Journal of Lie Theory Volume 10 (2000) 463–489 C ◯2000 Heldermann Verlag

Capelli elements for the orthogonal Lie algebras

Minoru Itoh^{∗}

Communicated by E. Zelmanov

Abstract. For the central elements in the universal enveloping algebra of the orthogonal Lie algebra given by R. Howe and T. Umeda, Newton type and Cayley-Hamilton type identities are given, and their images under the Harish-Chandra homomorphism are determined. These results show that these central elements can be regarded as the counterparts of the Capelli elements. The calculation of the images under the Harish-Chandra homomorphism clearly reveals the fact that these central elements coincide with those given with the Sklyanin determinant.

Introduction

The Capelli elements are well-known central elements in the universal enveloping algebra of the Lie algebra 𝔤𝔩_{n}, which played an important role in the classical invariant theory. They are defined as sums of determinants of the generators of 𝔤𝔩_{n}. Also in the universal enveloping algebra U(𝔬_{n}) of the orthogonal Lie algebra 𝔬_{n}, a set of central elements is given with determinant in a similar way as the Capelli elements (the Appendix of [6]). Their properties have not been studied so minutely as the Capelli elements so far. In this paper, we give the following results for these central elements: (i) Newton type and Cayley-Hamilton type identities, and (ii) the calculation of the images under the Harish-Chandra homomorphism. These results show that these central elements have strong similarities to the Capelli elements not only in their form but properties. From (ii), we also clearly see that they are essentially identical to the central elements given with the Sklyanin determinant in [10].

We work over a fixed field 𝕂 of characteristic 0. We realize the orthogonal Lie algebra 𝔬_{n} as the Lie algebra consisting of the alternating matrices. Put A_{ij} = E_{ij} − E_{ji}, and arrange a matrix A = (A_{ij})_{1≤i,j≤n} ∈ Mat(n, U(𝔬_{n})) from these generators of 𝔬_{n}. Furthermore, we consider the submatrix A_{I} = (A_{ij})_{i,j∈I} for an index set I ⊆ {1, 2, …, n}. The sum of determinants of submatrices

C_{2k} = ∑_{|I|=2k} det(A_{I} + diag(k, k − 1, …, −k + 1))

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag is known to be invariant under the adjoint action of the orthogonal Lie group O_{n}, hence central in U(𝔬_{n}) ([6], [9]). More precisely, the set {C_{2k} | 0 ≤ 2k ≤ n} generates the subalgebra U(𝔬_{n})^{O_{n}} consisting of the invariants under the adjoint action of O_{n}. This definition is very similar to that of the Capelli elements. Here we define the determinant of a matrix Φ = (Φ_{ij})_{1≤i,j≤n} whose entries are not necessarily commutative by the following alternating sum, which is often called the “column-determinant”:

∗ Partially supported by JSPS Research Fellowships for Young Scientists

(1)  det(Φ) = ∑_{σ∈𝔖_{n}} sign(σ)Φ_{σ(1)1}Φ_{σ(2)2} ⋯ Φ_{σ(n)n}.

Another set of generators of U(𝔬_{n})^{O_{n}} is given with the trace. Take the trace of the power of the matrix A:

tr(A^{r}) = ∑_{i_{1},i_{2},... ,i_{r}} A_{i_{1}i_{2}}A_{i_{2}i_{3}} ⋯ A_{i_{r}i_{1}}.

This tr(A^{r}) is known to be invariant under the adjoint action of O_{n} ([3]). Furthermore the set {tr(A^{r}) | 0 ≤ r ≤ n} generates U(𝔬_{n})^{O_{n}}.

The first result of this paper is the following relation between these two sets of generators of U(𝔬_{n})^{O_{n}}:

Theorem A. (Newton’s formula for 𝔬_{n}) The following equality holds as U(𝔬_{n})^{O_{n}} -coefficient formal power series in λ:

\frac{N(λ)}{D(λ)} = ∑_{r=0}^{∞} λ^{−1−r} tr ((A − (\frac{n}{2} − 1) I)^{r}) .

Here D(λ) and N(λ) are defined by the following polynomials in λ with coefficients C_{2k}:

(2)  \begin{aligned}D(λ) & = ∑_{k=0}^{m} C_{2m−2k}λ\overline{k}λ\underline{k}, \\ N(λ) & = ∑_{k=0}^{m−1}(2k + 2)C_{2m−2k−2}λ\overline{k}λ\underline{k+1} \end{aligned}

for n = 2m, and

(3)  \begin{aligned}D(λ) & = ∑_{k=0}^{m} C_{2m−2k}(λ − 1/2)\overline{k+1}(λ − 1/2)\underline{k}, \\ N(λ) & = ∑_{k=0}^{m}(2k + 1)C_{2m−2k}(λ − 1/2)\overline{k}(λ − 1/2)\underline{k} \end{aligned}

for n = 2m + 1. Here the symbols λ\overline{k} and λ\underline{k} mean respectively the rising and falling factorial functions:

λ\overline{k} = λ(λ + 1) ⋯ (λ + k − 1), λ\underline{k} = λ(λ − 1) ⋯ (λ − k + 1).

Theorem A is a counterpart of Newton’s formula for symmetric functions and its analogues for 𝔤𝔩_{n} given in [21] (see also [15], [14]). In fact, considering the case that the matrix in question is replaced by a matrix with commutative entries, we can regard our C_{2k} ’s and tr(A^{r})’s as analogues of the elementary symmetric functions and the power sum symmetric functions in the eigenvalues respectively. In classical Newton’s formula, the denominator on the left-hand side is the characteristic polynomial of a matrix, and the numerator is its differentiation. In the 𝔤𝔩_{n} case, these are given respectively as a Capelli type determinant with parameter λ, and its backward difference. In our case of 𝔬_{n}, the relation between N(λ) and D(λ) is a bit more complicated:

N(λ) = \frac{λ}{λ − 1/2}(D(λ) − D(λ − 1))

for n = 2m, and

N(λ) = D̃(λ) − D̃(λ − 1)

for n = 2m + 1 with D̃(λ) = \frac{λ}{λ−1/2}D(λ). More explicit relations between C_{2k} ’s and tr(A^{r})’s are given in Theorem 5.2 and Corollary 5.3.

In the case of 𝔤𝔩_{n}, an analogue of the Cayley-Hamilton theorem is known ([10], [21], [17]). Also in our case of 𝔬_{n}, we have a similar result:

Theorem B. (The Cayley-Hamilton theorem for 𝔬_{n}) The following equality holds in Mat(n, U(𝔬_{n})):

D (A − (\frac{n}{2} − 1) I) = 0.

These results imply that D(λ) is the “characteristic polynomial” of the matrix A. It is interesting that D(λ) is expressed as a “symmetrized determinant” of the matrix λI − A (Proposition 2.2).

The proofs of Theorems A and B are similar to those in the 𝔤𝔩_{n} case given in [21]. Namely our main subject is to construct the “cofactor matrix” Δ(λ), which satisfies the following relation (Proposition 4.2):

(4)  Δ(λ) (A − (\frac{n}{2} − 1) I − λI) = D(λ)I.

Theorems A and B are immediately shown from this relation (§5).

The cofactor matrix Δ(λ) is explicitly described with the Pfaffian by progressing the exterior calculus in [9]. This calculation is based on the following two results: the first is the expression

(5)  C_{2k} = ∑_{|I|=2k} Pf(A_{I})^{2}

given in [9], and the second is an explicit description of the cofactor matrix for Pf(A) (§3). Such a computation utilizing the Pfaffian is one of the key points of this paper.

Theorems A and B can be extended to the orthogonal Lie algebra realized with any non-degenerate symmetric matrix (§6). In fact, we can naturally define the counterparts of A and D(λ), which satisfy Theorems A and B. This generalization is based on the fact that our C_{2k} ’s and D(λ) are written with the “symmetrized determinant”, which is invariant under the conjugation (§§1 and 2).

For the characteristic polynomial D^{†}(λ) for the split realization of the orthogonal Lie algebra, we have the following theorem as another application of the cofactor matrix:

Theorem C. The image of D^{†}(λ) under the Harish-Chandra homomorphism \overline{γ} is given by

\overline{γ}(D^{†}(λ)) = \{ \begin{matrix}(λ^{2} − F_{11}^{2}) ⋯ (λ^{2} − F_{mm}^{2}), & (n = 2m), \\ (λ − \frac{1}{2}) (λ^{2} − F_{11}^{2}) ⋯ (λ^{2} − F_{mm}^{2}), & (n = 2m + 1). \end{matrix}

Here, {F_{11}, …, F_{mm}} is the canonical basis of the Cartan subalgebra.

From these Theorems A, B and C, we clearly see that our D^{†}(λ) is identical to a central element C(λ) given with the Sklyanin determinant in [10]. In fact, the counterparts of these theorems in terms of C(λ) are given in the framework of the twisted Yangians (see [14] and [10]). It is interesting that such similar results are deduced from two different motives.

It should be noted that this equality D^{†}(λ) = C(λ) is also seen from the expression (5) of C_{2k} above and the recent result in [13]. Once we have this identity, our theorems can be deduced from the known results in [14] and [10]. However, our method of cofactors is quite elementary and produces these theorems directly from a unified point of view. We also remark that the techneque in this paper can be applied to obtain a new Cayley-Hamilton type formula for differential operators related with the Skew Capelli elements in U(𝔤𝔩_{n}) [8].

The author is deeply grateful to Professor Tôru Umeda for his valuable advice, guidance and encouragement. The author also thanks Professor Hiroyuki Ochiai and Professor Masato Wakayama for fruitful discussions.

