns r and r + 1. Thus the full determinant is skew-symmetric in these columns. The assumption of the third part implies that the analogous 2 × 2 minors in the middle terms of the products in the expansion vanish, so det A = 0.

Proposition 3.3.7 is obvious from the definition of the determinant. To prove Proposition 3.3.8, note that under our assumptions all nonzero summands in the expansion (3.3.2) of det grA have the same degree. Denote them by a_{1}, …, a_{n}. From the definition of the principal symbol it is clear that if a_{i} ∈ A all have the same degree and ∑_{i=1}^{n} gra_{i} is not 0 then it coincides with gr(∑_{i=1}^{n} a_{i}). In our case this means that det grA = ∑_{i=1}^{n} gra_{i} = gr(∑_{i=1}^{n} a_{i}) = gr det A, as we claimed.

We point out some distinctions between column determinants discussed here and ordinary determinants over a commutative ring. The column determinant is not skew-symmetric in columns, since switching columns changes the order of multiplication in the expansion (3.3.2). For the same reason, expansions in rows do not hold, and in general there is no connection between the determinant of a matrix and of its transpose.

3.4 Some computations in U(𝔤𝔩_{n})

Let us define

E_{ij}(a) = E_{ij} + aδ_{ij} · 1 ∈ U(𝔤𝔩_{n}).  (3.4.1)

We claim that the map E_{ij} ↦ E_{ij}(a) extends to a 𝔤𝔩_{n}-equivariant automorphism of the algebra U(𝔤𝔩_{n}). The inverse of this map is given by E_{ij} ↦ E_{ij}(−a). The 𝔤𝔩_{n}-equivariance amounts to the following lemma.

Lemma 3.4.2. For any 1 ≤ p, q, i, j ≤ n and a ∈ ℂ

[E_{pq}, E_{ij}(a)] = δ_{qi}E_{pj}(a) − δ_{jp}E_{iq}(a).  (3.4.3)

Proof. For a = 0 this is the standard commutation relation in 𝔤𝔩_{n}. For arbitrary a both sides are the same as for a = 0.

Now let us check that the commutation relations in 𝔤𝔩_{n} are preserved under the map E ↦ E(a).

Lemma 3.4.4. For any 1 ≤ p, q, i, j ≤ n and a ∈ ℂ

[E_{pq}(a), E_{ij}(a)] = δ_{qi}E_{pj}(a) − δ_{jp}E_{iq}(a).  (3.4.5)

Proof. The expression in the left hand side is equal to [E_{pq}, E_{ij}(a)], because E_{pq}(a) − E_{pq} = aδ_{pq} · 1 commutes with everything in U(𝔤𝔩_{n}). So the statement reduces to the previous lemma.

Remark 3.4.6. In fact, the equivariance of the map E ↦ E(a) and compatibility with the commutation relations both follow from the fact that E_{ij} ↦ aδ_{ij} is a character of the Lie algebra 𝔤𝔩_{n}.

The next proposition states a skew-symmetry property for certain 2 × 2 determinants with entries in U(𝔤𝔩_{n}). These determinants are quantum minors of order 2 (defined in the next section).

Proposition 3.4.7. For any a, the following identity holds:

det | \begin{matrix}E_{ij}(a) & E_{im}(a − 1) \\ E_{lj}(a) & E_{lm}(a − 1) \end{matrix} | + det | \begin{matrix}E_{im}(a) & E_{ij}(a − 1) \\ E_{lm}(a) & E_{lj}(a − 1) \end{matrix} | = 0.  (3.4.8)

Proof. This is a somewhat tedious computation that uses the commutation relation (3.4.3). The left hand side of Equation 3.4.8 is

(E_{ij}(a)E_{lm}(a − 1) − E_{lj}(a)E_{im}(a − 1)) + (E_{im}(a)E_{lj}(a − 1) − E_{lm}(a)E_{ij}(a − 1)),

or, changing the order of terms,

(E_{ij}(a)E_{lm}(a − 1) − E_{lm}(a)E_{ij}(a − 1))  (3.4.9a)
+(E_{im}(a)E_{lj}(a − 1) − E_{lj}(a)E_{im}(a − 1))  (3.4.9b)

Observe that (3.4.9b) may be obtained from (3.4.9a) by switching indices m and j, so that the left hand side of (3.4.8) is the symmetrization of (3.4.9a) in m and j. Expanding (3.4.9a), we see that it is equal to

E_{ij}E_{lm} + aδ_{ij}E_{lm} + (a − 1)δ_{lm}E_{ij} + a(a − 1)δ_{ij}δ_{lm}
−E_{lm}E_{ij} − aδ_{lm}E_{ij} − (a − 1)δ_{ij}E_{lm} − a(a − 1)δ_{lm}δ_{ij}
=[E_{ij}, E_{lm}] + δ_{ij}E_{lm} − δ_{lm}E_{ij}
=δ_{jl}E_{im} − δ_{mi}E_{lj} + δ_{ij}E_{lm} − δ_{lm}E_{ij}
=(δ_{lj}E_{im} − δ_{lm}E_{ij}) + (δ_{ij}E_{lm} − δ_{im}E_{lj}).

The last expression is a sum of two terms that are skew-symmetric in m and j, hence its symmetrization in m and j is indeed zero. Thus the sum of the determinants in Equation 3.4.8 is zero.

3.5 Quantum minors

We define certain elements of the enveloping algebra U(𝔤𝔩_{n}) which are noncommutative analogs of matrix minors. Their particular cases are involved in the formulation of the Capelli identities of Classical Invariant Theory (cf. [HU91, How89a, Wey39]).

Definition 3.5.1. Let I = (i_{0}, i_{1}, …, i_{k}) and J = (j_{0}, j_{1}, …, j_{k}) be two length k + 1 sequences of indices between 1 and n, and a ∈ ℂ. The corresponding quantum minor of order k + 1 is Δ_{I,J}(a) = det |E_{i_{r}j_{s}}(a + \frac{k}{2} − s)|_{0≤r,s≤k}. More explicitly, it is given by the following column determinant:

Δ_{I,J}(a) = det | \begin{matrix}E_{i_{0}j_{0}}(a + \frac{k}{2}) & E_{i_{0}j_{1}}(a + \frac{k}{2} − 1) & … & E_{i_{0}j_{k}}(a − \frac{k}{2}) \\ E_{i_{1}j_{0}}(a + \frac{k}{2}) & E_{i_{1}j_{1}}(a + \frac{k}{2} − 1) & … & E_{i_{1}j_{k}}(a − \frac{k}{2}) \\ ⋮ & ⋮ & ... & ⋮ \\ E_{i_{k}j_{0}}(a + \frac{k}{2}) & E_{i_{k}j_{1}}(a + \frac{k}{2} − 1) & … & E_{i_{k}j_{k}}(a − \frac{k}{2}) \end{matrix} | .  (3.5.2)

We write Δ_{I,J} for Δ_{I,J}(0).

To avoid confusion, we often refer to conventional minors as classical minors. Here are some properties of quantum minors, reinforcing their similarity to classical minors.

Proposition 3.5.3. Consider the set Q_{k+1}(a) of quantum minors Δ_{I,J}(a) of order k +1 with fixed a ∈ ℂ and varying I and J.

(1) Quantum minors Δ_{I,J}(a) are skew-symmetric with respect to row indices i_{r} ∈ I.

(2) Quantum minors Δ_{I,J}(a) are skew-symmetric with respect to column indices j_{s} ∈ J.

