ORTHOGONAL POLYNOMIALS OF DISCRETE VARIABLE AND LIE ALGEBRAS OF COMPLEX SIZE MATRICES

DIMITRY LEITES, ALEXANDER SERGEEV^{1}

Abstract. We uniformly interprete the classical continuous Tchebyshev’s and Hahn’s orthogonal polynomials of discrete variable in terms of 𝔤𝔩(λ) for λ ∈ ℂ. One can similarly interprete Hahn’s q-polynomials and orthogonal polynomials corresponding to Lie superlagebras. We also describe quasifinite modules over 𝔤𝔩(λ) and its real forms.

§1. Introduction

Among Saveliev-Vershik’s continum Lie algebras, 𝔤𝔩(λ) is one of the first examples. We observed that the definition and properties of the continuous Tchebyshev’s and Hahn’s orthogonal polynomials of discrete variable (and their q-analogs) follow from the properties of the Casimir elements for 𝔤𝔩(λ) (and its q-analog). Our starting point was the attempt to calculate explicitely the value of the quadratic Casimir operator on quasifinite modules — a source of orthogonality relations and various identities the polynomials satisfy. (For a more detailed than ours description of quasifinite modules of level 1 see [Sh].)

Summary. In 𝔰𝔩(2), consider the basis Y, H, X with the commutation relations

[X, Y] = H, [H, Y] = −2Y, [H, X] = 2X.  (1.1)

Then the quadratic Casimir operator

Ω = 2Y X + \frac{1}{2}H^{2} + H  (1.2)

lies in the center of U(𝔰𝔩(2)). Recall the definition of the Lie algebra 𝔤𝔩(λ) introduced by Feigin in [F]. Let I_{λ} be the two-sided ideal in the associative algebra U(𝔰𝔩(2)) generated by Ω − \frac{1}{2}(λ^{2} − 1). It turns out that the associative algebra 𝔄̃_{λ} = U(𝔰𝔩(2))/I_{λ} is simple for λ ∉ ℤ \ {0}, otherwise 𝔄̃_{λ} contains an ideal such that the quotient is isomorphic to the matrix algebra Mat(|λ|). Set ([Di1])

𝔄_{λ} = \begin{cases}𝔄̃_{λ} & if λ ∉ ℤ \ \{0\} \\ Mat(|λ|) & otherwise. \end{cases}  (1.3)

As associative algebra with unit, 𝔄_{λ} is generated by X, Y and H subject to the relations (1.1) and

XY = \frac{1}{4}(n^{2} − (H − 1)^{2}) and X^{|λ|} = 0 if λ ∈ ℤ \ \{0\}.  (1.4)

In what follows we set 𝔤𝔩(λ) := L(𝔄_{λ}), the Lie algebra associated with the associative algebra 𝔄_{λ}.

1991 Mathematics Subject Classification. 33C45, 17B01, 17A70 (Primary) 17B35, 17B66 (Secondary).

Key words and phrases. Trace, matrices of complex size, orthogonal polynomials.

We are thankful to NFR for financial support, M. Vasililev for a stimulating question, P. Grozman and, especially, V. Gerdt for help (sec.9.3).

2 DIMITRY LEITES, ALEXANDER SERGEEV^{1}

Consider the principal embedding of 𝔰𝔩(2) into 𝔊 = 𝔤𝔩(n) (it corresponds to the n-dimensional irreducible 𝔰𝔩(2)-module):

Y ↦ ∑_{i=1}^{n} E_{i+1,i}, H ↦ ∑_{i=1}^{n}(n − 2i + 1)E_{ii}, X ↦ ∑_{i=1}^{n} i(n − i)E_{i,i+1}.  (1.5)

Then 𝔊 considered as an 𝔰𝔩(2)-module with respect to the principal embedding (1.5) is of the

form 𝔊 = ⊕_{|i|≤n−1} 𝔊_{i}, where 𝔊_{i} = {X^{i}𝔊_{0}} and 𝔊_{−i} = {𝔊_{0}Y^{i}} for i > 0 and 𝔊_{0} = ℂ[H]/(P_{n}(H)),

where P_{n}(H) = ∏_{1≤i≤n−1}(H − n + 2i − 1), is the Cartan subalgebra.