1. Expressions of Pfaffians and determinants with the exterior algebra

We start with the expressions of the Pfaffians and the determinants in the exterior calculus, which are valid for matrices with non-commutative entries. From these expressions, some fundamental properties of the Pfaffians and the determinants are naturally deduced. Most of contents of the present and the next sections are discussed in [9].

Consider an n × n matrix Φ whose entries are elements of an associative algebra A. The exterior algebra Λ_{n} is the associative algebra generated by the n elements e_{1}, e_{2}, …, e_{n} subject to the relations e_{i}e_{j} + e_{j}e_{i} = 0. Hereafter, we calculate in the extended algebra Λ_{n} ⊗ A, in which the two subalgebras Λ_{n} and A commute with each other. The elements e_{1}, e_{2}, …, e_{n} are thus considered to be formal (anti-commuting) variables to make “generating functions” with coefficients in A.

We often use the symbol of the divided power: x^{(k)} = \frac{1}{k!}x^{k}. Note that the binomial expansion is simply expressed with this symbol: (x + y)^{(n)} = ∑_{k=0}^{n} x^{(k)}y^{(n−k)} .

1.1. For an alternating matrix Φ = (Φ_{ij})_{1≤i,j≤2m} of size n = 2m, the Pfaffian Pf(Φ) is defined by

Pf(Φ) = \frac{1}{2^{m}m!} ∑_{σ∈𝔖_{2m}} sign(σ)Φ_{σ(1)σ(2)}Φ_{σ(3)σ(4)} ⋯ Φ_{σ(2m−1)σ(2m)}.

To express this in the exterior calculus, we form an element θ_{Φ} in Λ_{n} ⊗ A as

θ_{Φ} = \frac{1}{2}eΦ^{t}e = \frac{1}{2} ∑_{i,j=1}^{n} e_{i}e_{j}Φ_{ij}

with e = (e_{1}, …, e_{n}). The Pfaffian Pf(Φ) is expressed with this θ_{Φ} as follows:

(1.1)  e_{1} ⋯ e_{n} Pf(Φ) = θ_{Φ}^{(m)}.

This relation is easy to see from the definition of the Pfaffian.

1.2. The determinant defined by (1) is often called “column-determinant” compared with the name “row-determinant” ([9], [19]). This can be expressed with the element ψ_{j} = ∑_{i=1}^{n} e_{i}Φ_{ij} as follows:

(1.2)  e_{1} ⋯ e_{n} det(Φ) = ψ_{1}ψ_{2} ⋯ ψ_{n}.

We introduce another kind of determinant, which should be called the “symmetrized determinant” or the “double-determinant”:

Det(Φ) = \frac{1}{n!} ∑_{(σ,σ′)∈𝔖_{n}×𝔖_{n}}sign(σ) sign(σ′)Φ_{σ(1)σ′(1)}Φ_{σ(2)σ′(2)} ⋯ Φ_{σ(n)σ′(n)}.

Furthermore consider the following quantity with scalar parameters u_{1}, …, u_{n}:

Det(Φ ; u_{1}, u_{2}, …, u_{n})

= \frac{1}{n!} ∑_{(σ,σ′)∈𝔖_{n}×𝔖_{n}}sign(σ) sign(σ′)Φ_{σ(1)σ′(1)}(u_{1}) ⋯ Φ_{σ(n)σ′(n)}(u_{n}),

where Φ_{ij}(u) = Φ_{ij} + δ_{ij}u. To express this new determinant in terms of the exterior calculus, we must double the anti-commutative variables. Let Λ_{2n} be the exterior algebra generated by the anti-commutative variables e_{i}, e′_{i} (i = 1, …, n). Form the elements Ξ_{Φ} and τ in Λ_{2n} ⊗ A as

Ξ_{Φ} = eΦ^{t}e′ = ∑_{i,j=1}^{n} e_{i}e′_{j}Φ_{ij}, τ = e^{t}e′ = ∑_{i=1}^{n} e_{i}e′_{i}

with e′ = (e′_{1}, …, e′_{n}). Then we can express the symmetrized determinant as

(1.3)  e_{1}e′_{1} ⋯ e_{n}e′_{n} Det(Φ) = Ξ_{Φ}^{(n)},

(1.4)

e_{1}e′_{1} ⋯ e_{n}e′_{n} Det(Φ; u_{1}, u_{2}, …, u_{n}) = \frac{1}{n!}(Ξ_{Φ} + u_{1}τ)(Ξ_{Φ} + u_{2}τ) ⋯ (Ξ_{Φ} + u_{n}τ).

Since the factors (Ξ_{Φ} + u_{i}τ)’s are commutative, we see that Det(Φ ; u_{1}, …, u_{n}) does not depend on the order of the parameters. Note that det(Φ) = Det(Φ), when the entries of Φ are commutative.

Remark. The expressions of the Pfaffians and the determinants above are generalized to the following expressions of the subpfaffians and the principal minors. For an alternating matrix Ψ and an n × n matrix Φ, we have

(1.5)  ∑_{|I|=2k} e_{i_{1}} ⋯ e_{i_{2k}} Pf(Ψ_{I}) = θ_{Ψ}^{(k)}

with I = {i_{1}, …, i_{2k}} , and

\begin{matrix}e_{1}e′_{1} ⋯ e_{n}e′_{n} ∑_{|I|=ν} Det(Φ_{I} ; u_{1}, u_{2}, …, u_{ν}) \\ = \frac{1}{ν!}(Ξ_{Φ} + u_{1}τ)(Ξ_{Φ} + u_{2}τ) ⋯ (Ξ_{Φ} + u_{ν}τ)τ^{(n−ν)}. \end{matrix} (1.6)

1.3. The Pfaffians and the symmetrized determinant have the following invariances under the actions of GL_{n} = GL_{n}(𝕂):

Proposition 1.1. The following equality holds for g ∈ GL_{n}:

Pf(gΦ^{t}g) = det(g) Pf(Φ).

Proposition 1.2. The symmetrized determinant is invariant under the conjugation by g ∈ GL_{n}:

Det(gΦg^{−1} ; u_{1}, …, u_{n}) = Det(Φ ; u_{1}, …, u_{n}).

More generally we have

∑_{|I|=ν} Det((gΦg^{−1})_{I} ; u_{1}, …, u_{ν}) = ∑_{|I|=ν} Det(Φ_{I} ; u_{1}, …, u_{ν}).

Proposition 1.3. The following equality holds:

Det(^{t}Φ ; u_{1}, …, u_{n}) = Det(Φ ; u_{1}, …, u_{n}).

These propositions are naturally deduced from the expressions of the Pfaffian and the determinant in the exterior calculus by using the invariance of the exterior algebra given in Lemma 1.4 below.

Consider the natural action of g ∈ GL_{n} on the vector space 𝕂^{n} spanned by the anti-commutative variables e_{1}, …, e_{n}. This action is naturally extended to an automorphism g_{∗} of the exterior algebra Λ_{n}, and Λ_{n} ⊗ A. Similarly, the action of α ∈ GL_{2n} on the vector space 𝕂^{2n} spanned by {e_{i}, e′_{i} | 1 ≤ i ≤ n} is extended to an automorphism α_{∗} of the exterior algebra Λ_{2n} and Λ_{2n} ⊗ A. Note that the exterior algebras Λ_{n} and Λ_{2n} have standard graded structures: Λ_{n} = ⨁_{k=0}^{n} Λ_{n}^{(k)} , Λ_{2n} = ⨁_{k=0}^{2n} Λ_{2n}^{(k)} . The following is an elementary fact for the exterior algebras:

Lemma 1.4.

(i) For φ ∈ Λ_{n}^{(n)} ⊗ A and g ∈ GL_{n}, the equality g_{∗}(φ) = det(g)φ holds. (ii) For φ ∈ Λ_{2n}^{(2n)} ⊗ A and α ∈ GL_{2n}, the equality α_{∗}(φ) = det(α)φ holds.

Proof of Proposition 1.1. By definition, the following relation holds:

θ_{gΦ^{t}g} = egΦ^{t}g^{t}e = (eg)Φ^{t}(eg) = g_{∗}eΦ^{t}e = g_{∗}θ_{Φ}.

In particular, since g_{∗} is an automorphism, we have

θ_{gΦ^{t}g}^{(m)} = g_{∗}θ_{Φ}^{(m)} = det(g)θ_{Φ}^{(m)}.

Here, the second equality is seen from Lemma 1.4, because θ_{Φ}^{(m)} is of top-degree in Λ_{n} ⊗ A. Then our formula is immediate from the expression (1.1) of the Pfaffian.

Propositions 1.2 and 1.3 are similarly proved by considering the following two types of elements of GL_{2n}. We put

α_{g} = diag(g,^{t}g^{−1}) = ( \begin{matrix}g & 0 \\ 0 & ^{t}g^{−1} \end{matrix} ) ∈ GL_{2n}

for g ∈ GL_{n}, and define an involution ι ∈ GL_{2n} by ι(e_{i}) = e′_{i} and ι(e′_{i}) = e_{i}. We often write the action of this ι on φ ∈ Λ_{2n} ⊗ A as ι_{∗}(φ) = φ′ simply. Note that det(α_{g}) = 1 and det(ι) = (−)^{n}. Proof of Proposition 1.2. A direct calculation leads us to the following relations:

Ξ_{gΦg^{−1}} = (α_{g})_{∗}Ξ_{Φ}, (α_{g})_{∗}τ = τ.

