A Generalized Poincaré Theorem

for Dual Lie Transformation Groups

Tuong Ton-That

Abstract. Let k and n be integers such that k > 2n > 0. Let M be the complex analytic manifold defined by M = {x ∈ ℂ^{n×k} : xx^{t} = 0, rank (x) = n}. Let G = SO(k, ℂ) and G′ = GL(n, ℂ), then Witt’s theorem on quadratic forms implies that G is a maximal connected Lie group acting transitively on M by right multiplication. Also, G′ is a maximal connected Lie group acting freely on M by left multiplication. If f ∈ C^{∞}(M), x ∈ M, g ∈ G, and g′ ∈ G′ define R(g)f (resp. L(g′)f) by

(R(g)f)(x) = f(xg) and (L(g)f)(x) = f(g^{−1}x).

If D^{ω}(M) denotes the algebra of all analytic differential operators on M then an element D ∈ D^{ω}(M) is called right (resp. left)-invariant if DR(g) = R(g)D, ∀ g ∈ G (resp. DL(g′) = L(g′)D, ∀ g′ ∈ G′).

THEOREM: Let D_{l}^{ω}(M) (resp. D_{r}^{ω}(M)) denote the subalgebra of D^{ω}(M) of all left (resp. right)-invariant analytic differential operators on M. Let Ũ(𝔤) (resp. Ũ(𝔤′)) denote the universal enveloping algebra generated by the infinitesimal action of R(g) (resp. L(g′)). Then we have

D_{l}^{ω}(M) = Ũ(𝔤) and D_{r}^{ω}(M) = Ũ(𝔤′).

Moreover, the commutant of D_{l}^{ω}(M) in D^{ω}(M) is D_{r}^{ω}(M), and viceversa.

This theorem also holds for other types of dual Lie transformation groups acting on analytic manifolds.

2000 Mathematics Subject Classification: Primary 15A63, 16S32; Secondary 16S30, 14L35.

1 Introduction

In 1900 H. Poincaré established the existence of the universal enveloping algebra of a Lie algebra and proved one of the most fundamental results in the theory of Lie groups and Lie algebras. This theorem which is valid for a Lie algebra over an arbitrary field is usually called the Poincaré-Birkhoff-Witt theorem; however for the case of a real or complex Lie algebra it is entirely due to Poincaré as shown in [TT-T].

Theorem 1.1 (Poincaré). Let G be a real or complex Lie group with Lie algebra 𝔤. Let U(𝔤) denote the universal enveloping algebra of 𝔤. If {X_{i} : 1 ≤ i ≤ n} is a basis of 𝔤 then the ordered monomials 1 and X_{i_{1}} ⋯ X_{i_{s}}(s ≥ 1, i_{1} ≤ ⋯ ≤ i_{s}) form a basis for U(𝔤).

Assume that G is a real or complex connected Lie group. For each g ∈ G, the translations l_{g}, r_{g} : G → G defined by l_{g}(x) = gx and r_{g}(x) = xg, x ∈ G, are analytic diffeomorphisms of G onto itself. Let D^{ω}(G) denote the algebra of all analytic differential operators on G. A differential operator D of D^{ω}(G) is said to be left (resp. right)-invariant if it is invariant under all left (resp. right) translations. Let [·, ·] denote the commutator product of D^{ω}(G). If A is a subalgebra of D^{ω}(G) then the centralizer (or commutant) of A in D^{ω}(G) is defined as the set {D′ ∈ D^{ω}(G) : [D′, D] = 0, ∀ D ∈ A}, and the centre of A is defined as the set {D′ ∈ A : [D′, D] = 0, ∀ D ∈ A}. Then the following can be easily deduced from the Poincaré theorem.

Corollary 1.2 (To Poincaré Theorem). If D_{l}^{ω}(G) (resp. D_{r}^{ω}(G)) denotes the subalgebra of D^{ω}(G) of all left (resp. right)-invariant analytic differential operators on G then D_{l}^{ω}(G) (resp. D_{r}^{ω}(G)) is isomorphic to the universal enveloping algebra U(𝔤). Moreover, the centralizer of D_{l}^{ω}(G) in D^{ω}(G) is D_{r}^{ω}(G), and viceversa. Finally, the centres of D_{l}^{ω}(G) and D_{r}^{ω}(G) coincide with D_{l}^{ω}(G)∩D_{r}^{ω}(G).