The invariant functional, trace, on 𝔊 gives rise to the nondegenerate bilinear form (A, B) = tr AB. The homogeneous components 𝔊_{i} — the space of matrices with support on the ith over- (for i > 0) and under-diagonals (for i < 0) — are orthogonal with respect to this form and the form is nondegenerate on 𝔊_{−i} ⊕ 𝔊_{i} and on 𝔊_{0}. Reduction of the form to the canonical form on these spaces leads to the classical Hahn and Tchebyshev polynomials of discrete variable, respectively. By expressing the Casimir elements — the generators of the center of U(𝔊) — in terms of these polynomials in the standard basis one gets all the identities the polynomials satisfy. By applying this approach to 𝔤𝔩(λ) we clarify the known results on the co-called continuous versions of the classical Hahn and Tchebyshev polynomials of discrete variable, cf. [NU], [NSU].

By replacing in (1.3) the universal enveloping of 𝔰𝔩(2) with its q-deformation we get q-versions of the classical polynomials, identities, etc. The details will be given elsewhere. In another paper we also consider superprincipal embeddings of 𝔬𝔰𝔭(1|2) — the closest relative of 𝔰𝔩(2) among Lie superalgebras — and the corresponding orthogonal polynomials with respect to the nondegenerate indefinite bilinear form, the supertrace. To replace 𝔰𝔩(2) with ANY Lie (super)algebra or Lie superalgebra 𝔥 in the above scheme is also possible provided 𝔥 possesses Verma modules with finitely many generators and several finite dimensional representations. The theory seems to be reachest if 𝔥 is simple or “close” to simple (nontrivial central extension, the derivation algebra, etc.). Such generalizations of 𝔤𝔩(λ) appear in spin > 2 models, see [V]. The description of 𝔤𝔩(λ)-modules in §§5–7 is our answer to Vasiliev’s question.

§2. The trace formula

2.1. Lemma . Let A be an associative algebra generated by the Lie algebra 𝔤, conisdered as a subspace. Then [A, A] = [𝔤, A].

Proof. It suffices to show that [x_{1} … x_{n}, a] ∈ [𝔤, A] for any a ∈ A and x_{1}, …, x_{n} ∈ 𝔤. The induction on n: for n = 1 the statement is obvious. Let n > 1. Let us make use of the identity [ab, c] = [a, bc] + [b, ca] true in any associative algebra, cf. [M]. Then

[x_{1}(x_{2} … x_{n}), a] = [x_{1}, x_{2} … x_{n}a] + [x_{2} … x_{n}, ax_{1}].

By inductive hypothesis, the second summand in the right hand side lies in [𝔤, A].

2.1.1. Corollary . ([Di]) Let 𝔤 be a simple Lie algebra. Then U(𝔤) = Z(U(𝔤)) ⊕ [U(𝔤), U(𝔤)].

Proof. Let U(𝔤) = Z(U(𝔤)) ⊕ ( ⊕_{λ≠0} n_{λ}L^{λ}) be the decomposition with respect to the adjoint repre-

sentation. Since the modules L^{λ} are irreducible, [𝔤, U(𝔤)] = ( ⊕_{λ≠0} n_{λ}L^{λ}).

2.1.2. Corollary . Let 𝔥 be a Cartan subalgebra in 𝔤, which is either a simple finite dimensional Lie algebra; let λ ∈ 𝔥^{∗} and M^{λ−ρ} the corresponding Verma module; let J_{λ} be the (left) ideal of U(𝔤) corresponding to the kernel of the representation of U(𝔤) in M^{λ−ρ}. Set 𝔄_{λ} = U(𝔤)/J_{λ}. Then 𝔄_{λ} = ℂ ⊕ [𝔄_{λ}, 𝔄_{λ}].

Proof. By Lemma 2.1 [𝔄_{λ}, 𝔄_{λ}] = [𝔤, 𝔄_{λ}]; hence, as in Corollary 2.1.1, 𝔄_{λ} = Z(𝔄_{λ}) ⊕ [𝔄_{λ}, 𝔄_{λ}]. But Z(𝔄_{λ}) is a homomorphic image of Z(U(𝔤)); hence, is equal to ℂ.

If 𝔤 is a simple Lie algebra and λ is its highest weight, then P (λ) = dim L^{λ} is a polynomial in λ = (λ_{1}, …, λ_{n}). Consider P (λ) for any λ ∈ 𝔥^{∗} such that ∏_{i≠j}(λ_{i} − λ_{j}) ≠ 0. By Corollary 2.1.2,

there is a unique functional tr on 𝔄_{λ} such that tr(ab) = tr(ba) and tr(1) = P (λ).