Since (α_{g})_{∗} is an automorphism, we have

\begin{aligned}(Ξ_{gΦg^{−1}} + u_{1}τ) ⋯ (Ξ_{gΦg^{−1}} + u_{ν}τ)τ^{n−ν} & = (α_{g})_{∗}(Ξ_{Φ} + u_{1}τ) ⋯ (Ξ_{Φ} + u_{ν}τ)τ^{n−ν} \\ & = (Ξ_{Φ} + u_{1}τ) ⋯ (Ξ_{Φ} + u_{ν}τ)τ^{n−ν}. \end{aligned}

Here, the second equality is seen from Lemma 1.4, because det(α_{g}) = 1. This implies our formula by (1.6). Proof of Proposition 1.3. The following relations are easily seen:

Ξ′_{Φ} = −Ξ_{^{t}Φ}, τ′ = −τ.

Noting these, we apply the involution ′ = ι_{∗} to φ = (Ξ_{Φ} + u_{1}τ) ⋯ (Ξ_{Φ} + u_{n}τ). Then we have on one hand φ′ = (−)^{n}(Ξ_{^{t}Φ} + u_{1}τ) ⋯ (Ξ_{^{t}Φ} + u_{n}τ) by a direct calculation. On the other hand, since φ is of top-degree, we see φ′ = det(ι)φ = (−)^{n}φ from Lemma 1.4. Thus our formula holds by the equality (1.4).

2. Determinants for the matrix A

In this section, we apply the consideration in the previous section to the matrix A, and give two expressions of D(λ) respectively with the symmetrized determinant and the Pfaffian.

2.1. The following relation holds between the column-determinant and the symmetrized determinant for the matrix A:

Proposition 2.1. ([9, Proposition 3.2]) The following equality holds:

det(A + diag(u, u − 1, …, u − n + 1)) = Det(A; u, u − 1, …, u − n + 1).

This is easily checked by using the following commutation relation [9, Lemma 3.1]:

(2.1)  ψ_{i}(u + 1)ψ_{j}(u) + ψ_{j}(u + 1)ψ_{i}(u) = −δ_{ij} · 2θ_{A}.

Here we put ψ_{j}(u) = ∑_{i=1}^{n} e_{i}A_{ij}(u) = ψ_{j} + ue_{j}.

Applying Propositions 1.3 and 2.1 to the submatrices A_{I}, we obtain the following expressions of C_{2k}, because A_{I} is alternating:

(2.2)  C_{2k} = ∑_{|I|=2k} det(A_{I} + diag(k, k − 1, …, −k + 1))

= ∑_{|I|=2k} det(A_{I} + diag(k − 1, k − 2, …, −k))

= ∑_{|I|=2k} Det(A_{I} ; k, k − 1, …, −k + 1)

= ∑_{|I|=2k} Det(A_{I} ; k − 1, k − 2, …, −k).

It is reasonable to put C_{2k+1} = 0. In fact, we have

∑_{|I|=2k+1} det(A_{I}+diag(k, k−1, …, −k)) = ∑_{|I|=2k+1} Det(A_{I} ; k, k−1, …, −k) = 0.

Our D(λ) defined as a linear combination of C_{2k} ’s in (2) and (3) can be expressed simply with the symmetrized determinant just like a characteristic polynomial:

Proposition 2.2. The following equality holds:

D(λ) = \{ \begin{matrix}Det(λI − A; m − 1, m − 2, …, −m + 1, 0), & (n = 2m), \\ \frac{λ − 1/2}{λ} Det(λI − A; m − \frac{1}{2}, m − \frac{3}{2}, …, −m + \frac{1}{2}, 0), & (n = 2m + 1). \end{matrix}

Here, the dots indicate a sequence of numbers descending by 1.

Proof. Suppose that n = 2m. By induction, we see the binomial expansion

f_{2m}^{±}(u + w; h) = ∑_{ν=0}^{2m} \binom{2m}{ν}f_{2m−ν}^{±}(u; h)f_{ν}^{±}(w; h)

for the polynomials f_{ν}^{±}(u; h) defined by

f_{2k}^{+}(u; h) = (u + kh)(u + (k − 1)h) ⋯ (u − (k − 1)h),
f_{2k}^{−}(u; h) = (u + (k − 1)h)(u + (k − 2)h) ⋯ (u − kh),
f_{2k+1}^{±}(u; h) = (u + kh)(u + (k − 1)h) ⋯ (u − kh).

Using this, we have

(2.3)  f_{2m}^{±}(Ξ + λτ; τ) = ∑_{ν=0}^{2m} \binom{2m}{ν}f_{2m−ν}^{±}(Ξ; τ)f_{ν}^{±}(λτ; τ)

= ∑_{ν=0}^{2m} \binom{2m}{ν}f_{2m−ν}^{±}(Ξ; τ)τ^{ν}f_{ν}^{±}(λ; 1)

= (2m)! ∑_{k=0}^{m} e_{1}e′_{1} ⋯ e_{n}e′_{n}C_{2m−2k} · f_{2k}^{±}(λ; 1).

Here, we used the relation

f_{ν}^{±}(Ξ; τ)τ^{2m−ν} = ν!(2m − ν)!e_{1}e′_{1} ⋯ e_{n}e′_{n}C_{ν},

which follows from (1.6) and (2.2). Sum up the both sides of this (2.3) over the indices +, −, and divide by 2. Then, we obtain

(Ξ + (λ + m − 1)τ)(Ξ + (λ + m − 2)τ) ⋯ (Ξ + (λ − m + 1)τ) · (Ξ + λτ)

= (2m)! ∑_{k=0}^{m} e_{1}e′_{1} ⋯ e_{n}e′_{n}C_{2m−2k} · λ\overline{k}λ\underline{k}.

This means our formula by (1.6). In the case of n = 2m + 1, since the proof is almost the same, we omit it.

From the proof, we see another expression of D(λ) with two columndeterminant:

Corollary 2.3. The following equality holds:

2D(λ) = \{ \begin{matrix}H(λ) + H(λ − 1), & (n = 2m), \\ \frac{λ − 1/2}{λ}\{H(λ) + H(λ − 1)\}, & (n = 2m + 1). \end{matrix}

Here we put

H(λ) = det(A + diag(\frac{n}{2}, \frac{n}{2} − 1, …, −\frac{n}{2} + 1) + λI).

Remark. Theorem A can be written with this H(λ) in a form which does not depend on the parity of n (§5).

Remark. It is known that Pf(A) and ∑_{|I|=ν} Det(A_{I} ; u_{1}, u_{2}, …, u_{ν}) are invariant under the adjoint actions of SO_{n} and O_{n} respectively. This fact is easily seen from Propositions 1.1 and 1.2 by using the following relation for the adjoint action of the orthogonal Lie group O_{n} on the matrix A [9]:

Ad(g)A = (gA_{ij}g^{−1})_{1≤i,j≤n} = ^{t}gA^{t}g^{−1} = ^{t}gAg,

where g ∈ O_{n}. This follows from the relation

(2.4)  Ad(g)E = (gE_{ij}g^{−1})_{1≤i,j≤n} = ^{t}gE^{t}g^{−1}

for E = (E_{ij})_{1≤i,j≤n} ∈ Mat(n, U(𝔤𝔩_{n})) and g ∈ GL_{n}, because A = E −^{t}E.

2.2. The following expression of C_{2k} is given in [9]:

(2.5)  C_{2k} = ∑_{|I|=2k} Pf(A_{I})^{2}.

Using the expression (1.5) of the subpfaffians, we can rewrite this as the expression with θ = θ_{A} and θ′ = θ′_{A} = \frac{1}{2}e′A^{t}e′ = \frac{1}{2} ∑_{i,j=1}^{n} e′_{i}e′_{j}A_{ij}:

(2.6)  e_{1}e′_{1} ⋯ e_{n}e′_{n}C_{2k} = (−)^{k}θ^{(k)}θ^{′(k)}τ^{(n−2k)}.

The cofactor matrix for D(λ) is constructed in §4 by combining this expression of C_{2k} and the cofactors for Pf(A) given in the next section.

3. The cofactor matrix for the Pfaffian Pf(A)

In this section, we introduce the cofactor matrix for the Pfaffian Pf(A) as a clue to the construction of the cofactors of the characteristic polynomial D(λ).

We first recall the cofactor matrix for the Pfaffian in the classical situation [4]. For an alternating matrix Φ of size n = 2m whose entries are commutative, we define a matrix Γ_{Φ} by

(3.1)  e_{1} ⋯ e_{2m}(Γ_{Φ})_{ij} = −θ_{Φ}^{(m−1)}e_{i}e_{j}.

Here (Γ_{Φ})_{ij} is the (i, j)-entry of the matrix Γ_{Φ} . From (1.5), we see that (Γ_{Φ})_{ij} is equal to the subpfaffian Pf(Φ_{(i,j)}) up to sign. Here Φ_{(i,j)} is the alternating submatrix obtained by deleting the ith and j th rows and ith and j th columns from Φ. We note that this definition is rewritten as

e_{1} ⋯ e_{2m}Γ_{Φ} = −^{t}e (\frac{1}{2}eΦ^{t}e)^{(m−1)} e.

We call this matrix Γ_{Φ} the cofactor matrix for the Pfaffian Pf(Φ). In fact, the product of Γ_{Φ} and Φ is known to equal to the “scalar matrix” Pf(Φ)I:

(3.2)  Γ_{Φ}Φ = ΦΓ_{Φ} = Pf(Φ)I.

For the Pfaffian Pf(A) of our matrix A = (A_{ij})_{1≤i,j≤2m}, whose entries are non-commutative, we also define its cofactor matrix Γ_{A} by the same formula (3.1). Then the following relation holds corresponding to (3.2):

Proposition 3.1. The following equality holds in Mat(2m, U(𝔬_{2m})):

Γ_{A}A′ = A′Γ_{A} = Pf(A)I,

where A′ = A − (m − 1)I. This relation is rewritten in the exterior calculus as

−θ^{(m−1)}e_{β}ψ_{α}(−m + 1) = δ_{αβ}θ^{(m)},

where ψ_{α}(u) = ∑_{i=1}^{n} e_{i}A_{iα}(u) = ψ_{α} + ue_{α}.