In the context of Lie transformation groups on analytic manifolds the corollary above can be phrased as follows: Consider G as a G-transformation group acting on the analytic manifold M = G to the right and as a G′-transformation acting on M to the left; then the subalgebras of all left (resp. right)-invariant analytic differential operators on the analytic manifold M are mutual commutants in D^{ω}(M). We shall generalize this result to dual transformation groups acting on analytic manifolds. The simplest case with G = GL(k, ℂ), G′ = GL(n, ℂ), M = {x ∈ ℂ^{n×k} : x of maximum rank} was considered in [TT5]. In this article three more cases are considered. They are more intricate and Witt’s theorems on quadratic forms and skew-symmetric bilinear forms play a crucial role in their resolution. The general case will be considered in a future publication.

2 A Duality Theorem for Commutants in D^{ω}(M)

Let E = ℂ^{n×k}, G = SO(k, ℂ), G′ = GL(n, ℂ). Then it is clear that G′ (resp. GL(k, ℂ)) is the maximum linear group acting on E by left (resp. right) multiplication. As a subgroup of GL(k, ℂ), G acts on E by right multiplication and leaves the nondegenerate symmetric bilinear form (x, y) → tr(xy^{t}), x, y ∈ ℂ^{n×k}, invariant. If S(E^{∗}) is the symmetric algebra of all polynomial functions on E then the action of G on E induces an action of G on S(E^{∗}), denoted by g · p, for g ∈ G, p ∈ S(E^{∗}). We say that p ∈ S(E^{∗}) is G-invariant if g · p = p, for all g ∈ G. The G-invariant polynomial functions form a subalgebra J(E^{∗}) of S(E^{∗}). If J_{+}(E^{∗}) is the subset of all elements in J(E^{∗}) without constant term we let J_{+}(E^{∗})S(E^{∗}) denote the ideal in S(E^{∗}) generated by J_{+}(E^{∗}). Recall ([We, Theorem 2.9A]) that J_{+}(E^{∗})S(E^{∗}) is generated by the n(n + 1)/2 algebraically independent polynomials

p_{ij}(x) = ∑_{s=1}^{k} x_{is}x_{js},  1 ≤ i ≤ j ≤ n, x ∈ E,  (2.1)

together with the (k×k) minors of the matrix x (which are 0 when k > n). If P is the null cone of the common zeros of polynomial functions in J_{+}(E^{∗})S(E^{∗}) then by the Hilbert Nullstellensatz the ideal in S(E^{∗}) of all polynomial functions which vanish on P is \sqrt{J_{+}(E^{∗})S(E^{∗})}. By [D-TT, Theorem 2.1] the ideal J_{+}(E^{∗})S(E^{∗}) is prime if and only if k > 2n, and the scheme P which is then equal to the set {x ∈ E : xx^{t} = 0} is a complete intersection, with one open dense orbit. Henceforth we assume that k > 2n. Let M = {x ∈ E : xx^{t} = 0, rank(x) = n} then obviously M is dense in P. Since (g′x)(g′x)^{t} = g′(xx^{t})(g′)^{t} it follows immediately that G′ is the maximum linear group acting on M by left multiplication. For γ ∈ GL(k, ℂ) and p ∈ S(E^{∗}) define R(γ)p by (R(γ)p)(x) = p(xγ), then clearly γ leaves M, and hence P, invariant if and only if R(γ)p_{ij} ∈ J_{+}(E^{∗})S(E^{∗}) for all 1 ≤ i ≤ j ≤ n. Obviously, R(γ)p_{ij} are quadratic polynomials, and since the p_{ij} form a basis for the quadratic polynomials in J_{+}(E^{∗})S(E^{∗}) we have

R(γ)p_{ij} = ∑_{r,s} C_{rs}^{ij}p_{rs},  1 ≤ r ≤ s ≤ n,  (2.2)

where C_{rs}^{ij} ∈ ℂ are constants depending on γ. For 1 ≤ i ≤ n, 1 ≤ t ≤ k let x(i, t) denote the element of E which has the (i, t)-entry equal to 1 and all other entries equal 0. Then an easy computation shows that

(R(γ)p_{ii})(x(i, t)) = ∑_{l=1}^{k} γ_{tl}^{2} = ∑_{r,s} C_{rs}^{ii}p_{rs}(x(i, t)) = C_{ii}^{ii}.