2.2. Lemma . Let u ∈ U(𝔤), ũ ∈ 𝔄_{λ} its image. Then tr(ũ) is also a polynomial in λ.

Proof. Let ♯ : Z(U(𝔤)) ⊕ [U(𝔤), U(𝔤)] ⟶ Z(U(𝔤)) be the natural projection and˜ : U(𝔤) ⟶ 𝔄_{λ} be given by the formula u ↦ ũ. By definition of ♯ we have:

tr(ũ) = χ_{λ}(u^{♯})P (λ) = φ(u^{♯})(λ)P (λ),  (2.2)

where φ : Z(U(𝔤)) ⟶ ℂ[𝔥] is the Harish-Chandra homomorphism.

2.3. Theorem . Let us identify the space (ℂ[𝔥])^{∗} with the algebra of formal power series on indeterminates t = (t_{1}, …, t_{n}) by the following formula:

f ↦ ∑_{ν=(ν_{1},...,ν_{n})} f (\frac{h_{1}^{ν_{1}}}{ν_{1}!} … \frac{h_{n}^{ν_{n}}}{ν_{n}!} ) t_{1}^{ν_{1}} … t_{n}^{ν_{n}},  (2.3.1)

where ν_{i} ∈ ℤ_{+} for all i and h_{1}, …, h_{n} is a basis of 𝔥.

Then to the functional tr there corresponds the series

ψ(λ, t) = \frac{⊕_{w∈W} ε(w)e^{w(λ)}}{⊕_{w∈W} ε(w)e^{w(ρ)}} (t),  (2.3.2)

where ρ is the half-sum of positive roots and e^{λ}(t) = e^{λ_{1}t_{1}+···+λ_{n}t_{n}}. (Recall that the λ_{i} are supposed to be distinct.)

Proof. By Lemma 1.2 tr (\frac{h_{1}^{ν_{1}}}{ν_{1}!} … \frac{h_{n}^{νn}}{ν_{n}!} ) is a polynomial in λ. If λ ∈ P_{++} (the set of highest weights of finite dimensional modules), then

ψ̂(λ, t) = ∑_{ν=(ν_{1},...,ν_{n})} tr (\frac{h_{1}^{ν_{1}}}{ν_{1}!} … \frac{h_{n}^{ν_{n}}}{ν_{n}!} ) t_{1}^{ν_{1}} … t_{n}^{ν_{n}}

coinsides with (2.3.2) thanks to the Weyl’s character formula. But ψ(λ, t) = ∑_{ν} P_{ν}t^{ν}, where P_{ν} are

some polynomials. Indeed, since P_{ν}(λ = tr (\frac{h_{1}^{ν_{1}}}{ν_{1}!} … \frac{h_{n}^{νn}}{ν_{n}!} ) for λ ∈ P_{++}, this is true for any λ ∈ 𝔥^{∗}. Hence, ψ̂(λ, t) = ψ(λ, t).

4 DIMITRY LEITES, ALEXANDER SERGEEV^{1}

§3. Highest weights of quasifinite modules over 𝔤𝔩(λ)

Let L^{i} be the irreducible 𝔰𝔩(2)-module with highest weight i. The Lie algebra 𝔤𝔩(λ) considered as 𝔰𝔩(2)-module with respect to the adjoint action of the latter is of the form

𝔤𝔩(λ) = \begin{cases}L^{0} ⊕ L^{2} ⊕ … L^{2|λ|−2} for λ ∈ ℤ \ \{0\} \\ L^{0} ⊕ L^{2} ⊕ … otherwise. \end{cases}  (3.0.1)

The Lie algebra 𝔤𝔩(λ) possesses a ℤ-grading 𝔤𝔩(λ) = ⊕_{i∈ℤ}𝔤𝔩(λ)_{i}, where

𝔤𝔩(λ)_{i} = {z ∈ 𝔤𝔩(λ) : [H, z] = 2iz}  (3.0.2)

and the increasing filtration: 𝔤𝔩(λ)_{(i)} = ⊕_{k≤i} 𝔤𝔩(λ)_{k}. Therefore, we can speak about the triangular decomposition 𝔤 = 𝔤_{−} ⊕ 𝔤_{0} ⊕ 𝔤_{+} of 𝔤 = 𝔤𝔩(λ), where 𝔤_{±} = ⊕_{i≥0} 𝔤𝔩(λ)_{±i}, parabolic subalgebras, highest weights, etc., see [KR].

Following [KR] we say that the 𝔤-module V is quasifinite if V = ⊕_{j∈ℤ}V_{j} and dim V_{j} < ∞ for every j. (We only consider graded 𝔤-modules, i.e., such that 𝔤_{i}V_{j} ⊂ V_{i+j}.)

The well-known fact that 𝔤𝔩(n) has a nontrivial one-dimensional representation (A ↦ tr A) has its counterpart for 𝔤𝔩(λ).