Proof. (i) The case of α = β. We put θ_{(α)} = \frac{1}{2} ∑_{i,j≠α} e_{i}e_{j}A_{ij}, so that θ = θ_{(α)} −e_{α}ψ_{α}. By a direct calculation we see that θ_{(α)} and e_{α}ψ_{α}(u) commute. Our assertion is deduced from the binomial expansion of θ^{(m)} = (θ_{(α)}−e_{α}ψ_{α})^{(m)} . In fact, since θ_{(α)}^{m} = 0 and (e_{α}ψ_{α})^{2} = 0, we have

θ^{(m)} = (θ_{(α)} − e_{α}ψ_{α})^{(m)} = −θ_{(α)}^{(m−1)}e_{α}ψ_{α} = −θ^{(m−1)}e_{α}ψ_{α}(−m + 1)

as desired. The last equality is easily seen, because the influence of the terms of θ and ψ_{α}(u) containing the variable e_{α} is killed by the factor e_{α}:

(3.3)  θe_{α} = θ_{(α)}e_{α},  e_{α}ψ_{α} = e_{α}ψ_{α}(u)  for any u.

(ii) The case of α ≠ β. The following expansion holds in a similar way as in (i):

θ^{(m−1)} = (θ_{(α)} − e_{α}ψ_{α})^{(m−1)} = θ_{(α)}^{(m−1)} − θ_{(α)}^{(m−2)}e_{α}ψ_{α}.

Hence, we have

(3.4)  θ^{(m−1)}e_{β}ψ_{α}(u) = θ_{(α)}^{(m−1)}e_{β}ψ_{α}(u) − θ_{(α)}^{(m−2)}e_{α}ψ_{α}e_{β}ψ_{α}(u).

Here, on one hand, the first term on the right-hand side is equal to uθ_{(α)}^{(m−1)}e_{β}e_{α}.

In fact, since this is of top-degree, and the factor θ_{(α)}^{(m−1)}e_{β} does not contain the variable e_{α}, we see that, in the factor ψ_{α}(u) = ∑_{p=1}^{n} e_{p}A_{pα}(u), only the term e_{α}A_{αα}(u) = ue_{α} survives. On the other hand, the second term is computed as follows. Putting i = j = α in the formula (2.1), we have

ψ_{α}(u + 1)ψ_{α}(u) = −θ,

and in particular

e_{α}ψ_{α}ψ_{α}(u) = −e_{α}θ_{(α)}

by the relations (3.3). Hence the second term on the right-hand side of (3.4) is equal to θ_{(α)}^{(m−2)}θ_{(α)}e_{β}e_{α} = (m − 1)θ_{(α)}^{(m−1)}e_{β}e_{α}. Thus, we have

θ^{(m−1)}e_{β}ψ_{α}(u) = (u + m − 1)θ_{(α)}^{(m−1)}e_{β}e_{α},

and our goal θ^{(m−1)}e_{β}ψ_{α}(−m + 1) = 0 in particular.

The second equality in Proposition 3.1 is generalized to the following formula for arbitrary natural number n not necessarily even:

Corollary 3.2. The following formula holds as an equality in Λ_{n} ⊗ U(𝔬_{n}) for distinct indices β_{1}, …, β_{n−2k+1} :

θ^{(k−1)}e_{β_{1}}e_{β_{2}} ⋯ e_{β_{n−2k+1}}ψ_{α}(−k + 1)

= \{ \begin{matrix}(−)^{n+i}θ^{(k)}e_{β_{1}} ⋯ ê_{β_{i}} ⋯ e_{β_{n−2k+1}}, & α = β_{i}, \\ 0, & α ∉ \{β_{1}, β_{2}, …, β_{n−2k+1}\}. \end{matrix}

Here ê_{β_{i}} means that e_{β_{i}} is omitted. Proof. We can assume that α ≠ β_{p} for p ≠ i. Applying the second equality in Proposition 3.1 to the submatrix A_{I} with

I = {1, 2, …, n} \ {β_{1}, …, β̂_{i}, …, β_{n−2k+1}},

we have

θ̃^{(k−1)}e_{β_{i}} ψ̃_{α}(−k + 1) = −δ_{αβ_{i}} θ̃^{(k)}

with θ̃ = ∑_{p,q∈I} e_{p}e_{q}A_{pq} and ψ̃_{α}(u) = ∑_{p∈I} e_{p}A_{pα}(u). Then, we obtain our formula multiplying the both sides of this equality by e_{β_{1}} ⋯ ê_{β_{i}} ⋯ e_{β_{n−2k+1}} . In fact, we have

θ̃e_{β_{1}} ⋯ ê_{β_{i}} ⋯ e_{β_{n−2k+1}} = θe_{β_{1}} ⋯ ê_{β_{i}} ⋯ e_{β_{n−2k+1}},
ψ̃_{α}(u)e_{β_{1}} ⋯ ê_{β_{i}} ⋯ e_{β_{n−2k+1}} = ψ_{α}(u)e_{β_{1}} ⋯ ê_{β_{i}} ⋯ e_{β_{n−2k+1}}.

4. The cofactor matrix for D(λ)

In this section, we construct the cofactor matrix Δ(λ) for D(λ) using the cofactors for Pf(A) given in the previous section. The main results in this paper are obtained as applications of this cofactor matrix.

Put the matrices B_{ν} ’s in Mat(n, U(𝔬_{n})) for 0 ≤ ν ≤ n − 1 as

e_{1}e′_{1} ⋯ e_{n}e′_{n}(B_{ν})_{ij} = \{ \begin{matrix}(−)^{k+1}θ^{(k)}θ^{′(k)}τ^{(n−2k−1)}e_{i}e′_{j}, & (ν = 2k), \\ (−)^{k}θ^{(k+1)}θ^{′(k)}τ^{(n−2k−2)}e′_{i}e′_{j}, & (ν = 2k + 1). \end{matrix}

Here (B_{ν})_{ij} is the (i, j)-entry of the matrix B_{ν}. Moreover we put B_{n} = 0. Note that this definition is rewritten as

(B_{2k})_{ij} = ∑_{|I|=2k+1, i,j∈I} ± Pf(A_{I\{i}}) Pf(A_{I\{j}}),

(B_{2k+1})_{ij} = \{ \begin{matrix}∑_{|I|=2k+2, i,j∈I} ± Pf(A_{I}) Pf(A_{I\\{i,j\}}), & (i ≠ j) \\ 0, & (i = j). \end{matrix}

Here, the sign in each term is determined by i, j and I. We define the matrix Δ(λ) by the following linear combination of these B_{ν} ’s:

Δ(λ) = \{ \begin{matrix}∑_{ν=0}^{n−1} B_{n−ν−1}h_{ν}^{−}(λ), & (n = 2m), \\ ∑_{ν=0}^{n−1} B_{n−ν−1}h_{ν}^{+}(λ − 1/2), & (n = 2m + 1). \end{matrix}

Here h_{ν}^{±}(λ) are polynomials of degree ν defined by

h_{ν}^{+}(λ) = \{ \begin{matrix}λ\overline{k}λ\underline{k}, & (ν = 2k), \\ λ\overline{k+1}λ\underline{k}, & (ν = 2k + 1), \end{matrix} h_{ν}^{−}(λ) = \{ \begin{matrix}λ\overline{k}λ\underline{k}, & (ν = 2k), \\ λ\overline{k}λ\underline{k+1}, & (ν = 2k + 1). \end{matrix}

We easily see the following relations from the expression (2.6) of C_{2k}:

Proposition 4.1. The following relations hold:

tr(B_{2k}) = −(n − 2k)C_{2k}, tr(B_{2k+1}) = 0.

In particular, the trace of the matrix Δ(λ) is equal to −N(λ):

tr(Δ(λ)) = −N(λ).

We call this Δ(λ) the cofactor matrix for D(λ). In fact, the product of Δ(λ) and A − (n/2 − 1)I − λI is equal to the “scalar matrix” D(λ)I as is seen in the following proposition. Newton’s formulas and the Cayley-Hamilton theorem are obtained as corollaries of this relation (§5).

Proposition 4.2. The following equality holds:

Δ(λ)(A′ − λI) = (A′ − λI)Δ(λ) = D(λ)I,

where A′ = A − (n/2 − 1)I.

For the proof of Proposition 4.2, we use the following three lemmas. In these, Lemmas 4.3 and 4.5 are based on the cofactors for Pf(A).

Lemma 4.3. We have

θ^{′(k−1)}τ^{(n−2k+1)}ψ′_{j}(−k + 1) = −θ^{′(k)}τ^{(n−2k)}e_{j},
θ^{′(k−1)}τ^{(n−2k)}e′_{i}ψ′_{j}(−k + 1) = −θ^{′(k)}τ^{(n−2k)}δ_{ij} − θ^{′(k)}τ^{(n−2k−1)}e′_{i}e_{j}.

where ψ′_{j}(u) = ∑_{i=1}^{n} e′_{i}A_{ij}(u) = ψ′_{j} + ue′_{j}.

Lemma 4.4. We have

θ^{(k)}θ^{′(k−1)}τ^{(n−2k)}e′_{i}e′_{j} = θ^{(k−1)}θ^{′(k)}τ^{(n−2k)}e_{i}e_{j}.

Lemma 4.5. We have

θ^{(k)}θ^{′(k)}τ^{(n−2k−1)}(e_{i}e′_{j} + e′_{i}e_{j}) + (n − 2k)θ^{(k)}θ^{′(k−1)}τ^{(n−2k)}e′_{i}e′_{j} = 0.