It follows that ∑_{l=1}^{k} γ_{tl}^{2} = C ∈ ℂ for all t, and i. Choose x of the form x(i, t) + x(i, t′) with t ≠ t′ then we obtain

(R(γ)p_{ii})(x(i, t) + x(i, t′)) = ∑_{l=1}^{k}(γ_{tl} + γ_{t′l})^{2}

= ∑_{r,s} C_{rs}^{ii}p_{rs}(x(i, t) + x(i, t′))

= 2C_{ii}^{ii} = 2C.

Thus

∑_{l=1}^{k}(γ_{tl} + γ_{t′l})^{2} = ∑_{l=1}^{k} γ_{tl}^{2} + ∑_{l=1}^{k} γ_{t′l}^{2} + 2 ∑_{l=1}^{k} γ_{tl}γ′_{t′l} = 2C

= C + C + 2 ∑_{l=1}^{k} γ_{tl}γ_{t′l}.

It follows that we have the system of equations

∑_{l=1}^{k} γ_{tl}^{2} = C,  ∑_{l=1}^{k} γ_{tl}γ_{t′l} = 0  for all t, t′ = 1, …, k, t ≠ t′,  (2.3)

or equivalently, γ^{t}γ = CI_{k}. Since (det(γ))^{2} = C^{k} and γ is invertible it follows that C ≠ 0. Let λ be a fixed square root of C and set g = \frac{1}{λ}γ, then g^{t}g = I_{k}, or g ∈ O(k, ℂ). It follows that the largest group acting on M by right multiplication is ℂ^{∗}O(k, ℂ) = {λg : λ ∈ ℂ, g ∈ O(k, ℂ)}, and G is a maximal connected linear group acting on M by right multiplication. By Witt’s theorem on symmetric bilinear forms (see, e.g., [Ar] and [TT1, Lemma 2.8]) G acts analytically and transitively on M. More precisely, if x_{0} ∈ M then M is the G-orbit of x_{0}, and if G_{x_{0}} is the stability subgroup at x_{0}, then it is easy to verify that G_{x_{0}} is isomorphic to SO(k − n, ℂ). Moreover, the map G_{x_{0}}g → x_{0}g is an analytic diffeomorphism of G_{x_{0}}\G onto M (see, e.g., [Va, Theorem 2.9.4]). Thus M is an analytic manifold of complex dimension nk − n(n + 1)/2 (this also follows from [TT1, Lemma 2.9] and the implicit function theorem for analytic functions [Hö, Theorem 2.1.2]). Now let us show that G′ acts freely on M, i.e., the stability subgroup G′_{x} is {1_{G′}} at every x ∈ M. Indeed, if x ∈ M then by the assumption rank(x) = n there exist n columns x_{i_{1}} ⋯ x_{i_{n}}, i_{1} < ⋯ < i_{n}, of x such that the n × n matrix x_{n} formed by them is invertible. So g′x = x implies that g′x_{n} = x_{n} or g′ = x_{n}x_{n}^{−1} = 1_{G′}. Now let us recall the definition of differential operators on a complex manifold M of dimension m (see, e.g., [He, Chapter 10]). If (φ, U) is a local chart on M with φ(p) = (x_{1}(p), …, x_{m}(p)) ∈ ℂ^{m}, p ∈ U, and f ∈ C^{∞}(M), set f^{∗} = f ∘ φ^{−1}: φ(U) ⊂ ℂ^{m} → ℂ. Set ∂_{i} = ∂/∂x_{i} (1 ≤ i ≤ m) and if α = (α_{1}, …, α_{m}) is an m-tuple of indices α_{i} ≥ 0 we put D^{(α)} = ∂_{1}^{α_{1}} ⋯ ∂_{m}^{α_{m}}. Then a linear transformation D: C_{c}^{∞}(M) → C_{c}^{∞}(M) is called a differential operator on M if the following condition is satisfied: For each p ∈ M and each chart (φ, U), p ∈ U, there exists a locally finite set of functions h_{(α)} ∈ C^{∞}(U) such that for each f ∈ C_{c}^{∞}(M) with support contained in U,