A subalgebra 𝔭 ⊂ 𝔤 is called a parabolic one if it contains 𝔤_{0} ⊕ 𝔤_{+} as a proper subalgebra. For example, for any P ∈ ℂ[H] set 𝔭_{−1} = P ℂ[H]Y and let 𝔭 be generated by 𝔭_{−1} and 𝔤_{0} ⊕ 𝔤_{+}. This is the minimal parabolic subalgebra corresponding to P.

3.1. Lemma . The minimal parabolic subalgebra 𝔭 (corresponding to a polynomial P) is of the form 𝔭 = ⊕_{k∈ℤ}𝔭_{k}, where 𝔭_{k} = 𝔤_{k} for k ≥ 0 and 𝔭_{−k} = I_{k} · Y^{k}, where I_{k} is the ideal of ℂ[H] generated by P (H)P (H + 2) … P (H + 2k − 2).

Proof. See [KR], mutatis mutandis.

Let λ ∈ 𝔤_{0}^{∗}, 𝔤_{+}v_{λ} = 0 and M^{λ} = ind_{𝔤_{0}⊕𝔤_{+}}^{𝔤}(ℂv_{λ}) the corresponding Verma module; let L^{λ} be the irreducible 𝔤-module with highest weight λ.

3.2. Lemma . ([KR]) The following conditions are equivalent:

i) M^{λ} contains a singular vector from M_{(−1)}^{λ}, the space of filtration −1;

ii) The module L^{λ} is quasifinite;

iii) L^{λ} is a quotient of the generalized Verma module M^{λ,P} = ind_{𝔭}^{𝔤}(ℂ), where P ∈ ℂ[H] and 𝔭 is the minimal parabolic subalgebra corresponding to P.

Shoikhet calls deg P the level of M^{λ,P}. In [Sh] he described modules of level 1 in more details than we give here.

We have already established that 𝔤_{0}^{∗} ≃ (ℂ[H])^{∗} ≃ ℂ[[t]] with the algebra isomorphism established by the rule

F : (ℂ[H])^{∗} ⟶ ℂ[[t]], θ ↦ F_{θ}(t) = ∑_{k=0}^{∞} \frac{θ(H^{k})}{k!}t^{k}.

The following theorem describes the set of formal power series corresponding to the highest weights of quasifinite modules. Recall that a quasipolynomial is an expression of the form ∑ R_{i}(t)e^{α_{i}t}, where the R_{i}(t) are polynomials.)

Theorem . The formal power series corresponding to the highest weight of any quasifinite module over 𝔤𝔩(λ) is of the form \frac{R(t)}{1 − e^{−2t}}, where in R(t) is a quasipolynomial such that R(0) = 0.

Proof. First, observe the following easy to verify statements: if θ ∈ (ℂ[H])^{∗} and F_{θ}(t) = ∑_{k=0}^{∞} \frac{θ(H^{k})}{k!}t^{k} the corresponding series, then

∑_{k=0}^{∞} \frac{θ((H + a)^{k})}{k!}t^{k} = e^{at}F_{θ}(t);

and for any R(H) ∈ ℂ[H] we have

∑_{k=0}^{∞} \frac{θ(R(H)H^{k})}{k!}t^{k} = R(\frac{d}{dt})F_{θ}(t) and ∑_{k=0}^{∞} \frac{θ(R(H + a)(H + a)^{k})}{k!}t^{k} = R(\frac{d}{dt})(e^{at}F_{θ}(t)).

If θ is the highest weight of a quasifinite module over 𝔤𝔩(λ), then by Lemma 3.2 there exists a polynomial P (H) ∈ ℂ[H] such that θ can be extended to a one-dimensional representation of the minimal parabolic subalgebra corresponding to P. It is not difficult to verify that [𝔭, 𝔭] ∩ 𝔤_{0} = [𝔤_{1}, 𝔭_{−1}]. Hence, θ([𝔤_{1}, 𝔭_{−1}]) = 0.

Denote:

T (H) = XY = \frac{1}{4}(λ^{2} − (H + 1))^{2}.

Then the condition θ([𝔤_{1}, 𝔭_{−1}]) = 0 can be expressed as θ([X, P (H)H^{k}Y]) = 0, or, equivalently, as θ(XP (H)H^{k}Y − P (H)H^{k}Y X]) = 0, or, as

θ(T (H − 2)P (H − 2)(H − 2)^{k} − P (H)H^{k}T (H)) = 0.