Proof of Lemma 4.3. We can reduce this lemma to Corollary 3.2 by applying the involution ι_{∗} = ′ and expanding the powers of τ = ∑_{p=1}^{n} e_{p}e′_{p}.

Proof of Lemma 4.4. Our assertion is deduced from the special case of n = 2k by applying to the submatrices A_{I} of A for |I| = 2k. In fact, we have

θ^{(k)}θ^{′(k−1)}τ^{(n−2k)}e′_{i}e′_{j} = ∑_{|I|=2k, i,j∈I} θ_{I}^{(k)}θ_{I}^{′(k−1)}e′_{i}e′_{j} · τ_{I^{c}}^{(n−2k)},
θ^{(k−1)}θ^{′(k)}τ^{(n−2k)}e_{i}e_{j} = ∑_{|I|=2k, i,j∈I} θ_{I}^{(k−1)}θ_{I}^{′(k)}e_{i}e_{j} · τ_{I^{c}}^{(n−2k)},

where θ_{I}, θ′_{I} and τ_{I^{c}} are defined by

θ_{I} = ∑_{p,q∈I} e_{p}e_{q}A_{pq}, θ′_{I} = ∑_{p,q∈I} e′_{p}e′_{q}A_{pq}, τ_{I^{c}} = ∑_{p∉I} e_{p}e′_{p}.

Therefore, we may assume that n = 2k without loss of generality. We start with the application of the involution ι_{∗} = ′ to φ = θ^{(k)}θ^{′(k−1)}e′_{i}e′_{j}. On one hand, a direct calculation leads us to φ′ = θ^{′(k)}θ^{(k−1)}e_{i}e_{j}. On the other hand we see φ′ = (−)^{n}φ = φ from Lemma 1.4. Thus we have

(4.1)  θ^{(k)}θ^{′(k−1)}e′_{i}e′_{j} = θ^{′(k)}θ^{(k−1)}e_{i}e_{j}.

Here, the factor θ^{′(k)} on the right-hand side is central in Λ_{2n} ⊗ U(𝔬_{n}). In fact, as is seen in (1.1), the coefficient of θ^{′(k)} is equal to the Pfaffian Pf(A), which is central in U(𝔬_{n}). Hence, we can exchange θ^{′(k)} and θ^{(k−1)} on the right-hand side of (4.1). Thus our assertion has been proved.

Proof of Lemma 4.5. We assume that i ≠ j, because our assertion is obviously true in the case of i = j. Moreover we can assume that n = 2k + 1 as in the proof of Lemma 4.4. In this case, our formula is equivalent to

(4.2)  θ^{(k)}θ^{′(k)}e_{i}e′_{j} = θ^{′(k)}θ^{(k)}e_{i}e′_{j} − θ^{(k)}θ^{′(k−1)}τe′_{i}e′_{j}.

In fact, we see that θ^{(k)}θ^{′(k)}e′_{i}e_{j} = −θ^{′(k)}θ^{(k)}e_{i}e′_{j} by calculating the action of the involution ι_{∗} = ′ to θ^{(k)}θ^{′(k)}e′_{i}e_{j} in two ways as in the proof of Lemma 4.4. This (4.2) is proved by computing the element φ = θ^{(k)}θ^{′(k)}e_{i}e′_{j} as follows. Note the relations

(4.3)  θ^{(k)}e_{j} = −θ^{(k−1)}e_{i}ψ_{i}e_{j}, θ^{′(k)}e′_{j} = −θ^{′(k−1)}e′_{i}ψ′_{i}e′_{j}.

These are immediate from Corollary 3.2, because we have e_{i}ψ_{i}(u) = e_{i}ψ_{i} for any u. Using this (4.3), we have

φ = θ^{(k)}θ^{′(k)}e_{i}e′_{j} = θ^{(k)}e_{i}θ^{′(k)}e′_{j} = −θ^{(k)}e_{i}θ^{′(k−1)}e′_{i}ψ′_{i}e′_{j}.

Here, the factors θ^{(k)}e_{i} and θ^{′(k−1)}e′_{i} are anti-commutative. This is seen from the fact that the coefficient of θ^{(k)}e_{i} is equal to the Pfaffian Pf(A_{(i)}) for the submatrix A_{(i)} = (A_{pq})_{p,q≠i}, which commutes with A_{pq} for p, q ≠ i. In fact, the coefficients of θ^{′(k−1)}e′_{i} are generated by {A_{pq} | p, q ≠ i}. Hence we have

φ = θ^{′(k−1)}e′_{i}θ^{(k)}e_{i}ψ′_{i}e′_{j}.

Furthermore we consider the commutation relation between θ^{(k)} and e_{i}ψ′_{i}:

[θ^{(k)}, e_{i}ψ′_{i}] = θ^{(k−1)}e_{i}ψ_{i}τ.

This is easily seen from the direct calculations [θ, e_{i}ψ′_{i}] = e_{i}ψ_{i}τ and [θ, e_{i}ψ_{i}] = 0. Thus, we have

φ = θ^{′(k−1)}e′_{i}θ^{(k)}e_{i}ψ′_{i}e′_{j} = θ^{′(k−1)}e′_{i}e_{i}ψ′_{i}θ^{(k)}e′_{j} + θ^{′(k−1)}e′_{i}θ^{(k−1)}e_{i}ψ_{i}τe′_{j}
= θ^{′(k)}θ^{(k)}e_{i}e′_{j} − θ^{′(k)}θ^{(k−1)}τe′_{i}e′_{j}.

On the last equality, we used (4.3) again. This implies (4.2), hence our formula.

Now we are ready to prove Proposition 4.2. Combining the three lemmas above, we have the following relation among B_{ν} ’s, C_{ν} ’s and A:

Proposition 4.6. The following equalities hold in Mat(n, U(𝔬_{n})):

B_{2k}(A − kI) = B_{2k}(A′ + (n/2 − k − 1)I) = B_{2k+1},

B_{2k+1}(A − (n − k − 2)I) = B_{2k+1}(A′ − (n/2 − k − 1)I) = B_{2k+2} + C_{2k+2}I.

Proof. Using Lemma 4.3 and then Lemma 4.4, we have the following equality:

(−)^{k+1}θ^{(k)}θ^{′(k)}τ^{(n−2k−1)}e_{i}ψ′_{j}(−k) = (−)^{k}θ^{(k)}θ^{′(k+1)}τ^{(n−2k−2)}e_{i}e_{j}
= (−)^{k}θ^{(k+1)}θ^{′(k)}τ^{(n−2k−2)}e′_{i}e′_{j}.

Our first formula is immediate from this together with the definition of B_{ν}. To show the second formula, we calculate the quantity

φ = (−)^{k}θ^{(k+1)}θ^{′(k)}τ^{(n−2k−2)}e′_{i}ψ′_{j}(−k).

In the case of i = j, we have the following by Lemma 4.3:

φ = (−)^{k+1}θ^{(k+1)}θ^{′(k+1)}τ^{(n−2k−2)} + (−)^{k}θ^{(k+1)}θ^{′(k+1)}τ^{(n−2k−3)}e_{i}e′_{i}.

In the case of i ≠ j, applying Lemma 4.3 and then Lemma 4.5, we have

φ = (−)^{k+1}θ^{(k+1)}θ^{′(k+1)}τ^{(n−2k−3)}e′_{i}e_{j}

= (−)^{k}θ^{(k+1)}θ^{′(k+1)}τ^{(n−2k−3)}e_{i}e′_{j} + (−)^{k}(n − 2k − 2)θ^{(k+1)}θ^{′(k)}τ^{(n−2k−2)}e′_{i}e′_{j}.

By the definition of B_{ν} and the expression (2.6) of C_{2k}, these equalities imply

B_{2k+1}(A − kI) = B_{2k+2} + C_{2k+2}I + (n − 2k − 2)B_{2k+1}.

Our second formula is immediate from this.

Proof of Proposition 4.2. The proposition is obtained as a linear combination of the relations in Proposition 4.6. Assume that n = 2m. By the definition of Δ(λ) and Proposition 4.6, we have

Δ(λ)(A′ − λI)

= \{ ∑_{k=0}^{m−1} B_{2m−2k−2}λ\overline{k}λ\underline{k+1} + ∑_{k=0}^{m−1} B_{2m−2k−1}λ\overline{k}λ\underline{k}\} (A − (m − 1)I − λI)

= ∑_{k=0}^{m−1}\{B_{2m−2k−2}(A − (m − k − 1)I) + B_{2m−2k−2} · (−k − λ)\}λ\overline{k}λ\underline{k+1}

+ ∑_{k=0}^{m−1}\{B_{2m−2k−1}(A − (n − (m − k − 1) − 2)I) + B_{2m−2k−1} · (k − λ)\}λ\overline{k}λ\underline{k}

= ∑_{k=0}^{m−1} \{B_{2m−2k−1}λ\overline{k}λ\underline{k+1} − B_{2m−2k−2}λ\overline{k+1}λ\underline{k+1}\}

+ ∑_{k=0}^{m−1} \{(B_{2m−2k} + C_{2m−2k}I)λ\overline{k}λ\underline{k} − B_{2m−2k−1}λ\overline{k}λ\underline{k+1}\}

= ∑_{k=0}^{m−1} C_{2m−2k}Iλ\overline{k}λ\underline{k} − B_{0}λ\overline{m}λ\underline{m} + B_{2m}.

This coincides with D(λ)I as desired, because B_{0} = −I and B_{2m} = 0. The proof for the case of n = 2m + 1 is similar.

In the remainder of this section, we present some supplementary formulas for the cofactor matrix Δ(λ).