\{ \begin{matrix}[Df] (p) = ∑_{(α)}h_{α} [D^{(α)}f^{∗}] (φ(p)) & if p ∈ U, \\ [Df] (p) = 0, & if p ∉ U. \end{matrix}  (2.4)

If M is a complex analytic manifold then a differential operator D is called a holomorphic or complex analytic differential operator if the functions h_{(α)} in Eq. (2.4) are holomorphic (or complex analytic). By Hilbert’s fifth problem G (resp. G′) can be equipped with an analytic structure (see, e.g., [M-Z]) so that they act analytically on M. Let D^{ω}(M) denote the algebra of (complex) analytic differential operators on M. Now consider a global G-transformation group on an analytic manifold M (see, e.g., [Pa] or [Va, 2.16]). Let φ: G × M → M ((g, x) → g · x, g ∈ G, x ∈ M) denote the global action of G on M. For x ∈ M, f ∈ C^{∞}(M) we define (Φ(g)f) (x) = f(g^{−1} · x). Let 𝔤 denote the Lie algebra of G and U(𝔤) the universal enveloping algebra of G. Then for X ∈ 𝔤 and x ∈ M we define

dΦ(X)_{x}(f) := (\frac{d}{dt}f(exp(−tX) · x))_{t=0}  (2.5)

for all f defined and C^{∞} in a neighborhood of x. The map X → dϕ(X) is a homomorphism of 𝔤 into the Lie algebra of analytic vector fields on M. Therefore it extends to a homomorphism a → d̃ϕ(a), a ∈ U(𝔤), of U(𝔤) into the algebra D^{ω}(M) of analytic differential operators on M (see [Va, Lemma 2.16.1]), where if a = X_{1} ⋯ X_{r} (X_{i} ∈ 𝔤) then

d̃ϕ(a)_{x}(f) = (\frac{∂}{∂t_{1}} ⋯ \frac{∂}{∂t}f(exp(−t_{r}X_{r}) ⋯ exp(−t_{1}X_{1}) · x)|_{0} ,  (2.6)

where the suffix 0 indicates that the derivatives are taken when t_{1} = ⋯ = t_{r} = 0. For our problem we consider the cases when Φ(g′) = L(g′) and Φ(g) = R(g), where (L(g′)f) (x) = f ((g′)^{−1}x) and (R(g)f)(x) = f(xg) for g′ ∈ G′, g ∈ G, and x ∈ M. Let Ũ(𝔤′) and Ũ(𝔤) denote the images of U(𝔤′) and U(𝔤) under the maps d̃L and d̃R, respectively.

Definition 2.1 A differential operator D of D^{ω}(M) is said to be right (resp. left)-invariant if D(R(g)f) = R(g)(Df) (resp. D(L(g′)f) = L(g′)(Df)) for all g ∈ G (resp. g′ ∈ G′), and for all f ∈ C^{∞}(M).

Theorem 2.2 Let D_{l}^{ω}(M) (resp. D_{r}^{ω}(M)) denote the subalgebra of D^{ω}(M) of all left (resp. right)-invariant analytic differential operators on M. Then (i) D_{l}^{ω}(M) = Ũ(𝔤) and D_{r}^{ω}(M) = Ũ(𝔤′),

(ii) the commutant of D_{l}^{ω}(M) in D^{ω}(M) is D_{r}^{ω}(M), and vice-versa. Moreover, the centres of D_{l}^{ω}(M) and D_{r}^{ω}(M) coincide with the subalgebra D_{l}^{ω}(M) ∩D_{r}^{ω}(M).

Proof. (i) Let X ∈ 𝔤′, g ∈ G, x ∈ M and f ∈ C^{∞}(M). Then

dL(X)(R(g)f)(x) = \frac{d}{dt} ((R(g)f)(exp(−tX)x))_{t=0}

= \frac{d}{dt}(f(exp(−tX)xg))_{t=0},

while

R(g)(dL(X)f)(x) = (dL(X)f)(xg)

= \frac{d}{dt}(f(exp(−tX)xg))_{t=0}.