Therefore,

∑_{k=0}^{∞} \frac{θ(T (H − 2)P (H − 2)(H − 2)^{k})}{k!}t^{k} − ∑_{k=0}^{∞} \frac{θ(T (H)P (H)(H)^{k})}{k!}t^{k} =

T (\frac{d}{dt})P (\frac{d}{dt})(e^{−2t}F_{θ}(t)) − T (\frac{d}{dt})P (\frac{d}{dt})(F_{θ}(t)) =

T (\frac{d}{dt})P (\frac{d}{dt}) ((e^{−2t} − 1)F_{θ}(t)) = 0.

Thus, the function (e^{−2t} − 1)F_{θ}(t) is a solution of an ordinary differential equation with constant coefficients, hence, is a quasipolynomial. Obviously, R(0) = 0.

Conversely, if R(t) is a quasipolynomial, R(0) = 0 and P (\frac{d}{dt})R(t) = 0, then T (\frac{d}{dt})P (\frac{d}{dt})R(t) = 0

and, by setting F_{θ}(t) = \frac{R(t)}{e^{−2t} − 1} (which is well defined since R(0) = 0) we satisfy the condition θ([𝔤_{1}, 𝔭_{−1}]) = 0 for 𝔭_{−1} = P (H)𝔤_{0}Y. Hence, θ is the highest weight of a quasifinite module.

3.3. The trace on 𝔤𝔩(0). If P = 1, then the parabolic subalgebra of 𝔤 = 𝔤𝔩(0) coincides with the whole algebra and, therefore, [𝔤, 𝔤]∩𝔤_{0} = [𝔤_{1}, 𝔤_{−1}]∩𝔤_{0} ∌ 1. This proves that 𝔤 ≠ [𝔤, 𝔤] and, therefore, proves also existence of an invariant functional θ on 𝔤 = 𝔤𝔩(λ). Therefore, R = (1 − e^{−2t})F_{θ}(t) satisfies the equation T (\frac{d}{dt})R(t) = 0, more exactly,

(λ^{2} − (\frac{d}{dt} + 1)^{2})R(t) = 0.

If λ ≠ 0, then solutions are of the form R(t) = c_{1}e^{(λ−1)t} + c_{2}e^{−(λ+1)t} and the initial condition R(0) = 0 implies that

F_{θ}(t) = c\frac{e^{(λ−1)t} − e^{−(λ+1)t}}{1 − e^{−2t}} .

6 DIMITRY LEITES, ALEXANDER SERGEEV^{1}

If we norm the functional θ (trace) naturally, i.e., by setting θ(1) = λ, then c = 1.

Clearly, if λ = 0, then the characteristic equation has multiple roots and

F_{θ}(t) = c\frac{te^{−t}}{1 − e^{−2t}} .

Thus, 𝔰𝔩(0) is an infinite dimensional Lie algebra with an ∞-dimensional identity module.

§4. The Lie algebras 𝔤𝔩(λ) and 𝔤𝔩^{−}(∞), 𝔤𝔩^{+}(∞), 𝔤𝔩(∞)

Let V be a vector space with a fixed basis v_{i} for i ∈ ℤ, let V^{+} be its subspace spanned by v_{i} for i ≥ 0 and V^{−} its subspace spanned by v_{i} for i < 0. Let 𝔤𝔩^{−}(∞), 𝔤𝔩^{+}(∞), and 𝔤𝔩(∞) be the Lie algebras of linear transformations of V^{−}, V^{+} and V, respectively, whose matrices in the fixed bases are supported on a finite number (depending on the matrix) of diagonals parallel to the main one.

On 𝔤𝔩(∞), there is a (unique) nontrivial 2-cocycle

c(A, B) = tr ([J, A]B) where J = ∑_{i≤0} E_{ii} − ∑_{i>0} E_{ii}.  (4.0)

Denote the corresponding central extension by 𝔤̂𝔩(∞); the bracket in 𝔤̂𝔩(∞) is of the form

[A, B] = AB − BA + tr ([J, A]B) · z, where z is the new central element.

In particular, for matrix units we have

[E_{ij}, E_{kl}] = δ_{jk}E_{il} − δ_{il}E_{kj} + δ_{il}δ_{jk}(κ(i) − κ(j)) · z,

where κ(i) = 1 if i ≤ 0 and 0 otherwise.