(3) The principal symbol of a quantum minor is the corresponding classical minor: grΔ_{I,J}(a) = det |grE_{i_{r}j_{s}}|_{0≤r,s≤k}, which we denote by δ_{IJ}.

Proof. Switching two row indices in a quantum minor is equivalent to switching the corresponding two rows in the determinant. Since the column determinant is skew-symmetric in rows (Proposition 3.3.4), a quantum minor is skew-symmetric with respect to row indices. By Proposition 3.3.5, a quantum minor can be expanded in a pair of consecutive columns. The 2 × 2 determinants involved in this decomposition are quantum minors of order 2, which are skew-symmetric in column indices by Proposition 3.4.7. Therefore, quantum minors are skew-symmetric under switching the indices of two consecutive columns, j_{s} ↔ j_{s+1}. Since an arbitrary permutation is a product of such transpositions, quantum minors are skewsymmetric with respect to any permutation of column indices. If either I or J contains repeating indices, both the quantum minor Δ_{I,J}(a) and the classical minor δ_{IJ} vanish because of the skew-symmetry. If both I and J consist of k + 1 distinct terms, then the classical minor δ_{IJ} is a non-zero homogeneous polynomial of degree k + 1. Moreover, all entries of the (k + 1) × (k + 1) matrix involved in the definition of a quantum minor have degree 1, and for any i ∈ I, j ∈ J we have grE_{ij}(a) = grE_{ij}. By Proposition 3.3.8, we conclude that grΔ_{I,J}(a) = det(grE_{IJ}).

The proof of the proposition makes it clear that the skew-symmetry of Δ_{I,J}(a) with respect to the column indices is due to the fact that the “twists” a + \frac{k}{2}, a + \frac{k}{2} − 1, …, a − \frac{k}{2} of the columns 0, 1, …, k in the definition of Δ_{I,J}(a) (see Equation 3.5.2) form an arithmetic progression with the difference −1. Let us fix a ∈ ℂ and consider the set Q_{k+1}(a) of quantum minors of order k +1 with varying I and J as the set of components of a tensor in ℂ^{n} with k + 1 covariant and k + 1 contravariant indices. Reinforcing the tensor interpretation, let us introduce the following notation. Let I = (i_{0}, i_{1}, …, i_{k}) is a sequence of indices of length k + 1. Denote by Iî_{s} the sequence (i_{0}, …, i_{s−1}, i_{s+1}, …, i_{k}), obtained from I by skipping the sth term of I and by I(î_{r}q) the sequence (i_{0}, …, i_{r−1}, q, i_{r+1}, …, i_{k}) obtained from I by replacing the rth term i_{r} with q. In both cases, the order of the other terms is preserved. The operations of skipping an index and replacing another index commute and may be combined in an obvious way.

Theorem 3.5.4. For any 1 ≤ p, q ≤ n and two sequences of indices I and J of length k + 1, we have the following formula for the adjoint action of 𝔤𝔩_{n} on the linear span in U(𝔤𝔩_{n}) of quantum minors Δ_{I,J}(a) of order k + 1:

[E_{pq}, Δ_{I,J}(a)] = ∑_{r=0}^{k} δ_{qi_{r}}Δ_{I(î_{r}p), J}(a) − ∑_{r=0}^{k} δ_{j_{r}p}Δ_{I, J(ĵ_{r}q)}(a).  (3.5.5)

In particular, the linear span in U(𝔤𝔩_{n}) of the quantum minors of order k + 1 is stable under the adjoint action of 𝔤𝔩_{n}.

Proof. For 0 ≤ s ≤ k, let a_{s} = a + \frac{k}{2} − s. By the definition of the quantum minor, we have Δ_{I,J}(a) = ∑_{σ∈S_{k+1}} sgn(σ)T_{I,J}(σ, a), where we set

T_{I,J}(σ, a) = \overrightarrow{∏}_{0≤s≤k}E_{i_{σ(s)}j_{s}}(a_{s}).  (3.5.6)

We claim that

[E_{pq}, T_{I,J}(σ, a)] = ∑_{r=0}^{k} δ_{qi_{r}}T_{I(î_{r}p),J}(σ, a) − ∑_{r=0}^{k} δ_{j_{r}p}T_{I,J(ĵ_{r}q)}(σ, a).  (3.5.7)

Multiplying by sgn(σ) and summing over σ ∈ S_{k+1} would yield Equation 3.5.5. By the Leibniz rule,

[E_{pq}, T_{I,J}(σ, a)] = ∑_{r=0}^{k} (\overrightarrow{∏}_{s<r}E_{i_{σ(s)}j_{s}}(a_{s})) [E_{pq}, E_{i_{σ(r)}}(a_{r})]\overrightarrow{∏}_{s>r}E_{i_{σ(s)}j_{s}}(a_{s}),  (3.5.8)

which by Equation (3.4.3) is equal to

∑_{r=0}^{k} (\overrightarrow{∏}_{s<r}E_{i_{σ(s)}j_{s}}(a_{s})) (δ_{qi_{σ(r)}}E_{pj_{r}}(a_{r}) − δ_{j_{r}p}E_{i_{σ(r)}q}(a_{r}))\overrightarrow{∏}_{s>r}E_{i_{σ(s)}j_{r}}(a_{s})

= ∑_{r=0}^{k} δ_{qi_{σ(r)}} (\overrightarrow{∏}_{s<r}E_{i_{σ(s)}j_{s}}(a_{s})) E_{pj_{r}}(a_{r})\overrightarrow{∏}_{s>r}E_{i_{σ(s)}j_{r}}(a_{s})

− ∑_{r=0}^{k} δ_{j_{r}p} (\overrightarrow{∏}_{s<r}E_{i_{σ(s)}j_{s}}(a_{s})) E_{i_{σ(r)}q}(a_{r})\overrightarrow{∏}_{s>r}E_{i_{σ(s)}j_{r}}(a_{s}).

The second terms in the last expression and in the right hand side of (3.5.7) are the same. We claim that the first terms are also the same. Indeed, observe that

δ_{qi_{r}}T_{I(î_{r}p),J}(σ, a) = δ_{qi_{r}} (\overrightarrow{∏}_{s<σ^{−1}(r)}E_{i_{σ(s)}j_{s}}(a_{s})) E_{pj_{σ^{−1}(r)}}(a_{σ^{−1}(r)})\overrightarrow{∏}_{s>σ^{−1}(r)}E_{i_{σ(s)}j_{s}}(a_{s}),

and the summation over r in (3.5.7) corresponds to the summation over σ^{−1}(r) here.

Note that the properties of quantum minors used in the proof of the theorem also hold for classical minors. Due to Proposition 3.5.3, the assignment Δ_{I,J}(a) ↦ δ_{IJ} uniquely extends to a linear isomorphism between the linear spans of the quantum minors in Q_{k+1}(a) and of the classical minors of order k + 1. We see that this isomorphism is equivariant with respect to the adjoint actions of 𝔤𝔩_{n} on U(𝔤𝔩_{n}) and on P(M_{n,n}).

Corollary 3.5.9. The Capelli elements, defined as

C_{k}(a) = ∑_{{I: card I=k}} Δ_{I,I}(a),

belong to the center of U(𝔤𝔩_{n}).