In the case that n = 2m is even, we can still factorize the cofactor matrix Δ(λ). Define a matrix Y (λ) by

Y (λ) = ∑_{k=0}^{m−1} B_{2m−2k−2}λ\overline{k}λ\underline{k}.

Then we have the following relation:

Proposition 4.7. When n = 2m, we have

Y (λ)(A′ + λI) = Δ(λ),

and in particular

Y (λ)(A′ + λI)(A′ − λI) = D(λ)I.

This can be checked in a similar way as Proposition 4.2 by using Proposition 4.6.

For odd n = 2m + 1, such a factorization of the cofactor matrix Δ(λ) is not available. We have the following proposition instead of Proposition 4.7. Define a matrix Y (λ) by

Y (λ) = ∑_{k=0}^{m−1} B_{2m−2k−2} (λ − 1/2)\overline{k+1} (λ − 1/2)\underline{k} .

Proposition 4.8. When n = 2m + 1, the following equalities hold:

Y (λ)(A′ + λI) + Δ(1/2) = Δ(λ),

(A − mI)Y (λ)(A′ + λI)(A′ − λI) = D(λ)(A − mI).

Here, the second equality is immediately follows from the first, because we have (A − mI)Δ(1/2) = D(1/2)I = 0.

5. Newton’s formulas and the Cayley-Hamilton theorem for 𝔬_{n}

As corollaries of Proposition 4.2 for the cofactor matrix Δ(λ) for D(λ), we have Newton’s formulas and the Cayley-Hamilton theorem for 𝔬_{n}.

Theorem 5.1. (Newton’s formula for 𝔬_{n} (1)) The following equality holds as U(𝔬_{n})-coefficient formal power series in λ:

\frac{N(λ)}{D(λ)} = ∑_{r=0}^{∞} λ^{−1−r} tr(A^{′r}).

Here, we put A′ = A − (n/2 − 1)I.

Remark. This Theorem 5.1 is rewritten with a column-determinant H(λ) by using Corollary 2.3:

\frac{λ}{λ − 1/2} · \frac{H(λ) − H(λ − 2)}{H(λ) + H(λ − 1)} = ∑_{r=0}^{∞} λ^{−1−r} tr(A^{′r}).

This expression does not depend on the parity of n.

Theorem 5.2. (Newton’s formula for 𝔬_{n} (2)) A relation between C_{2k} ’s and tr(A^{r})’s is given in the form

∑_{i=0}^{k} C_{2k−2i} tr ((A′ + (\frac{n}{2} − k) I)\overline{i} (A′ − (\frac{n}{2} − k) I)\underline{i}) = (n − 2k)C_{2k},

∑_{i=0}^{k} C_{2k−2i} tr ((A′ + (\frac{n}{2} − k − 1) I)\overline{i+1} (A′ − (\frac{n}{2} − k) I)\underline{i}) = 0.

From this Theorem 5.2, we can deduce an explicit expression of C_{2k} with tr(A^{r})’s by induction:

Corollary 5.3. The central element C_{2k} is expressed as the following determinant of degree k:

(−)^{k}2^{k}k!C_{2k} = det ( \begin{matrix}Q_{1,1} & 2 & & & 0 \\ Q_{2,2} & Q_{2,1} & 4 \\ Q_{3,3} & Q_{3,2} & Q_{3,1} & ... \\ ⋮ & ⋮ & ⋮ & ... & 2k − 2 \\ Q_{k,k} & Q_{k,k−1} & Q_{k,k−2} & … & Q_{k,1} \end{matrix} ) .

Here, we put Q_{α,β} = tr((A′ + (n/2 − α)I)\overline{β}(A′ − (n/2 − α)I)\underline{β}). Proof of Theorem 5.1. Multiplying the both sides of the equality in Proposition 4.2 by (λI − A′)^{−1} from right, we have

−Δ(λ) = D(λ)(λI − A′)^{−1} = D(λ) ∑_{r=0}^{∞} λ^{−1−r}A^{′r}.

Then, taking the trace of the both sides, we obtain the theorem. In fact, the trace of Δ(λ) is equal to −N(λ) as is seen in Proposition 4.1.

Proof of Theorem 5.2. Define B̃_{ν} ∈ Mat(n, U(𝔬_{n})) by

B̃_{2k} = B_{2k} + ∑_{i=0}^{k} C_{2k−2i}(A′ + (n/2 − k)I)\overline{i}(A′ − (n/2 − k)I)\underline{i},

B̃_{2k+1} = B_{2k+1} + ∑_{i=0}^{k} C_{2k−2i}(A′ + (n/2 − k − 1)I)\overline{i+1}(A′ − (n/2 − k)I)\underline{i}.

Using Proposition 4.6, we see the relations

B̃_{2k}(A′ + (n/2 − k − 1)I) = B̃_{2k+1},
B̃_{2k+1}(A′ − (n/2 − k − 1)I) = B̃_{2k+2}.

Then, since B̃_{0} = B_{0} + I = 0, we see that B̃_{ν} = 0 by induction. Taking the trace, we come to the theorem by Proposition 4.1.

Since B_{n} = 0, we see that B̃_{n} = D(A′) from the definition of B̃_{ν}. Thus, the equality B̃_{n} = 0 in the proof of Theorem 5.2 implies the following theorem:

Theorem 5.4. (The Cayley-Hamilton theorem for 𝔬_{n}) The following equality holds in Mat(n, U(𝔬_{n})):

D(A′) = 0.

Remarks. (1) Theorems 5.1 and 5.2 are equivalent. In fact, Theorem 5.2 is directly deduced from Theorem 5.1 by comparing the coefficients of the powers in λ.

(2) Another Newton type identity in U(𝔬_{n}) is given in [12]. This has the same form as Theorem 5.1 except that N(λ) and D(λ) are given with another certain “single” determinant d̃et instead of the column-determinant. This means that our C_{2k} ’s are equal to this determinant d̃et. This fact will be also deduced more clearly in §7 by calculating the image of D(λ) under the Harish-Chandra homomorphism.

6. Relation to other realizations of the orthogonal Lie algebras

We have discussed on the orthogonal Lie algebra 𝔬_{n} realized as the Lie algebra consisting of the alternating matrices, and constructed some identities in its universal enveloping algebra. In this section, we extend these results to the realization of the orthogonal Lie algebra with any non-degenerate symmetric matrix. For this general realization, we can naturally arrange a matrix F in a similar way as A, and will see that this F satisfies Newton type and Cayley-Hamilton type identities, which are described with the symmetrized determinant.

Fix a non-degenerate symmetric matrix S ∈ Mat_{n}(𝕂), and consider the orthogonal Lie algebra defined with this S:

𝔬(S) = {X ∈ 𝔤𝔩_{n} |^{t}XS + SX = 0}.

According to the embedding of the Lie algebras 𝔬(S) and 𝔬_{n} into 𝔤𝔩_{n}, we regard the universal enveloping algebras U(𝔬(S)) and U(𝔬_{n}) as subalgebras of U(𝔤𝔩_{n}). Consider an involution i_{S} : X ↦ S^{−1}^{t}XS of 𝔤𝔩_{n}, so that X − i_{S}(X) ∈ 𝔬(S) for any X ∈ 𝔤𝔩_{n}. Then the elements F_{ij} = E_{ij} − i_{S}(E_{ij}) span 𝔬(S). We arrange a matrix F = (F_{ij})_{1≤i,j≤n} from these. This F is expressed with E = (E_{ij})_{1≤i,j≤n} as

F = E − Ad(S^{−1})^{t}E = E − S^{t}ES^{−1}.

Here we used the relation (2.4). In particular, the matrices F S and S^{−1}F are alternating.

The two orthogonal Lie algebras 𝔬_{n} and 𝔬(S) are related with each other as follows. Fix a matrix s ∈ Mat_{n}(𝕂) such that S = ^{t}ss. (Such an s can be taken by extending the ground field 𝕂.) Then the restriction of the automorphism Ad(s^{−1}): X ↦ s^{−1}Xs of 𝔤𝔩_{n} gives a natural isomorphism 𝔬_{n} ∼_{→} 𝔬(S). By this isomorphism, the matrix A is mapped to Ad(s^{−1})A = ^{t}s^{−1}F^{t}s as is seen from the relation (2.4). Then, we obtain the images of Pf(A), C_{2k} and tr(A^{r}): (6.1)

Ad(s^{−1}) Pf(A) = det(s)^{−1} Pf(F S) = det(s) Pf(S^{−1}F),

Ad(s^{−1})C_{2k} = ∑_{|I|=2k} Det(F_{I} ; k, k − 1, …, −k + 1), Ad(s^{−1}) tr(A^{r}) = tr(F^{r}).

These are easily checked by using Propositions 1.1, 1.2 and the invariance of the trace under the conjugation. We can express D_{F} (λ) = Ad(s^{−1})D(λ) and N_{F} (λ) = Ad(s^{−1})N(λ) as linear combinations of the symmetrized determinants in similar ways as (2) and (3). Moreover D_{F} (λ) is written as follows by Propositions 1.2 and 2.2:

D_{F} (λ)=\{ \begin{matrix}Det(λI − F ; m − 1, m − 2, …, −m + 1, 0), & (n = 2m), \\ \frac{λ − 1/2}{λ}Det(λI − F; m − \frac{1}{2}, m − \frac{3}{2}, …, −m + \frac{1}{2}, 0), & (n = 2m + 1). \end{matrix}

Newton’s formulas and the Cayley-Hamilton theorem given in §5 still hold by replacing D(λ), N(λ) and A respectively by D_{F} (λ), N_{F} (λ) and F :

Theorem 6.1. (Newton’s formula for 𝔬(S)) The following equality holds as U(𝔬(S))-coefficient formal power series in λ:

\frac{N_{F} (λ)}{D_{F} (λ)} = ∑_{r=0}^{∞} λ^{−1−r} tr(F^{′r}),

where F′ = F − (n/2 − 1)I.