4.0. Harish-Chandra modules over 𝔰𝔩(2). Harish-Chandra modules M^{λ,s} = Span(v_{i} : i ∈ ℤ) for any λ, s ∈ ℂ are defined to satisfy the following relations

\begin{matrix}Hv_{i} = (s + 2i)v_{i} \\ Xv_{i} = \frac{1}{2} \sqrt{(λ − s − 2i − 1)(λ + s + 2i + 1)}v_{i+1} \\ Y v_{i} = \frac{1}{2} \sqrt{(λ − s − 2i + 1)(λ + s + 2i − 1)}v_{i−1} \end{matrix}  (4.0.1)

It is not difficult to observe that Ω acts on M^{λ,s} as a scalar operator of multiplication by \frac{1}{2}(λ^{2}−1) and M^{λ,s} ≃ M^{λ,s′} if s − s′ ∈ 2ℤ.

Suppose λ ∈ ℤ \ {0}. Then by [Di] if λ − s ∉ 2ℤ + 1, then M^{λ,s} is irreducible, whereas if λ − s ∈ 2ℤ + 1, then M^{λ,s} = M_{+}^{λ,s} ⊕ M_{−}^{λ,s}, where M_{+}^{λ,s} is an irreducible 𝔰𝔩(2)-module with the lowest weight λ + 1 and M_{−}^{λ,s} is an irreducible 𝔰𝔩(2)-module with the highest weight λ − 1 and these module exhaust all irreducible 𝔰𝔩(2)-modules with a diagonal H-action on which Ω acts as a scalar operator of multiplication by \frac{1}{2}(λ^{2} − 1).

The module M^{λ,s} over 𝔤𝔩(λ) determines an embedding ϕ : 𝔤𝔩(λ) ⟶ 𝔤𝔩(∞). Since H^{2}(𝔤𝔩(λ)) = 0 (see [F]), there exists a lift of the embedding ϕ to a homomorphism ϕ̂ : 𝔤𝔩(λ) ⟶ 𝔤̂𝔩(∞) which is of the form ϕ̂(u) = ϕ(u) + θ(u) · z, where θ is a linear functional on 𝔤𝔩(λ) such that θ([u, v]) = c(ϕ(u), ϕ(v)) for the cocycle on 𝔤𝔩(∞) which makes it into 𝔤̂𝔩(∞). Observe that θ is determined uniquely up to a functional proportional to the trace on 𝔤𝔩(λ).

The following theorem describes ϕ̂ more explicitely.

4.1. Theorem . Let 𝔤 = 𝔤𝔩(λ). Then ϕ̂(𝔤_{i}) = ϕ(𝔤_{i}) for i ≠ 0.

ϕ̂(e^{tH}) = ϕ(e^{tH}) − \frac{e^{st} − e^{−(λ+1)t}}{1 − e^{−2t}} z.  (4.1.1)

Proof.

ϕ(X) = ∑ α_{i}E_{i+1,i}, where α_{i} = \frac{1}{2} \sqrt{(λ − s − 2i − 1)(λ + s + 2i + 1)},
ϕ(H) = ∑ γ_{i}E_{i,i}, where γ_{i} = s + 2i,
ϕ(Y) = ∑ β_{i}E_{i−1,i}, where β_{i} = \frac{1}{2} \sqrt{(λ − s − 2i + 1)(λ + s + 2i − 1)}.

(4.1.2)

Therefore,

[J, ϕ(X)] = ∑(κ(i) − κ(i + 1))α_{i}E_{i+1,i}

and ϕ(H^{k})ϕ(Y) = ∑ γ_{i}^{k}β_{i+1}E_{i,i+1}. Hence,

[J, ϕ(X)]ϕ(H^{k})ϕ(Y) = ∑ γ_{i}^{k}β_{i+1}α_{i}(κ(i) − κ(i + 1))E_{i+1,i+1}

and

\begin{matrix}c(ϕ(X), ϕ(H^{k}Y)) = tr([J, ϕ(X)]ϕ(H^{k}Y)) = ∑ γ_{i}^{k}β_{i+1}α_{i}(κ(i) − κ(i + 1)) = \\ γ_{0}^{k}β_{1}α_{0} = \frac{1}{4}s^{k}(λ^{2} − (s + 1)^{2}) = T (s)s^{k}. \end{matrix}

Therefore,

T (\frac{d}{dt})(e^{st}) = T (s)e^{st} = ∑_{k≥0} \frac{T (s)s^{k}}{k!} t^{k} =

∑_{k≥0} \frac{c(ϕ(X), ϕ(H^{k}Y))}{k!}t^{k} =

∑_{k≥0} \frac{θ([X, H^{k}Y])}{k!}t^{k} = ∑_{k≥0} \frac{θ(XH^{k}Y − H^{k}Y X)}{k!}t^{k} =

∑_{k≥0} \frac{θ(T (H − 2)(H − 2)^{k} − H^{k}T (H))}{k!}t^{k} =

T (\frac{d}{dt}) ((e^{−2t} − 1)θ(e^{tH})) .