Proof. An element of U(𝔤𝔩_{n}) is in the center if and only if it is fixed by the adjoint action of 𝔤𝔩_{n}. The theorem implies that there is a 𝔤𝔩_{n}-equivariant isomorphism between the span of quantum minors Δ_{I,J}(a) of order k into the span of the corresponding classical minors, and the Capelli element C_{k}(a) is the lift of c_{k} = ∑_{I} δ_{II}. Under the adjoint action of 𝔤𝔩_{n} the minors δ_{II} transform as v_{I}^{∗} ⊗ v_{I}, where {v_{I}} and {v_{I}^{∗}} are the standard dual bases in ∧^{k}ℂ^{n} and ∧^{k}ℂ^{n∗}, respectively (see Proposition 3.1.2). Two copies of the group GL_{n} act on ∧^{k}ℂ^{n} and ∧^{k}ℂ^{n∗} in the mutually contragredient ways, and the adjoint action of GL_{n} is the restriction of this GL_{n} ×GL_{n}-action to the diagonal GL_{n}. Thus ∑_{I} v_{I}^{∗} ⊗v_{I} is invariant under the diagonal GL_{n}-action, and hence ad 𝔤𝔩_{n}-invariant. Therefore, c_{k} and C_{k} are also ad 𝔤𝔩_{n}- invariant. Moreover, a little thought further reveals that c_{k} is nothing but the coefficient of λ^{n−k} in the characteristic polynomial of n × n matrices (up to a sign), which is obviously invariant under the conjugation by GL_{n}.

3.6 Behavior of rank ideals under tensor products

For any a ∈ ℂ we have defined above an automorphism of the algebra U(𝔤𝔩_{n}) given on the generators E_{ij} ∈ 𝔤𝔩_{n} by the formula E_{ij} ↦ E_{ij}(a). We call these automorphisms twists. Composing a twist with the map R leads to the following generalization of the kth rank ideal J_{k} ⊆ U(𝔤𝔩_{n}). Consider a homomorphism R^{k}(a) : U(𝔤𝔩_{n}) → PD(M_{k,n}) given by the formula

R^{k}(a)(E_{ij}) = R^{k}(E_{ij}(a)) = ∑_{t=1}^{k} x_{ti}∂_{tj} + (\frac{k}{2} + a)δ_{ij}  (3.6.1)

(cf. Equation 2.2.5).

Definition 3.6.2. The ideal J_{k}(a) ⊆ U(𝔤𝔩_{n}) is the kernel of the map R^{k}(a) : U(𝔤𝔩_{n}) → PD(M_{k,n}).

Denote by P(M_{k,n})(a) the module P(M_{k,n}) with the twisted action of 𝔤𝔩_{n} that we have just defined. Since the algebra PD(M_{k,n}) acts faithfully on P(M_{k,n}), the ideal J_{k}(a) is the annihilator of the module P(M_{k,n})(a). Any enveloping algebra U(𝔤) is equipped with a coproduct δ : U(𝔤) → U(𝔤) ⊗ U(𝔤). The coproduct is an algebra homomorphism defined on the generators E ∈ 𝔤 by the formula δ(E) = E ⊗ 1 + 1 ⊗ E, and δ(1) = 1 ⊗ 1.

Theorem 3.6.3. Let k, l ≥ 0 and a, b, c ∈ ℂ such that a = b + c. Then we have the following identity involving twisted rank ideals in U(𝔤𝔩_{n})

J_{k+l}(a) = δ^{−1}(J_{k}(b) ⊗ U(𝔤𝔩_{n}) + U(𝔤𝔩_{n}) ⊗ J_{l}(c)).

Moreover, if M and N are U(𝔤𝔩_{n})-modules such that Ann M ⊇ J_{k}(b) and Ann N ⊇ J_{l}(c), then Ann(M ⊗ N) ⊇ J_{k+l}(a).

Proof. The idea behind the proof is to exploit the following Exponential Property of algebras of polynomial coefficient differential operators: for two vector spaces U and V, we have PD(U ⊕V) ≃ PD(U)⊗PD(V). Let us show that the U(𝔤𝔩_{n})-module P(M_{k+l,n})(a) is isomorphic to the the tensor product of the U(𝔤𝔩_{n})-modules P(M_{k,n})(b) and P(M_{l,n})(c). We have a direct sum decomposition M_{k+l,n} = M_{k,n} ⊕ M_{l,n} of the vector space of (k + l) × n matrices. Concretely, split a (k +l)×n matrix into the upper k ×n part and the lower l ×n part. In the formula (3.6.1) for the action of E_{ij} on P(M_{k+l,n})(a) the sum over 1 ≤ t ≤ k + l correspondingly splits into two parts, the sum over 1 ≤ t ≤ k and the sum over k + 1 ≤ t ≤ k + l. Split also aδ_{ij} into bδ_{ij} and cδ_{ij} and add them to the first and second sum, respectively. We see that the action of any E_{ij} on the module P(M_{k+l,n})(a) is the same as on the tensor product of P(M_{k,n})(b) and P(M_{l,n})(c). Now we apply the following standard proposition describing the behavior of annihilators of modules under tensor products.

Proposition 3.6.4. Suppose that M and N are U(𝔤)-modules. Let I = Ann M and J = Ann N be their annihilators in U(𝔤). Then

Ann_{U(𝔤)}(M ⊗ N) = δ^{−1}(I ⊗ U(𝔤) + U(𝔤) ⊗ J).

Proof. For the sake of completeness, we give the proof of the proposition. We need to show the following:

Ann_{U(𝔤)⊗U(𝔤)}(M ⊗ N) = I ⊗ U(𝔤) + U(𝔤) ⊗ J.  (3.6.5)

This holds in a very general setup of countably-dimensional modules over algebras. Let us denote the first copy of U(𝔤) by B and the second U(𝔤) by C. Then we have a B-module M, a C-module N and maps ϕ : B → End M and ψ : C → End N, defining the module structures. We also have the tensor product of algebras B ⊗ C and a module M ⊗ N over it. Equation 3.6.5 in this notation becomes

Ker(ϕ ⊗ ψ) = Ker ϕ ⊗ C + B ⊗ Ker ψ.

The right inclusion ⊇ is clear. We prove the opposite inclusion using the fact that the canonical map End M ⊗ End N → End(M ⊗ N) is injective. Let us choose a linear complement B′ to Ker ϕ in B and a basis {b′_{i}} in B′. Then any x ∈ B⊗C may be written as ∑_{i} b′_{i}⊗c′_{i}+∑_{j} b′′_{j}⊗c′′_{j}, where b′_{i} are elements of the basis in B′ and b′′_{j} ∈ Ker ϕ. Suppose that x ∈ Ker ϕ ⊗ ψ, then 0 = ϕ⊗ψ(x) = ∑ ϕ(b′_{i})⊗ψ(c′_{i}) (the second summand is annihilated by ϕ). By the construction, the restriction of ϕ to B′ is injective, therefore all ϕ(b′′_{i}) are linearly independent. Hence all ψ(c′_{j}) must be 0, which means that x ∈ B ⊗ Ker ψ + Ker ϕ ⊗ C, as claimed.

Taking 𝔤 = 𝔤𝔩_{n}, M = P(M_{k,n})(b), and N = P(M_{l,n})(c), we obtain the first statement of the theorem. The second statement follows by an application of Proposition 3.6.4.

Corollary 3.6.6. For any k we have J_{k+1}(a) ⊆ J_{k}(a + \frac{1}{2}) and J_{k+1}(a) ⊆ J_{k}(a − \frac{1}{2}).