Thus any vector field X̃ = dL(X) ∈ Ũ(𝔤′) is right-invariant, and it follows immediately that Ũ(𝔤′) ⊂ D_{r}^{ω}(M). Similarly we have Ũ(𝔤) ⊂ D_{l}^{ω}(M). Let us show that D_{r}^{ω}(M) ⊂ Ũ(𝔤′) and D_{l}^{ω}(M) ⊂ Ũ(𝔤). Let L denote the Lie algebra of all right-invariant analytic vector fields on M. Then L is an involutive analytic system (see [Va, p. 25] for the definition), i.e., if U is an open subset of M and X, Y are right-invariant vector fields on M then [X, Y] is (obviously) right-invariant. Then the Global Frobenius Theorem (see, e.g., [Va, Theorem 1.3.6]) implies that: given any point of M, there is one and exactly one maximal integral manifold S of L containing that point, i.e., S is a connected analytic submanifold of M and for each y ∈ S, L_{y} is the tangent space T_{y}(S). In fact since L is an infinitesimal group [Pa, Theorem IV, p. 98] implies that S is the image of a unique connected Lie transformation group H of M. Since G′ is the largest linear group acting on M by left multiplication and dL(𝔤′) ⊂ L it follows that G′ ⊂ H, and hence, G′ = H. It follows that if {X_{1}, …, X_{n^{2}}} is a basis of 𝔤′ then {X̃_{1}, …, X̃_{n^{2}}}, where X̃_{i} = dL(X_{i}), 1 ≤ i ≤ n^{2}, is a basis for right-invariant analytic vector fields on M. Therefore if D ∈ D_{r}^{ω}(M) then we can find a unique set of locally finite functions h_{(α)} ∈ C^{ω}(M) such that

D = ∑_{(α)} h_{(α)}X̃^{(α)}.  (2.7)

Since the X̃^{(α)} are right-invariant, we have

D = D^{r_{g}} = ∑_{(α)} h_{(α)}^{r_{g}}X̃^{(α)}  (g ∈ G).  (2.8)

The relations (2.7) and (2.8) imply that all the h_{(α)} are right-invariant, and since G acts transitively on M, they must be all constant. Thus D ∈ Ũ(𝔤′) for all D ∈ D_{r}^{ω}(M), and hence, D_{r}^{ω}(M) ⊂ Ũ(𝔤′). To show that D_{l}^{ω}(M) ⊂ Ũ(𝔤) we need the following

Lemma 2.3 For each x ∈ M let G′x = {g′x : g′ ∈ G′} denote the orbit of x. Let X = M/G′ be the set of all orbits G′x, x ∈ M, and define π: M → X by assigning to each x ∈ M its orbit G′x. Then (M, X , π, G′) is a principal G′-bundle.

Proof. Define γ: M×G′ → M×M by γ(x, g′) = (x, g′x) and Γ = γ(M×G′) = {(x, g′x) : x ∈ M, g ∈ G}. Then since G′ acts freely on M, γ is injective. Now suppose that lim_{n→∞} x_{n} = x, and lim_{n→∞} g′_{n}x_{n} = y for x_{n}, x, y ∈ M, g′_{n} ∈ G. The same argument used in the proof that G′ acts freely on M implies that there exists a submatrix s[x] of x such that s[x] ∈ G′. If s[x_{n}] denotes the corresponding submatrix of x_{n} then clearly lim_{n→∞} s[x_{n}] = s[x]. So for n sufficiently large we may assume that s[x_{n}] ∈ G′. Then clearly lim_{n→∞} s^{−1}[x_{n}] = s^{−1}[x] and lim_{n→∞} g′_{n}s[x_{n}] = s[y]. By the continuity of the action of G′ on M we have s[y] ∈ G′. Write g′_{n} = (g′_{n}s[x_{n}])s^{−1}[x_{n}] for sufficiently large n then lim_{n→∞} g′_{n} = s[y]s^{−1}[x]. Set g′ = s[y]s^{−1}[x], then g′ ∈ G′, lim_{n→∞} g′_{n} = g′, and lim_{n→∞} g′_{n}x_{n} = g′x = y. Thus Γ is closed in M × M, and γ is a homeomorphism of M × G onto Γ. Now all the hypotheses of [Va, Theorem 2.9.10] are met, and we can conclude that there exists an analytic structure on X such that π is an analytic immersion (i.e., (dπ)_{x} is injective for all x ∈ M). Moreover for each p ∈ X we can select an open subset Y of X containing p and an analytic diffeomorphism ξ_{Y} = ξ of G′ × Y onto π^{−1}(Y), such that

ξ(h′g′, y) = h′ξ(g′, y) (g′, h′ ∈ G, y ∈ Y).