Therefore, T (\frac{d}{dt}) ((e^{−2t} − 1)θ(e^{tH}) − e^{st})) = 0 and

(e^{−2t} − 1)θ(e^{tH}) = \begin{cases}e^{st} + c_{1}e^{−(λ+1)t} + c_{2}e^{(λ−1)t} & if λ ≠ 0 \\ e^{st} + c_{1}e^{−t} + c_{2}te^{−t} & if λ = 0. \end{cases}

In both cases, subtracting the summand proportional to the trace we get

(e^{−2t} − 1)θ(e^{tH}) = e^{st} − e^{−(λ+1)t}.

4.2. 𝔤𝔩(∞) over the algebra of truncated polynomials. Let R_{m} = ℂ[ε], where ε^{m+1} = 0. By extending the module M^{λ,s} to a module M_{R_{m}}^{λ,s} over R_{m}, i.e., by assuming that s ∈ R_{m} we may define the action of 𝔰𝔩(2) by the same formulas. If the v_{i} is the initial basis, then for the fixed basis of the extended module M_{R_{m}}^{λ,s} we take ε^{j}v_{i} for all i ∈ ℤ and j = 1, …, m.

Observe that in this basis of M_{R_{m}}^{λ,s} the action of H is not a diagonal one.

The algebras 𝔤𝔩(∞; R_{m}), its subalgebras 𝔤𝔩^{±}(∞; R_{m}) are naturally defined and the central extension 𝔤̂𝔩(∞; R_{m}) is determined by the same formula.

By considering s + ε instead of s ∈ ℂ, we get the following

8 DIMITRY LEITES, ALEXANDER SERGEEV^{1}

Corollary . The homomorphism ϕ̂ : 𝔤𝔩(λ) ⟶ 𝔤̂𝔩(∞; R_{m}) induced by the embedding ϕ : 𝔤𝔩(λ) ⟶ 𝔤𝔩(∞; R_{m}) satisfies ϕ̂(𝔤𝔩(λ)_{i}) = ϕ(𝔤𝔩(λ)_{i}) for i ≠ 0 and

ϕ̂(e^{tH}) = ϕ(e^{tH}) − (\frac{e^{st} − e^{−(λ+1)t}}{1 − e^{−2t}} + \frac{e^{st}}{1 − e^{−2t}} ∑_{j=1}^{m} \frac{ε^{j}t^{j}}{j!} ) z.  (4.2)

4.3. Remark . We are “forced” to consider the central extension 𝔤̂𝔩(∞) in order to describe quasifinite modules over 𝔤𝔩(λ) because 𝔤̂𝔩(∞) has more quasifinite representations than 𝔤𝔩(∞) and it turns out to be insufficient to consider only 𝔤𝔩(∞)-modules.

The following statement describes the restrictions onto the higheset weight of an irreducible module necessary and sufficient for the module to be quasifinite.

4.4. The highest weights of quasifinite modules. Denote by 𝔤𝔩_{f}(∞) and 𝔤𝔩_{f}^{±}(∞) the subalgebras of 𝔤𝔩(∞) and 𝔤𝔩^{±}(∞), respectively, consisting of matrices with finite support. Let 𝔥 denote the Cartan subalgebra of any of these Lie algebras; consider it spanned by the diagonal elements E_{ii}. Observe that the restriction of the embedding ϕ̂ : 𝔤𝔩_{f}(∞) ⟶ 𝔤̂𝔩(∞) onto 𝔥 is of the form E_{ii} ↦ ϕ(E_{ii}) + κ(i)z, where ϕ is the natural embedding of 𝔤𝔩_{f}(∞) into 𝔤𝔩(∞). If θ is a weight of a (highest weight) module over one of the above Lie algebras, both 𝔤𝔩(∞) and its subalgebras and 𝔤𝔩_{f}(∞) and its subalgebras, we may, considering 𝔥 in the above way, determine the coordinates θ_{i} = θ(E_{ii}) of θ.

4.4.1. Proposition . The 𝔤𝔩_{f}(∞)-module with highsest weight Λ is quasifinite if and only if there are only finitely many distinct coordinates λ_{i}, where λ_{i} = Λ(E_{ii}).

Proof. If V is quasifinite and generated by the highest weight vector v. Since dim V_{−1} < ∞, it follows that V_{−1} = Span(E_{i+1,i}v)_{i∈I} for a finite set I. Let k ∉ I and k + 1 ∉ I. Then

E_{k+1,k}v = ∑_{i∈I} α_{i}E_{i+1,i}v, so E_{k,k+1}(E_{k+1,k}v) = 0 or, equivalently, (E_{k,k} − E_{k+1,k+1})v = 0, i.e.,

λ_{k} = λ_{k+1}.