Proof. It is enough to prove that J_{1}(±\frac{1}{2}) ⊆ J_{0}, for then we can apply the theorem with l = 1, b = a and c = ±\frac{1}{2}. By Equation 2.2.8, R^{1}(\frac{1}{2})(E_{ij}) = ∂_{j}x_{i} and R^{1}(−\frac{1}{2})(E_{ij}) = x_{i}∂_{j}. Consider the module ℂ[x_{1}, …, x_{n}] over PD(ℂ^{n}). The subspace of polynomials without the constant term is invariant under x_{i}∂_{j} for any 1 ≤ i, j ≤ n, so it is a submodule under the R^{1}(−\frac{1}{2})-action of 𝔤𝔩_{n}. The quotient of the module of all polynomials by this submodule is the trivial one-dimensional module over U(𝔤𝔩_{n}), which has the annihilator J_{0}. Hence J_{0} contains the annihilator in U(𝔤𝔩_{n}) of the module of all polynomials, which is J_{1}(−\frac{1}{2}). To prove that J_{0} ⊇ J_{1}(\frac{1}{2}), consider the module ℂ[∂_{1}, …, ∂_{n}] with the R^{1}(\frac{1}{2})-action of 𝔤𝔩_{n}. Similarly to the above, the constant coefficient differential operators without the constant term form a 𝔤𝔩_{n}- submodule of codimension one, and the quotient by it is the trivial one-dimensional module over 𝔤𝔩_{n}, which has the annihilator J_{0}.

We see that unlike the determinantal ideals, the ideals J_{k} do not form a decreasing filtration. Nevertheless, we have J_{k+2} ⊆ J_{k}. The reason for this discrepancy between the rank ideals and the determinantal ideals is discussed below after the proof of Theorem 3.7.1.

3.7 Generators of the rank ideal J_{k}

Theorem 3.7.1. The set Q_{k+1} of the quantum minors Δ_{I,J} of order k +1 generates the rank k ideal J_{k} ⊆ U(𝔤𝔩_{n}). More generally, the quantum minors Δ_{I,J}(a) of order k + 1 generate the ideal J_{k}(a).

Proof. Applying the “twist” E_{ij} ↦ E_{ij}(a), we see that for a given k the second statement of the theorem is equivalent to the first one. We will prove the theorem by induction in k. The thrust of the proof is to show that the quantum minors in Q_{k+1} belong to the rank ideal J_{k}. The fact that the quantum minors, in fact, generate J_{k} would follow then by a standard argument with filtered algebras. By definition of the rank ideal, Δ_{I,J} ∈ J_{k} is equivalent to R^{k}(Δ_{I,J}) = 0. For k = 0 the quantum minors of order 1 are simply the generators E_{ij} of 𝔤𝔩_{n}. Clearly, they belong to the augmentation ideal J_{0} (and generate it). So suppose that k ≥ 1. Our strategy for proving that R^{k}(Δ_{I,J}) = 0 is as follows. For k = 1 we check the vanishing of the corresponding 2 × 2 determinants by a direct computation in PD(ℂ^{n}) (Proposition 3.7.2). Then we reduce the case k ≥ 2 to k = 1. We use the same idea as in the proof of Theorem 3.6.3: we decompose the vector space M_{k,n} into a direct sum of 𝔤𝔩_{n}-invariant vector subspaces, which leads to the decomposition of the 𝔤𝔩_{n}-action on P(M_{k,n}) into the sum of commuting 𝔤𝔩_{n}-actions. The determinant is a multilinear function of its columns, and we find a particular way to expand it (Proposition 3.7.4) so that every term of the expansion vanishes, due to properties of the determinant and the vanishing of 2 × 2 determinants for k = 1. We now realize the strategy outlined above. First, let us consider the case k = 1 (see Example 2.2.7).

Proposition 3.7.2. The quantum minors Δ_{I,J} of order 2 belong to the rank 1 ideal.

Proof. We need to check that for any quantum minor Δ_{I,J} of order 2, we have R(Δ_{I,J}) = 0. Let I = (i, l), J = (j, m), where 1 ≤ i, j, l, m ≤ n. Since R is an algebra homomorphism, Proposition 3.3.7 tells us that R(Δ_{I,J}) can be rewritten as a determinant with entries in PD(ℂ^{n}):

R(Δ_{I,J}) = det | \begin{matrix}R(E_{ij}(\frac{1}{2})) & R(E_{im}(−\frac{1}{2})) \\ R(E_{lj}(\frac{1}{2})) & R(E_{lm}(−\frac{1}{2})) \end{matrix} | .  (3.7.3)

Let us write down explicitly the entries of this matrix. We have R(E_{ij}(\frac{1}{2})) = (x_{i}∂_{j} + \frac{1}{2}δ_{ij}) + \frac{1}{2}δ_{ij} = x_{i}∂_{j} + δ_{ij} = ∂_{j}x_{i}, and R(E_{im}(−\frac{1}{2})) = (x_{i}∂_{m} + \frac{1}{2}δ_{im}) − \frac{1}{2}δ_{im} = x_{i}∂_{m}. Likewise, R(E_{lj}(\frac{1}{2})) = ∂_{j}x_{l}, and R(E_{lm}(−\frac{1}{2})) = x_{l}∂_{m}. Expanding the determinant (cf. Equation (3.3.3)), we get (∂_{j}x_{i})(x_{l}∂_{m})−(∂_{j}x_{l})(x_{i}∂_{m}) = 0, which can be written as ∂_{j}(x_{i}x_{l}−x_{l}x_{i})∂_{m} = 0. Thus the determinant in Equation 3.7.3 is zero, and the proposition follows.

The following combinatorial construction is the heart of the proof. It will be applied to a quantum determinant Δ_{I,J} of size k + 1 in order to show that R^{k}(Δ_{I,J}) = 0. Let A = (a, a−1, …, a−k), B = (b, b−1, …, b−k), C = (c, c−1, …, c−k) be three “formal arithmetic progressions” with a = b + c. This means that a − s, b − t, c − u (0 ≤ s, t, u ≤ k) are expressions which behave according to the formal rules of arithmetic, namely, a = b + c, a − 1 = (b − 1) + c = b + (c − 1), etc., and in general, a − s = (b − t) + (c − (s − t)), where 0 ≤ t ≤ s ≤ k.

Proposition 3.7.4. Keep the notation from the above. Let F be a multilinear function of k + 1 arguments. We interpret the set of arguments of F as a sequence of length k + 1. Then F (a, a − 1, …, a − k) may be expanded using the multilinearity of F into a sum of ∑_{t=0}^{k} \binom{k+1}{t} = 2^{k+1} terms of a special form. Each term in the expansion is the value of F on some shuffle of the initial segment of B of length t and the initial segment of C of length k + 1 − t (this is a sequence of length k + 1). The terms in the expansion correspond to paths of the length k + 1 on the lattice ℤ^{2}, connecting the point (0, 0) with a point on the line x + y = k + 1.

For example, for k = 0 we have F (a) = F (b) + F (c) and for k = 1 we have

F (a, a − 1) = F (b, a − 1) + F (c, a − 1) = (F (b, b − 1) + F (b, c)) + (F (c, b) + F (c, c − 1)).