That is, in other words, (M, X , π, G′) is a principal G′-bundle. Now let us finish the proof of part (i) of the theorem. Since G acts analytically and transitively on the analytic manifold M [Va, Lemma 2.9.2] implies that for each x ∈ M the map r: g → xg (g ∈ G) is an analytic submersion of G onto M (i.e., (dr)_{g} is surjective for all g ∈ G). It follows that if {Y_{1}, …, Y_{d}} is a basis of 𝔤 then there exists a basis for analytic vector fields of M of the form {Ỹ_{1}, …, Ỹ_{m}}, where m = dim(M), and each Ỹ_{i} = dR(Y_{j}) for some j, 1 ≤ j ≤ d. It follows that every D ∈ D_{l}^{ω}(M) can be expressed as

D = ∑_{(α)} k_{(α)}Ỹ^{(α)},  (2.9)

where {k_{(α)}} is a set of locally finite analytic functions. Since the Ỹ^{(α)} are left-invariant, we have

D = D^{l_{g′}} = ∑_{(α)} k_{(α)}^{l_{g′}}Ỹ^{(α)}  (g′ ∈ G′).  (2.10)

The relations (2.9) and (2.10) imply that all the k_{(α)} are left-invariant. By Lemma 2.3 a basic open set M is diffeomorphic to G′ × Y where Y is an open subset of X. A typical point in that basic open set is, for example, of the form x = (∗g′∗) ∈ M. A function k_{(α)} that is left-invariant will be independent of the n^{2} variables in the block containing g′, and since we can let g′ occupy any block in the matrix x it follows that k_{(α)} must be constant. Hence D ∈ Ũ(𝔤), and the proof of part (i) is completed. (ii) The proof of part (ii) depends on the following

Lemma 2.4 Let D ∈ D^{ω}(M) then the following statements hold.

(i) [dL(X), D] = 0 for all X ∈ 𝔤′ if and only if D(L(g′) = L(g′)D for all g′ ∈ G.

(ii) [dR(Y), D] = 0 for all Y ∈ 𝔤 if and only if D(R(g)) = R(g)D for all g ∈ G.

Proof. Since both G′ and G are connected the two statements are similar, so we will only prove (i). To prove (i) we first consider g′ = g′(t) = exp tX, X ∈ 𝔤′; then we have

L(g′)DL((g′)^{−1}) = (exp(dL(tX))D.

It follows that

L(g′)D = DL′(g) ⟺ [L(X), D] = 0.

Since G′ is connected, G′ is generated by the image of the exponential map (cf. [Go, Cor. I, p. 6.9]), i.e., an arbitrary element g′ of G′ can be expressed in the form g′ = exp(X_{1}) exp(X_{2}) ⋯ exp(X_{r}), X_{i} ∈ 𝔤′, it follows from [Na, Prop. 2.10.10] that

[dL(X), D] = 0, ∀ X ∈ 𝔤′ ⟺ L(g′)D = DL(g′), ∀ g′ ∈ G′.

Now part (i) of the theorem and Lemma 2.4 imply that the commutant of D_{l}^{ω}(M) in D^{ω}(M) is D_{r}^{ω}(M), and vice-versa. Finally, by definition the centre of D_{l}^{ω}(M) is the subalgebra of elements of D_{l}^{ω}(M) which commute with all elements of D_{l}^{ω}(M). So obviously the centre of D_{l}^{ω}(M) and similarly the centre of D_{r}^{ω}(M) coincide with D_{l}^{ω}(M) ∩ D_{r}^{ω}(M). Now let us consider the Lie transformation group G′ ×G acting on the analytic manifold M, where G′ = GL(n, ℂ), G = Sp(2k, ℂ), M = {x ∈ ℂ^{n×2k} : xs_{k}x^{t} = 0, rank(x) = n}, k ≥ n, and

s_{k} = [ \begin{matrix}0 & −I_{k} \\ I_{k} & 0 \end{matrix} ]

with I_{k} denoting the identity matrix of order k.