Theorem 6.2. (The Cayley-Hamilton theorem for 𝔬(S)) The following equality holds in Mat(n, U(𝔬(S))):

D_{F} (F′) = 0.

In the remainder of this section, we rewrite the cofactor matrices for Pf(A) and D(λ) and their equalities in terms of the matrix F . First, let us consider the cofactor matrix for Pf(S^{−1}F) in the case of n = 2m. We define the matrix Γ_{S^{−1}F} by

e_{1} ⋯ e_{n}(Γ_{S^{−1}F})_{ij} = −θ_{S^{−1}F}^{(m−1)}e_{i}e_{j}

with θ_{S^{−1}F} = \frac{1}{2}eS^{−1}F^{t}e = \frac{1}{2} ∑_{i,j=1}^{n} e_{i}e_{j}(S^{−1}F)_{ij}. We have the following relation corresponding to Proposition 3.1:

Proposition 6.3. The following equality holds:

Γ_{S^{−1}F} S^{−1}F′ = Pf(S^{−1}F)I.

Proof. We apply the isomorphism Ad(s^{−1}): 𝔬_{n} ∼_{→} 𝔬(S) to the following expression of Proposition 3.1 in the exterior calculus:

(6.2)  −^{t}e (\frac{1}{2}eA^{t}e)^{(m−1)} e · A′ = e_{1} ⋯ e_{n} Pf(A)I.

Then, since Ad(s^{−1})A = sS^{−1}F^{t}s, the left-hand side is mapped to

−^{t}s^{−1}^{t}(es) (\frac{1}{2}(es)S^{−1}F^{t}(es))^{(m−1)} (es) · S^{−1}F′^{t}s.

This is equal to the following by Lemma 1.4:

− det(s)^{t}s^{−1}^{t}e (\frac{1}{2}eS^{−1}F^{t}e)^{(m−1)} e · S^{−1}F′^{t}s
= −e_{1} ⋯ e_{n} det(s)^{t}s^{−1}Γ_{S^{−1}F} S^{−1}F′^{t}s.

This implies our assertion. In fact, the isomorphism Ad(s^{−1}) maps the righthand side of (6.2) to e_{1} ⋯ e_{n} det(s) Pf(S^{−1}F) as is seen in (6.1).

We can similarly rewrite the cofactor matrix for D(λ) in terms of F using Lemma 1.4. We put the matrices R_{ν} ’s for 0 ≤ ν ≤ n − 1 as (6.3)

e_{1}e′_{1} ⋯ e_{n}e′_{n}(R_{ν})_{ij} = \{ \begin{matrix}(−)^{k+1}θ_{S^{−1}F}^{(k)} θ_{S^{−1}F}^{′(k)} τ_{S^{−1}}^{(n−2k−1)}e_{i}e′_{j}, & (ν = 2k), \\ (−)^{k}θ_{S^{−1}F}^{(k+1)}θ_{S^{−1}F}^{′(k)} τ_{S^{−1}}^{(n−2k−2)}e′_{i}e′_{j}, & (ν = 2k + 1), \end{matrix}

and define Δ_{S^{−1}F} (λ) by

Δ_{S^{−1}F} (λ) = \{ \begin{matrix}∑_{ν=0}^{n−1} R_{n−ν−1}h_{ν}^{−}(λ), & (n = 2m), \\ ∑_{ν=0}^{n−1} R_{n−ν−1}h_{ν}^{+}(λ − 1/2), & (n = 2m + 1), \end{matrix}

where θ′_{S^{−1}F} = \frac{1}{2}e′S^{−1}F^{t}e′, and τ_{S^{−1}} = eS^{−1}^{t}e′. Then, we have the following proposition corresponding to Proposition 4.2. This is deduced from Proposition 4.2 in a similar way as Proposition 6.3.

Proposition 6.4. The following equality holds:

Δ_{S^{−1}F} (λ)S^{−1}(F′ − λI) = det(S^{−1})D_{F} (λ)I.
Moreover we define Y_{S^{−1}F} (λ) by

(6.4) Y_{S^{−1}F} (λ) = \{ \begin{matrix}∑_{k=0}^{m−1} R_{2m−2k−2}λ\overline{k}λ\underline{k}, & (n = 2m), \\ ∑_{k=0}^{m−1} R_{2m−2k−2} (λ − 1/2)\overline{k+1} (λ − 1/2)\underline{k} , & (n = 2m + 1). \end{matrix}

We have following propositions respectively corresponding to Propositions 4.7 and 4.8:

Proposition 6.5. When n = 2m, the following equality holds:

Y_{S^{−1}F} (λ)S^{−1}(F′ + λI)(F′ − λI) = det(S^{−1})D_{F} (λ)I.

Proposition 6.6. When n = 2m + 1, the following equality holds:

(F − mI)Y_{S^{−1}F} (λ)S^{−1}(F′ + λI)(F′ − λI) = det(S^{−1})D_{F} (λ)(F − mI).

Finally, we note the counterpart of (2.6) in terms of F , which we use in §7:

Proposition 6.7. The following equality holds:

e_{1}e′_{1} ⋯ e_{n}e′_{n} det(S^{−1}) ∑_{|I|=2k} Det(F_{I} ; k, k − 1, …, −k + 1)
= (−)^{k}θ_{S^{−1}F}^{(k)} θ_{S^{−1}F}^{′(k)} τ_{S^{−1}}^{(n−2k)}.

7. Image of D(λ) under the Harish-Chandra homomorphism

In this section, we work on the split realization of the orthogonal Lie algebra, and consider the image of the characteristic polynomial D^{†}(λ) = D_{F} (λ) under the Harish-Chandra homomorphism. From this result, it turns out that our D^{†}(λ) = D_{F} (λ) is identical to a central element C(λ) given with the Sklyanin determinant in [10].

For a reductive Lie algebra 𝔤, the Harish-Chandra homomorphism is defined as follows. Fix a triangular decomposition of 𝔤

𝔤 = 𝔫_{−} ⊕ 𝔥 ⊕ 𝔫_{+}.

This leads us to the following direct sum decomposition of U(𝔤) as a vector space by the Poincaré-Birkhoff-Witt theorem:

U(𝔤) = U(𝔥) ⊕ (𝔫_{−}U(𝔤) + U(𝔤)𝔫_{+}).

Denote by γ the projection of U(𝔤) onto U(𝔥) along this decomposition. The restriction of γ to the centralizer U(𝔤)^{𝔥} of 𝔥 gives an algebra homomorphism. Moreover we consider the rho-shift T_{−ρ}, the automorphism of U(𝔥) determined by 𝔥 ∋ H ↦ H − ρ(H). Here ρ is the element of 𝔥^{∗} defined by

ρ(H) = \frac{1}{2} tr_{𝔫_{+}}(ad H), for H ∈ 𝔥.

The composition \overline{γ} = T_{−ρ} ∘ γ gives an algebra homomorphism on U(𝔤)^{𝔥}. Moreover this \overline{γ} is known to give an algebra isomorphism ZU(𝔤) ≃ U(𝔥)^{W} , where ZU(𝔤) is the center of U(𝔤), and U(𝔥)^{W} is the subalgebra of U(𝔥) consisting of the invariants under the action of the Weyl group W. We call this \overline{γ} the Harish-Chandra homomorphism.

Hereafter we work on the orthogonal Lie algebra realized with the symmetric matrix S_{0} = (δ_{i,n+1−j})_{1≤i,j≤n}:

𝔬(S_{0}) = {X ∈ 𝔤𝔩_{n} |^{t}XS_{0} + S_{0}X = 0}.

This 𝔬(S_{0}) is spanned by F_{ij} = E_{ij} − E_{n+1−j,n+1−i}. Let 𝔫_{−}, 𝔥 and 𝔫_{+} be the Lie subalgebras spanned by the elements F_{ij} such that i > j, i = j, i < j respectively. Then these give a triangular decomposition of 𝔬(S_{0}):

𝔬(S_{0}) = 𝔫_{−} ⊕ 𝔥 ⊕ 𝔫_{+}.

The rho-shift T_{−ρ} acts on the basis {F_{ii} | 1 ≤ i ≤ n/2} of 𝔥 as

T_{−ρ}F_{ii} = F_{ii} − \frac{n}{2} + i.

The images of Pf(S_{0}^{−1}F) and D_{F} (λ) under \overline{γ} are given as follows. We note that Proposition 7.1 has been known. (For example see [25, §127] or [10, p. 942].)

Proposition 7.1. When n = 2m is even, the image of Pf(S_{0}^{−1}F) under the Harish-Chandra homomorphism is given by

\overline{γ}(Pf(S_{0}^{−1}F)) = (−)^{m}F_{11}F_{22} ⋯ F_{mm}.

Theorem 7.2. The image of D_{F} (λ) under the Harish-Chandra homomorphism is given by

\overline{γ}(D_{F} (λ)) = \{ \begin{matrix}(λ^{2} − F_{11}^{2}) ⋯ (λ^{2} − F_{mm}^{2}), & (n = 2m), \\ (λ − \frac{1}{2}) (λ^{2} − F_{11}^{2}) ⋯ (λ^{2} − F_{mm}^{2}), & (n = 2m + 1). \end{matrix}

These can be proved by induction in the following way. The Lie subalgebra generated by {F_{ij} | 2 ≤ i, j ≤ n−1} is regarded as the orthogonal Lie algebra 𝔬(S̃_{0}) realized with the symmetric matrix S̃_{0} = (S_{0})_{(1n)} = (δ_{i,n+1−j})_{2≤i,j≤n−1} . For this 𝔬(S̃_{0}), we can consider the matrix F̃ = F_{(1n)} = (F_{ij})_{2≤i,j≤n−1} and its characteristic polynomial DF̃(λ), or the Pfaffian Pf(S̃_{0}^{−1}F̃) as for 𝔬(S_{0}). The triangular decomposition of 𝔬(S̃_{0}) is naturally given by restricting that of 𝔬(S_{0}). We will prove Proposition 7.1 and Theorem 7.2 reducing their assertions to the case of 𝔬(S̃_{0}). For this reduction, we use the cofactors for Pf(S_{0}^{−1}F) and D_{F} (λ).