Proof. Let 1 ≤ s ≤ k+1. Consider the lattice ℤ^{2} on the Cartesian plane. Mark the x-axis with b, b − 1, …, b − k and the y-axis with c, c − 1, …, c − k. Consider a path P = (P_{0}, P_{1}, …, P_{s}) of length s connecting the origin P_{0} = (0, 0) with the point P_{s} = (t, s − t), where 0 ≤ t ≤ s. We construct a sequence of length s + 1 whose ith term (0 ≤ i ≤ s) is the mark on the x-axis below P_{i+1} if P_{i}P_{i+1} is horizontal and the mark on the y-axis to the left of P_{i+1} if P_{i}P_{i+1} is vertical. We extend this sequence to a sequence of length k + 1 by adding the terms a − (s + 1), …, a − k at the end. The sequence associated to a path P represents a term in the sth partial expansion of F (a, a − 1, …, a − k). The proposition is now proved by induction in s. Observe that a path of length s ≤ k can be extended to a path of length s + 1 in exactly two ways. If P_{s} = (b − t, c − (s − t)) then either P_{s+1} = (b − t − 1, c − (s − t)) (“step right”) or P_{s+1} = (b − t, c − (s − t) − 1) (“step up”). These correspond to expansions, respectively, a − (s + 1) = (b − t − 1) + (c − (s − t)) and a − (s + 1) = (b − t) + (c − (s − t) − 1). The (k + 1)st partial expansion is the full expansion of the proposition, involving 2^{k+1} terms.

The diagram illustrates the case k = 2. The thick line gives the term F (b, c, b − 1) in the expansion of F (a, a − 1, a − 2).

We are now in a position to prove that for any quantum minor Δ_{I,J} in Q_{k+1}, R(Δ_{I,J}) vanishes. We argue by induction in k. The case k = 1 was proved in Proposition 3.7.2. Let k ≥ 2. Let us use a direct sum decomposition M_{k,n} = M_{1,n} ⊕ M_{k−1,n} to decompose the representation R^{k}(b+c) of 𝔤𝔩_{n} in the vector space P(M_{k,n}) = P(M_{1,n})⊗P(M_{k−1,n}) into the sum of commuting representations R′(b) and R′′(c) on the factors of the tensor product (cf. Theorem 3.6.3).

Suppressing the indices in the notation, for any b, c ∈ ℂ we have the following identity for the actions of a generator E_{ij} of 𝔤𝔩_{n}:

R(b + c) = R′(b) + R′′(c).  (3.7.5)

Note that R′ is the map R^{1} and R′′ is the map R^{k−1} in the corresponding coordinates. We may interpret the column determinant of a matrix as a multilinear function of the columns. As in the proof of Proposition 3.7.2 we see that R(Δ_{I,J}) may be written as a column determinant with entries in R(U(𝔤𝔩_{n})) ⊆ PD(M_{k,n}). From the formula (3.5.2) for Δ_{I,J} we infer that the arguments of the determinant constitute a formal arithmetic progression A = (\frac{k}{2}, \frac{k}{2} −1, …, −\frac{k}{2}). This allows us to apply Proposition 3.7.4 to expand R(Δ_{I,J}) into the sum of determinants whose arguments are columns of the form R′(\frac{1}{2} − t) and R′′(\frac{k−1}{2} − u) with t, u ≥ 0. We are going to use the properties of the determinant from Section 3.3 to show that all terms in this expansion vanish. Consider one term of the expansion. First, since the R′-action commutes with the R′′-action, we can bring the columns containing R′ to the left and the column containing R′′ to the right, preserving the relative order of columns of the same “type” (possibly changing the sign of the determinant). If there are at least two columns of type R′, then the first two columns in the new expression for the determinant would be R′(\frac{1}{2}) and R′(−\frac{1}{2}). All the 2 × 2 determinants in these two columns are of the form (3.7.3), and hence vanish by Proposition 3.7.2. By the third part of Proposition 3.3.5, the full determinant vanishes. Consider one of the remaining terms of the expansion. Since it has at most one column of type R′, there are at least (k + 1) − 1 = k columns of type R′′, starting from R′′(\frac{k−1}{2}). These columns constitute a formal arithmetic progression (\frac{k−1}{2}, …, −\frac{k−1}{2}) of length k − 1. Recall that the kernel of R′′ = R^{k−1} is the rank k − 1 ideal. By the inductive assumption, every k × k determinant involving these columns vanishes. If there is a first column is of type R′, expand the determinant in the first column. Otherwise, expand the determinant in the last column. In either case the determinant vanishes. Thus all terms in the expansion of R(Δ_{I,J}) vanish. This concludes the proof of the fact that R(Δ_{I,J}) = 0 and all quantum minors Δ_{I,J} of size k + 1 belong to the kth rank ideal J_{k}. To finish the proof of the theorem, we apply the criterion of Proposition 2.3.3. Let us check that all conditions hold. We have just established that the quantum minors of order k + 1 belong to the rank ideal J_{k}. Their principal symbols are the classical minors δ_{IJ} of order k+1 (by Proposition 3.5.3). By Theorem 3.1.4, the classical minors of order k +1 generate the kth determinantal ideal I_{k}. We know from Theorem 3.2.4 that the ideal I_{k} the associated graded of the rank ideal J_{k}. Thus we conclude that the quantum minors of order k + 1 generate the rank ideal J_{k} ⊆ U(𝔤𝔩_{n}). In fact, since the associated graded algebra of U(𝔤𝔩_{n}) is commutative, we see that the quantum minors even generate J_{k} as a left ideal.

The last part of the proof of Theorem 3.7.1 makes clear the sense in which the quantum minors form a natural system of generators for the rank ideal J_{k}: they “quantize” a natural system of generators for the associated graded of J_{k}. We note that the part of the proof using Proposition 3.7.4 has a natural interpretation in terms of the Hopf algebra structure on U(𝔤𝔩_{n}). Instead of using an “external” decomposition of R^{k} into R′ and R′′, we can consider the coproduct δ(Δ_{I,J}) of the quantum determinant. In fact, this can be done for an arbitrary Lie algebra 𝔤 and a character χ : 𝔤 → ℂ. For any X ∈ 𝔤, α ∈ ℂ set X(α) = X + αχ(X). Now choose elements X_{ij} ∈ 𝔤 and let Δ′_{I,J}(α) = det |X_{IJ}(α − s)|. Since the coproduct is the algebra homomorphism, it can be applied to Δ′_{I,J}(α) entrywise (see Proposition 3.3.7), and δ(Δ′_{I,J}(a)) is thus expressed as a determinant with entries δ(X_{IJ}(α − s)). An application of Proposition 3.7.4 will expand this determinant into 2^{k+1} determinants with the columns of two types: X(β−t)⊗1 and 1⊗X(γ−u). Iterating the coproduct, we may obtain analogous identities for determinants with entries in U(𝔤)^{⊗m}. Theorem 3.7.1 clarifies why unlike the determinantal ideals, which satisfy I_{k} ⊆ I_{k−1}, the rank ideals only satisfy J_{k} ⊆ J_{k−1}(±\frac{1}{2}) (see Corollary 3.6.6). The reason for this discrepancy in the behavior of ideals can be traced back to the difference in the behavior of classical and quantum minors that generate them. As we saw in the proof of Theorem 3.1.4, the inclusion I_{k} ⊆ I_{k−1} of the determinantal ideals relies on the row or column expansion of a minor of size k involving minors of size k − 1. On the other hand, if we expand the generators of the rank ideal J_{k} in the first or last row, they will be expressed in terms of column determinants twisted by ±\frac{1}{2}, which belong to J_{k−1}(±\frac{1}{2}) rather than to J_{k−1}. This gives rise to a “twisted” inclusion of the rank ideals in U(𝔤𝔩_{n}), see Corollary 3.6.6.