Recall that Sp(2k, ℂ) is the group of all complex 2k × 2k matrices g satisfying gs_{k}g^{t} = s_{k}. Then by Witt’s theorem on skew-symmetric bilinear form (see, e.g.,

[Ar] and [TT2, Lemma 1.7]) it follows that G acts analytically and transitively by right multiplication on M. Obviously G′ acts freely by left multiplication on M, and G and G′ are both connected. Thus we have

Theorem 2.5 For k ≥ n let G = Sp(2k, ℂ), G′ = GL(n, ℂ), and M = {x ∈ ℂ^{n×2k} : xs_{k}x^{t} = 0, rank(x) = n}. Then Theorem 2.2 holds for this pair of Lie transformation groups acting on M.

Finally, let p, q, and k be positive integers such that k ≥ max(p, q) and consider G′ = GL(p, ℂ) × GL(q, ℂ), G = {(g, g^{✓}) : g ∈ GL(k, ℂ), g^{✓} = (g^{−1})^{t}} ≈ GL(k, ℂ), and

M = \{[ \begin{matrix}x_{1} \\ x_{2} \end{matrix} ] ∈ ℂ^{(p+q)×k} : x_{1} ∈ ℂ^{p×k}, x_{2} ∈ ℂ^{q×k},

x_{1}x_{2}^{t} = 0, rank(x_{1}) = p, rank(x_{2}) = q} .

Then by Witt’s theorem on quadratic forms, [TT3, Lemma 1.1] and [TT4, Theorem 5.1], it follows that G acts analytically and transitively on M via the action

([ \begin{matrix}x_{1} \\ x_{2} \end{matrix} ] , g) ⟶ [ \begin{matrix}x_{1}g \\ x_{2}g^{✓} \end{matrix} ] .

Obviously G′ acts on M freely via the action

((g′_{1}, g′_{2}) , [ \begin{matrix}x_{1} \\ x_{2} \end{matrix} ]) ⟶ [ \begin{matrix}g′_{1}x_{1} \\ g′_{2}x_{2} \end{matrix} ] , g′_{1} ∈ GL(p, ℂ), g′_{2} ∈ GL(q, ℂ).

Moreover, both G and G′ are connected. Thus we have

Theorem 2.6 Theorem 2.2 holds for the pair of Lie transformation groups G, G′ acting on the analytic manifold M described above.

3 Conclusion

In [TT5] we used the duality theorem for commutants in D^{ω}(M) with G = GL(k, ℂ), G′ = GL(n, ℂ), M = {x ∈ ℂ^{n×k} : x of maximum rank} to find the Casimir invariants of the infinite-dimensional group GL(∞, ℂ). In turn, a set of generators of these Casimir invariants determine the irreducible unitary representations of the group U(∞). We hope that Theorems 2.2, 2.5, and 2.6 will allow us to find the Casimir invariants of some other infinite-dimensional groups. Acknowledgements: The author was supported by a Carver Research Initiative Grant during the preparation of this paper. He also wishes to thank Ms. Cymie Wehr for her patience in preparing this manuscript.

References

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[TT1] T. Ton-That, Lie group representations and harmonic polynomials of a matrix variable, Trans. Amer. Math. Soc. 216 (1976), 1–46.

[TT2] T. Ton-That, Symplectic Stiefel harmonics and holomorphic representations of symplectic groups, Trans. Amer. Math. Soc. 232 (1977), 265– 277.

[TT3] T. Ton-That, Sur la décomposition des produits tensoriels des représentations irréductibles de GL(k, C), J. Math. Pures Appl. (9) 56 (1977), 251–261.

[TT4] T. Ton-That, Dual representations and invariant theory, Representation Theory and Harmonic Analysis (T. Ton-That, et al., eds.), Contemp. Math., vol. 191, American Mathematical Society, Providence, 1995, pp. 205–221.

[TT5] T. Ton-That, Poincaré-Birkhoff-Witt theorems and generalized Casimir invariants for some infinite-dimensional Lie groups: I, J. Phys. A 32 (1999), 5975–5991.

[TT-T] T. Ton-That and T.D. Tran, Poincaré’s proof of the so-called Birkhoff-Witt theorem, Rev. Histoire Math. 5 (1999), 249–284.

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Department of Mathematics The University of Iowa Iowa City, IA 52242-1419 USA tonthat@math.uiowa.edu