Proof of Proposition 7.1. Apply γ to the (n, n)-entry of the both sides of the equality in Proposition 6.3. Then, we have

γ(Pf(S_{0}^{−1}F)) = ∑_{i=1}^{n} γ((Γ_{S_{0}^{−1}F})_{ni}(F′)_{n−i+1,n}) = γ((Γ_{S_{0}^{−1}F})_{n1})(−F_{11} − (m − 1)).

In fact, (F′)_{n−i+1,n} belongs to 𝔫_{+} for i ≠ 1. Here, (Γ_{S_{0}^{−1}F})_{n1} is equal to the

Pfaffian Pf(S̃_{0}^{−1}F̃). This fact is easily seen from the following expressions in the exterior calculus by noting that θ_{S_{0}^{−1}F} e_{n}e_{1} = θ̃_{S_{0}^{−1}F} e_{n}e_{1} :

e_{1} ⋯ e_{n}(Γ_{S_{0}^{−1}F})_{n1} = −θ_{S_{0}^{−1}F}^{(m−1)}e_{n}e_{1}, e_{2} ⋯ e_{n−1} Pf(S̃_{0}^{−1}F̃) = θ̃_{S_{0}^{−1}F}^{(m−1)},

where θ̃_{S_{0}^{−1}F} = \frac{1}{2} ∑_{2≤i,j≤n−1} e_{i}e_{j}F_{n+1−i,j}. Hence, applying T_{−ρ}, we have

\overline{γ}(Pf(S_{0}^{−1}F)) = \overline{γ}(Pf(S̃_{0}^{−1}F̃)) · (−F_{11}).

Thus our task has been reduced to the calculation of the image of the Pfaffian Pf(S̃_{0}^{−1}F̃) of degree n − 2. The proposition is proved by induction.

To prove Theorem 7.2 similarly, we need the following relation between the (n, 1)-entry of the matrix Y_{S_{0}^{−1}F} (λ) given in (6.4) and the characteristic

polynomial DF̃(λ) for the submatrix F̃:

Lemma 7.3. The following equality holds:

γ((R_{2k})_{n1}) = det(S̃_{0}^{−1})γ( ∑_{|I|=2k} Det(F̃_{I} ; k, k − 1, …, −k + 1)).

In particular, we have

γ((Y_{S_{0}^{−1}F} (λ))_{n1}) = det(S̃_{0}^{−1})γ(DF̃(λ)).

Proof. By (6.3) and Proposition 6.7, the lemma is rewritten in terms of the exterior calculus as

(7.1) γ(θ_{S_{0}^{−1}F}^{(k)} θ_{S_{0}^{−1}F}^{′(k)} τ_{S_{0}^{−1}}^{(n−2k−1)}e_{n}e′_{1}) = γ(θ̃_{S_{0}^{−1}F}^{(k)} θ̃_{S_{0}^{−1}F}^{′(k)} τ̃_{S_{0}^{−1}}^{(n−2k−2)}e_{1}e′_{n}e_{n}e′_{1}),

where

θ̃′_{S_{0}^{−1}F} = \frac{1}{2} ∑_{2≤i,j≤n−1} e′_{i}e′_{j}F_{n+1−i,j}, τ̃_{S_{0}^{−1}} = ∑_{2≤i≤n−1} e_{i}e′_{n+1−i}.

We put ω′_{n} = ∑_{i=1}^{n} e′_{i}F_{n+1−i,n}, so that θ′_{S_{0}^{−1}F} e′_{1} = (θ̃′_{S_{0}^{−1}F} −e′_{n}ω′_{n})e′_{1} . By a direct

calculation, we see that θ̃′_{S_{0}^{−1}F} and e′_{n}ω′_{n} are commutative. Since (e′_{n}ω′_{n})^{2} = 0, we have the binomial expansion

θ_{S_{0}^{−1}F}^{′(k)} e_{n}e′_{1} = (θ̃′_{S_{0}^{−1}F} − e′_{n}ω′_{n})^{(k)}e_{n}e′_{1} = θ̃_{S_{0}^{−1}F}^{′(k)} e_{n}e′_{1} − θ̃_{S_{0}^{−1}F}^{′(k−1)}e′_{n}ω′_{n}e_{n}e′_{1}.

Here, the second term belongs to the subspace Λ_{2n} ⊗ U(𝔤)𝔫_{+} , so that we have

(7.2)  γ(θ_{S_{0}^{−1}F}^{(k)} θ_{S_{0}^{−1}F}^{′(k)} τ_{S_{0}^{−1}}^{(n−2k−1)}e_{n}e′_{1}) = γ(θ_{S_{0}^{−1}F}^{(k)} θ̃_{S_{0}^{−1}F}^{′(k)} τ_{S_{0}^{−1}}^{(n−2k−1)}e_{n}e′_{1}).

To complete the proof, we consider another binomial expansion

θ̃_{S_{0}^{−1}F}^{′(k)} τ_{S_{0}^{−1}}^{(n−2k−1)}e_{n}e′_{1} = θ̃_{S_{0}^{−1}F}^{′(k)} (τ̃_{S_{0}^{−1}} + e_{1}e′_{n})^{(n−2k−1)}e_{n}e′_{1}
= θ̃_{S_{0}^{−1}F}^{′(k)} τ̃_{S_{0}^{−1}}^{(n−2k−1)}e_{n}e′_{1} + θ̃_{S_{0}^{−1}F}^{′(k)} τ̃_{S_{0}^{−1}}^{(n−2k−2)}e_{1}e′_{n}e_{n}e′_{1}.

Here, the first term in the last line vanishes, because this is of degree n in the n − 1 variables e′_{1}, …, e′_{n−1} . Then, since θ_{S_{0}^{−1}F} e_{1}e_{n} = θ̃_{S_{0}^{−1}F} e_{1}e_{n}, we have

θ_{S_{0}^{−1}F}^{(k)} θ̃_{S_{0}^{−1}F}^{′(k)} τ_{S_{0}^{−1}}^{(n−2k−1)}e_{n}e′_{1} = θ̃_{S_{0}^{−1}F}^{(k)} θ̃_{S_{0}^{−1}F}^{′(k)} τ̃_{S_{0}^{−1}}^{(n−2k−2)}e_{1}e′_{n}e_{n}e′_{1}.

From this and (7.2), we have (7.1) as desired.

Proof of Theorem 7.2. The following is the key equality of the proof:

(7.3) \begin{matrix}det(S_{0}^{−1})γ(D_{F} (λ)) \\ = γ((Y_{S_{0}^{−1}F} (λ))_{n1}) (−F_{11} − (\frac{n}{2} − 1) + λ) (−F_{11} − (\frac{n}{2} − 1) − λ) . \end{matrix}

In the case of n = 2m, this is directly obtained by applying γ to the (n, n)- entry of the both sides of the equality in Proposition 6.5 as in the proof of Proposition 7.1. In the case of n = 2m+1, we can similarly deduce the following from Proposition 6.6:

det(S_{0}^{−1})(−F_{11} − m)γ(D_{F} (λ))

= (−F_{11} − m)γ((Y_{S_{0}^{−1}F} (λ))_{n1}) (−F_{11} − (\frac{n}{2} − 1) + λ) (−F_{11} − (\frac{n}{2} − 1) − λ) .

This leads us to (7.3) again. In fact, we can remove the factor (−F_{11} − m) from the both sides, because U(𝔬(S_{0})) is an integral domain.

Since γ((Y_{S_{0}^{−1}F} (λ))_{n1}) is equal to det(S̃_{0}^{−1})γ(DF̃(λ)) by Lemma 7.3, applying T_{−ρ} to (7.3), we have

det(S_{0}^{−1})\overline{γ}(D_{F} (λ)) = det(S̃_{0}^{−1})\overline{γ}(DF̃(λ))(−F_{11} + λ)(−F_{11} − λ).

Thus our task is reduced to the calculation of \overline{γ}(DF̃(λ)). The theorem is proved by induction.

Finally, we comment the relation between our D_{F} (λ) and the Sklyanin determinant in the twisted Yangians. In [10], a central element C(λ) in U(𝔬(S_{0})) is given with the Sklyanin determinant. The image of this C(λ) under the Harish-Chandra homomorphism is equal to the image of D_{F} (λ) given in Theorem 7.2, as is calculated in [10]. This implies that our D_{F} (λ) = Ad(s^{−1})D(λ) is identical to this C(λ). Thus, we obtain relations among the three type of determinants, i.e., the column-, symmetrized and Sklyanin determinants.

Remark. The fact that our D_{F} (λ) is identical to C(λ) is seen also from the following equality, which is obtained by applying the isomorphism Ad(s^{−1}) to (2.5):

det(S_{0}^{−1}) ∑_{|I|=2k} Det(F_{I} ; k, k − 1, …, −k + 1) = ∑_{|I|=2k} Pf((S_{0}^{−1}F)_{I})^{2}.

In fact, the coefficients of C(λ) have the same expressions [13].

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Department of Mathematics Faculty of Science Kyoto University Kyoto 606-8502, Japan minoru@kusm.kyoto-u.ac.jp

Received February 28, 2000