3.8 A localization property

In the theory of N-rank a prominent role played by the fact that an irreducible unitary representation of rank k remains irreducible when restricted to the kth maximal parabolic subgroup, see [How82]. The determinant expressions for generators of the kth rank ideal that we have found lead to an interesting analog of this property in the algebraic setting. Suppose that 2k ≤ n (this is an algebraic counterpart of the condition that a reductive dual pair be in stable range). Let 𝔭 be the kth maximal parabolic subalgebra of 𝔤𝔩_{n}, consisting of block upper triangular matrices with the blocks of orders k and n − k. We consider the images of the standard generators E_{ij} of 𝔤𝔩_{n} under R. Let k + 1 ≤ i ≤ n and 1 ≤ j ≤ k. Set i_{0} = i, j_{0} = j, and i_{r} = r for 1 ≤ r ≤ k. Choose any k + 1 ≤ j_{1} < … < j_{k} ≤ n, which is possible because of our restriction on k. Expanding the noncommutative determinant Δ_{I,J} in the first column, we get

Δ_{I,J} = E_{ij} det |E_{i_{r}j_{s}}(\frac{k}{2} − r)|_{1≤r,s≤k} + ∑_{a=1}^{k} E_{i_{a}j}(\frac{k}{2} ) det |E_{i_{r}j_{s}}(\frac{k}{2} − s)|_{1≤s≤k,0≤r≤k,r≠a}. (3.8.1)

Due to our choice of I and J, everything in the right hand side except E_{ij} belongs to the enveloping algebra U(𝔭). Since the quantum determinant Δ_{I,J} of order k + 1 is mapped by R^{k} to zero, we can express R^{k}(E_{ij}) for k + 1 ≤ i ≤ n and 1 ≤ j ≤ k as a right quotient of the images under R^{k} of two elements of the enveloping algebra U(𝔭). This means that the algebra U(𝔤𝔩_{n})/J_{k} = R^{k}(U(𝔤𝔩_{n})) is some noncommutative localization of its subalgebra U(𝔭)/J_{k} ∩ U(𝔭) = R^{k}(U(𝔭)). Since these two algebras are noncommutative noetherian domains, they admit right quotient rings by Goldie’s Theorem, and we obtain that these quotient rings coincide.

4 Rank of highest weight modules over 𝔤𝔩_{n}

Let us call a two-sided ideal I in U(𝔤𝔩_{n}) an ideal of rank at most k if I contains the kth rank ideal J_{k}. Primitive ideals satisfying this property will be classified in Part II. The classification uses a powerful theorem of M. Duflo claiming that every primitive ideal in the enveloping algebra of a complex reductive Lie algebra 𝔤 equals the annihilator of a simple highest weight module. This makes it natural to introduce and study the rank of highest weight modules, which is the objective of this section.

4.1 Characterization of rank in terms of the highest weight

We say that a U(𝔤𝔩_{n})-module M is a module of rank at most k if its annihilator Ann M is an ideal of rank at most k, i.e. contains J_{k}. We will use the description of generators of the kth rank ideal J_{k} provided by Theorem 3.7.1 to characterize the simple highest weight modules over 𝔤𝔩_{n} of rank at most k. Let us pick the Cartan subalgebra 𝔥 ⊆ 𝔤𝔩_{n} consisting of the diagonal matrices, and the Borel subalgebra 𝔟 ⊆ 𝔤𝔩_{n} consisting of the upper triangular matrices. The nilradical 𝔫 of 𝔟 then consists of the upper triangular matrices with zeros on the diagonal. Thus the subalgebra 𝔫 is spanned by the generators E_{ij} ∈ 𝔤𝔩_{n} with 1 ≤ i < j ≤ n. Let {ϵ_{i}|1 ≤ i ≤ n} be the basis in 𝔥^{∗} dual to the basis {E_{ii}|1 ≤ i ≤ n} of the Cartan subalgebra. Define a symmetric bilinear form on 𝔥^{∗} = ℂ^{n} making ϵ an orthonormal basis and use it to identify 𝔥^{∗} with 𝔥. For a weight λ ∈ 𝔥^{∗}, let λ_{1}, …, λ_{n} be its components in the basis ϵ. We thus view weights of 𝔤𝔩_{n} as sequences of length n. With these conventions, the roots are {ϵ_{i} − ϵ_{j}|1 ≤ i ≠ j ≤ n} The root subspace of ϵ_{i} − ϵ_{j} is ℂE_{ij}. A root ϵ_{i} − ϵ_{j} is positive for i < j and negative for i > j. The half-sum of the positive roots is the following sequence:

ρ = (\frac{n − 1}{2}, \frac{n − 3}{2}, …, −\frac{n − 1}{2}) .  (4.1.1)

Denote by L(λ) the simple highest weight module with highest weight μ = λ−ρ. The relation between the entries of λ and μ is given by the formula λ_{i} = μ_{i} + \frac{n+1}{2} − i. Denote by γ the the arithmetic progression of length n − k with first term \frac{n−k−1}{2} and difference −1:

γ = (\frac{n − k − 1}{2}, \frac{n − k − 3}{2}, …, −\frac{n − k − 1}{2}) .

The sequence γ numerically coincides with the half-sum of the positive roots ρ for the Lie algebra 𝔤𝔩_{n−k}.

Theorem 4.1.2. Let λ = μ + ρ be a weight of 𝔤𝔩_{n}. Then the following are equivalent:

(1) The simple highest weight module L(λ) is of rank at most k.

(2) The sequence λ contains γ as a subsequence.

(3) There exist a sequence of indices j_{0} = 0, 1 ≤ j_{1} < j_{2} < … < j_{k} such that for any 0 ≤ s ≤ k, if j_{s} < i < j_{s+1} then μ_{i} = −\frac{k}{2} + s. Put differently, μ has the form

μ = (−\frac{k}{2} , …, −\frac{k}{2} , ∗, −\frac{k}{2} + 1, …, −\frac{k}{2} + 1, ∗, …, ∗, \frac{k}{2} , …, \frac{k}{2} ),  (4.1.3)

where the stars represent the entries of μ with indices j_{1}, …, j_{k}.

Proof. Let us check the equivalence of the last two conditions. Denote by γ′ the subsequence of λ = μ + ρ consisting of the first n − k entries of λ with indices i ∉ {j_{1}, …, j_{k}}. We claim that μ has the form (4.1.3) if and only if γ′ = γ. Indeed, if j_{s} < i < j_{s+1} then γ′_{i−s} = λ_{i}, hence γ′_{i−s} −γ_{i−s} = λ_{i} −γ_{i−s} = (μ_{i} + \frac{n+1}{2} − i)−(\frac{n+1−k}{2} −(i−s)) = μ_{i} + \frac{k}{2} −s = μ_{i} −(−\frac{k}{2} +s). The claim is now obvious. It remains to show that the first condition of the theorem is equivalent to the second or third. We will prove that the first and third conditions are equivalent. Here is an outline of the proof. Proposition 4.1.4 below imposes restrictions on the highest weight μ of a highest weight module of rank at most k. These restrictions turn out to be precisely the third condition of the theorem. Conversely, for any μ of the form (4.1.3) we construct in Proposition 4.1.9 a module P over PD(M_{k,n}) and a vector w ∈ P which is a highest weight vector of weight μ with respect to the U(𝔤𝔩_{n})-action on P. By virtue of the construction, the 𝔤𝔩_{n}-module generated by w is a highest weight module of rank at most k with highest weight μ, demonstrating that the third condition of the theorem implies the first and completing the proof of the theorem. To see when a highest weight 𝔤𝔩_{n}-module M has rank at most k, let us analyze the effect of the generators of the rank k ideal on a highest weight vector of M. Our goal is to show that if the module M is of rank at most k then the highest weight has the special form (4.1.3).

Proposition 4.1.4. Suppose that M is a highest weight module with the highest weight vector v of weight μ. Let 1 ≤ i_{0} < i_{1} < … < i_{k} ≤ n. Then the “principal” quantum minor Δ_{I,I} multiplies v by the scalar

∏_{s=0}^{k}(μ_{i_{s}} + \frac{k}{2} − s).  (4.1.5)

Proof. To illustrate the idea behind the proof, let us consider the case k = 1 first. Since v is a highest weight vector of weight μ, it is annihilated by 𝔫 and is multiplied by μ(H) under the action of H ∈ 𝔥. With our conventions, 𝔫 consists of strictly upper-triangular matrices and 𝔥 consists of diagonal matrices. If I = (i, l), i < l then the principal quantum minor Δ_{I,I} is given by the formula

Δ_{I,I} = det | \begin{matrix}E_{ii}(\frac{1}{2}) & E_{il}(−\frac{1}{2}) \\ E_{li}(\frac{1}{2}) & E_{ll}(−\frac{1}{2}) \end{matrix} | = E_{ii}(\frac{1}{2})E_{ll}(−\frac{1}{2}) − E_{li}(\frac{1}{2})E_{il}(−\frac{1}{2})  (4.1.6)

(cf. Equations 3.3.3, 3.5.2). Since i < l, E_{il}(−\frac{1}{2}) = E_{il} is a strictly upper-triangular matrix, hence it annihilates v, and so the second term in the expansion (4.1.6) annihilates v. The first term in (4.1.6) is a product in U(𝔤𝔩_{n}) of diagonal matrices E_{ii}(\frac{1}{2}) and E_{ll}(−\frac{1}{2}) ∈ 𝔥 ⊂ U(𝔤𝔩_{n}). Hence it multiplies v by (μ_{i} + \frac{1}{2})(μ_{l} − \frac{1}{2}). Note that for k = 1 the action of Δ_{I,I} on v reduces to the action of the first term of the expansion (4.1.6), which is easily computed. An analogous fact is true in general. Now let k be arbitrary. Consider the full expansion of Δ_{I,I}. We have

Δ_{I,I} = ∑_{σ∈S_{k+1}} sgn(σ)\overrightarrow{∏}_{0≤s≤k}E_{i_{σ(s)}i_{s}}(\frac{k}{2} − s)  (4.1.7)

(cf. Equations 3.3.2, 3.5.2). We claim that for any σ ≠ id, the product in the expansion (4.1.7) annihilates v. Indeed, since v is a highest weight vector, any E ∈ 𝔥 multiplies v by a constant and E ∈ 𝔫 annihilates v. Suppose that σ ≠ id and let r be the largest integer between 0 and k such that σ(r) ≠ r. Since σ is a bijection and σ(s) = s for s > r, we see that σ(r) < r, hence i_{σ(r)} < i_{r}. Therefore E_{i_{σ}(s)i_{s}} = E_{i_{s}i_{s}} ∈ 𝔥 for s > r and E_{i_{σ(r)}i_{r}}(\frac{k}{2} − r) = E_{i_{σ(r)}i_{r}} ∈ 𝔫. Split the product in (4.1.7) into the product over 0 ≤ s < r, the term E_{i_{σ(r)}i_{r}}(\frac{k}{2} − r) corresponding to s = r, and the product over r < s ≤ k. By what we have just said, the last product multiplies v by a scalar and the middle term annihilates v. Therefore, the sum in (4.1.7) reduces to one term with σ = 1, which multiplies v by ∏_{s=0}^{k}(μ_{i_{s}} + \frac{k}{2} − s).

In the setting of Proposition 4.1.4, suppose additionally that M is a module of rank at most k. Let us inductively define a sequence of indices j_{0} < j_{1} < j_{2} < … < j_{k+1}: j_{0} = 0, j_{s+1} = min\{j > j_{s}|μ_{j} ≠ −\frac{k}{2} + s\}. For any 1 ≤ i ≤ n, if j_{s} < i < j_{s+1} then μ_{i} = −\frac{k}{2} + s. If j_{k+1} > n then μ and the sequence J satisfy the third condition of the theorem. We will show that this is always the case, namely, that j_{k+1} ≤ n contradicts the assumption that M is of rank at most k. Indeed, if j_{k+1} ≤ n then ∏_{s=0}^{k}(μ_{j_{s+1}} + \frac{k}{2} − s) ≠ 0, because μ_{j_{s+1}} ≠ −\frac{k}{2} + s for 0 ≤ s ≤ k. Let I be the sequence i_{s} = j_{s+1}, 0 ≤ s ≤ k, then by Proposition 4.1.4, Δ_{I,I}v ≠ 0. But Theorem 3.7.1 asserts that any quantum minor Δ_{I,J} of order k+1, and in particular Δ_{I,I}, belongs to J_{k}, hence acts on M by 0, which is the desired contradiction. Taking M = L(λ), we see that if L(λ) is of rank at most k then the third condition of the theorem holds. We now proceed to construct a highest weight module over U(𝔤𝔩_{n}) with the highest weight μ for any weight μ satisfying the third condition of the theorem.

Notation 4.1.8. To simplify the notation, let us agree for the rest of the section that indices t and i are varying in the limits 1 ≤ t ≤ k, 1 ≤ i ≤ n and all direct sums and tensor products are taken over 1 ≤ t ≤ k, 1 ≤ i ≤ n.

Recall that M_{k,n} is the vector space of k × n matrices and PD(M_{k,n}) is the algebra of polynomial coefficient differential operators on M_{k,n}.

Proposition 4.1.9. Let 1 ≤ j_{1} < j_{2} < … < j_{k}, a_{t} ∈ ℂ. There exists a pair (P, w) consisting of a non-zero module P over the algebra PD(M_{k,n}) of polynomial coefficient differential operators on M_{k,n} and a vector w ∈ P, with the following properties:

(i) x_{ti}w = 0 for 1 ≤ i < j_{t}; (ii) ∂_{ti}w = 0 for i > j_{t}; (iii) x_{ti}∂_{ti}w = a_{t}w for i = j_{t}.

Proof. For k = 1, n = 1 this amounts to the following proposition.

Proposition 4.1.10. Let V = ℂv be a one-dimensional vector space, a ∈ ℂ. There exists a pair (P, w) consisting of a module P over PD(V) and a vector w ∈ P satisfying any one of the following three conditions:

(i) ∂ · w = 0; (ii) x · w = 0; (iii) x∂ · w = aw.

Proof. In the first case let P = ℂ[x] and w = 1. In the second case let P = ℂ[d] and w = 1. In the the third case let P = x^{a}ℂ[x, x^{−1}] and w = x^{a}.

Remark 4.1.11. If a ∈ ℤ then the module x^{a}ℂ[x, x^{−1}] = ℂ[x, x^{−1}] is not irreducible. It has a unique nontrivial submodule ℂ[x] and a unique irreducible quotient, which is isomorphic to ℂ[∂]. Thus in the third case of the proposition, for a ∈ ℤ we can replace the module x^{a}ℂ[x, x^{−1}] by an irreducible module P as follows. When a ≥ 0, we let P = ℂ[x], and x^{a} ∈ P ⊂ ℂ[x, x^{−1}]. When a ≤ −1, x^{a} projects into a nonzero element proportional to ∂^{−a} of the quotient ℂ[∂] of ℂ[x, x^{−1}].

To treat the case of arbitrary k and n, let us exploit the exponential property of Weyl algebras once more (cf. Theorem 3.6.3). Decompose the vector space M_{k,n} of k × n matrices into a direct sum of one-dimensional subspaces spanned by matrices with one non-zero entry, M_{k,n} = ⊕ℂv_{ti}. We have a corresponding tensor product decomposition

PD(M_{k,n}) = ⊗PD(x_{ti}).

We define the module P and the vector w as tensor products, P = ⊗P_{ti}, w = ⊗w_{ti}, where P_{ti} is a module over PD(x_{ti}) and w_{ti} ∈ P_{ti}. For fixed t and i, precisely one of the conditions 1 ≤ i < j_{t}, i > j_{t}, i = j_{t} holds, and we take a pair (P_{ti}, w_{ti}) constructed in Proposition 4.1.10.

Let us use the homomorphism R : U(𝔤𝔩_{n}) → PD(M_{k,n}) to make a PD(M_{k,n})-module P into a U(𝔤𝔩_{n})-module. By definition, the U(𝔤𝔩_{n})-module obtained in this manner is of rank at most k. We claim that the vector w ∈ P constructed in Proposition 4.1.9 is in fact a highest weight vector for the 𝔤𝔩_{n}-action. Indeed, suppose that i < j. Then for any t we have either i < j_{t} or j > j_{t}. In the first case x_{ti}w_{ti} = 0, and so x_{ti}∂_{tj}w = ∂_{tj}x_{ti}w = 0. In the second case ∂_{tj}w_{tj} = 0, and x_{ti}∂_{tj}w = 0. Using Equation 2.2.5, we see that R(E_{ij})w = ∑_{t=1}^{k} x_{ti}∂_{tj}w = 0, because every term in the sum is 0, so w is annihilated by 𝔫 ⊂ 𝔤𝔩_{n}. Let us now compute the weight of w. First suppose that i ∉ {j_{1}, …, j_{k}} and choose s such that j_{s} < i < j_{s+1}. If t ≤ s then i > j_{s} ≥ j_{t}, so ∂_{ti}w_{ti} = 0 and x_{ti}∂_{ti}w = 0. If t ≥ s + 1 then i < j_{s+1} ≤ j_{t}, so x_{ti}w_{ti} = 0 and x_{ti}∂_{ti}w_{ti} = (∂_{ti}x_{ti} − 1)w = −w. Therefore R(E_{ii})w = ∑_{t=1}^{k} x_{ti}∂_{ti}w + \frac{k}{2}w = ∑_{t=1}^{s} x_{ti}∂_{ti}w + ∑_{t=s+1}^{k} x_{ti}∂_{ti}w + \frac{k}{2}w = 0 + (k − s)(−w) + \frac{k}{2}w = (−\frac{k}{2} + s)w. When i = j_{s}, the same argument yields R(E_{ii})w = (a_{s} − \frac{k}{2} + s)w. We conclude that the weight μ of w has the form (4.1.3) and conversely, setting a_{t} = μ_{j_{t}} + \frac{k}{2} − t we can get any μ of that form. Denote by M the U(𝔤𝔩_{n})-submodule of P generated by w. What we have established is that it is a highest weight module with the highest weight μ and Ann M ⊇ J_{k}, so Ann L(μ + ρ) ⊇ Ann M ⊇ J_{k}, and L(λ) = L(μ + ρ) is of rank at most k. This finishes the proof of the theorem.

4.2 Case of finite-dimensional modules

Let us use the results of the previous section to characterize the finite-dimensional 𝔤𝔩_{n}-modules of rank at most k. Finite-dimensional representations of 𝔤𝔩_{n} admit the following combinatorial parametrization (see [How95]). Let μ = (μ_{1}, …, μ_{n}) be an integral dominant weight for 𝔤𝔩_{n}. Represent μ in the form

μ_{1} ≥ … ≥ μ_{r} ≥ 0 ≥ … ≥ 0 ≥ μ_{n−s+1} ≥ … ≥ μ_{n},

so that μ_{i} = 0 for r < i ≤ n − s. We define the sequences d = (d_{1}, …, d_{r}) = (μ_{1}, …, μ_{r}) and e = (e_{1}, …, e_{s}) = (−μ_{n}, …, −μ_{n−s+1}). Since μ is a dominant weight, we get that d_{1} ≥ … d_{r} ≥ 0 and e_{1} ≥ … e_{s} ≥ 0, so that d and e are partitions. Denote by D and E the Young diagrams corresponding to d and e. They satisfy the conditions

depth(D) ≤ r, depth(E) ≤ s, and depth(D) + depth(E) ≤ n.  (4.2.1)

(The depth of a Young diagram is the number of nonzero parts of the corresponding partition). Denote by ρ^{D,E} the simple finite-dimensional highest weight module over 𝔤𝔩_{n} with highest weight μ = μ(D, E). More generally, one can consider a finite-dimensional 𝔤𝔩_{n}-module ρ = ρ^{D,E} ⊗ det^{l} (here the “twist” l may be an arbitrary complex number).

Theorem 4.2.2. Let ρ be a simple finite-dimensional module over 𝔤𝔩_{n}. Then ρ is of rank at most k if and only if there exist r, s ≥ 0 with r + s = k and Young diagrams D and E with depth(D) ≤ r, depth(E) ≤ s such that

ρ = ρ^{D,E} ⊗ det ^{\frac{r−s}{2}} .  (4.2.3)

Proof. Theorem 4.1.2 characterizes the highest weights of simple highest weight modules over 𝔤𝔩_{n} of rank at most k. We need to find which dominant weights satisfy the condition of the theorem. Suppose that μ is a dominant weight of the form (4.1.3). Then the sequence μ is weakly decreasing, and μ cannot contain two segments J_{t} = (−\frac{k}{2} + t, …, −\frac{k}{2} + t) and J_{r} = (−\frac{k}{2} + r, …, −\frac{k}{2} + r) with t < r. So μ contains exactly one such segment, let us say J_{r}. Then in the notation of Theorem 4.1.2, j_{t} = t for t ≤ r and J_{r} has length at least n − k. Let s = k − r, then −\frac{k}{2} + r = \frac{r−s}{2} and r + (n − k) = n − s. Denote μ′ = μ − \frac{r−s}{2}ω_{n}, where ω_{n} is the nth fundamental weight (1, …, 1). Then μ′_{i} = 0 for r + 1 ≤ i ≤ n − s, and μ′ is dominant, so by definition, the simple finite-dimensional module with highest weight μ′ is ρ^{D,E}, and D and E satisfy the condition (4.2.1). Therefore ρ = ρ^{D,E} ⊗ det^{\frac{r−s}{2}} .

By basic algebra, irreducible finite-dimensional representations of an associative algebra correspond bijectively to primitive ideals of finite codimension. Thus Theorem 4.2.2 describes those primitive ideals in U(𝔤𝔩_{n}) of rank at most k that have finite codimension. The general case will be studied in Part II.

Remark 4.2.4. Finite-dimensional representations of Theorem 4.2.2 arise in the dual pair correspondence for a reductive dual pair (U_{n}, U_{r,s}) (see [How83], [How95]). In fact, these representations (without a determinant twist) appear in Weyl’s analysis of the representation of GL_{n} in the space of polynomial functions P(V) on the vector space V = ℂ^{r} ⊗ℂ^{n} ⊕ℂ^{s} ⊗ℂ^{n∗} (see [Wey39]). The new point of Theorem 4.2.2 is that the representations of form (4.2.3) exhaust all finite-dimensional representations of 𝔤𝔩_{n} of rank at most k.

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