The opposite implication is obvious.

4.4.2. Proposition . The 𝔤𝔩(∞)-module V with highsest weight Λ, where λ_{i} = λ(E_{ii}), is quasifinite if and only if there are only finitely many nonzero coordinates λ_{i}, where λ_{i} = Λ(E_{ii}).

Proof. By considering V as 𝔤𝔩_{f}(∞)-module we deduce that there are only finitely many distinct coordinates λ_{i}, so we can set λ_{k} = λ_{+} and λ_{−k} = λ_{−} for sufficiently large k. Hence, E_{k,k+1}(E_{k+1,k}v) = 0 and, thanks to irreducibility of V, E_{k+1,k}v = 0 for all k but finitely many.

Let A = ∑_{i∈ℤ}E_{i,i+1} and B_{k} = ∑_{i≥k} E_{i+1,i}. Then

[A, B_{k}] = ∑_{i≥k} E_{i,i} − ∑_{i≥k} E_{i+1,i+1} = E_{k,k}.

Therefore,

[A, B_{k}]v = E_{k,k}v = λ_{k}v = AB_{k}v − B_{k}Av = AB_{k}v =

A( ∑_{i≥k, i∈I} E_{i+1,i}v) = ∑_{i≥k, i∈I}[A, E_{i+1,i}]v = ∑_{i≥k, i∈I}(E_{i,i} − E_{i+1,i+1})v =

( ∑_{i≥k, i∈I}(λ_{i} − λ_{i+1})) v = (λ_{k} − λ_{j})v,

where j is the largest index in the finite set I. Therefore, λ_{k} = λ_{k} − λ_{j} and λ_{j} = 0.

Similarly, if C_{k} = ∑_{i≤k} E_{i+1,i}, then

[A, C_{k}] = ∑_{i≤k} E_{i,i} − ∑_{i≤k} E_{i+1,i+1} = E_{k+1,k+1}.

Hence,

\begin{matrix}[A, C_{k}]v = λ_{k+1}v = AC_{k}v = A(∑_{i≤k} E_{i+1,i})v = ∑_{i≤k}[A, E_{i+1,i}]v \\ ∑_{i≤k}(E_{i,i} − E_{i+1,i+1})v = ∑_{i≤k}(λ_{i} − λ_{i+1})v = (λ_{j} − λ_{k+1})v. \end{matrix}

So, λ_{j} = 0 for sufficiently large j. The converse statement follows from Proposition 4.4.1.

4.4.3. Proposition . The 𝔤̂𝔩(∞)-module V with highest weight (Λ, c), where c is the value of the central charge on z, is quasifinite if and only if there are only finitely many nonzero coordinates λ_{i}, where λ_{i} = Λ(E_{ii}).

Proof. As in the proof of Proposition 4.4.2 we show that there are only finitely many distinct coordinates λ_{i}, and E_{k+1,k}v ≠ 0 for finetely many values of k.

Let A = ∑_{i∈ℤ}E_{i,i+1} and J = ∑_{i≤0} E_{i,i}. Then [A, J] = E_{01}, so c(A, B) = tr (E_{01}B) for any

B ∈ 𝔤𝔩(∞), so c(A, B_{k}) = κ(k). We have

[A, B_{k}]v = E_{k,k}v = (E_{k,k} + κ(k)z)v = AB_{k}v =

A( ∑_{i≥k, i∈I} E_{i+1,i}v) = ∑_{i≥k, i∈I}[A, E_{i+1,i}]v = ∑_{i≥k, i∈I}(E_{i,i} − E_{i+1,i+1} + δ_{i0}z)v =

κ(k)z + ( ∑_{i≥k, i∈I}(λ_{i} − λ_{i+1})) v = κ(k)cv + (λ_{k} − λ_{+})v = (κ(k)c + (λ_{k})v,

so λ_{+} = 0.

For C_{k} = ∑_{i≤k} E_{i+1,i} we have c(A, C_{k}) = κ(−k), so

[A, C_{k}]v = (−E_{k+1,k+1} − κ(−k))v =

(−λ_{k+1} − κ(−k)c)v = AC_{k}v = A ∑_{i≤k} E_{i+1,i}v = ∑_{i≤k}[A, E_{i+1,i}]v =

∑_{i≤k}(E_{i,i} − E_{i+1,i+1} + δ_{i0}z)v = (κ(−k)c + λ_{−} − λ_{k+1})v.

So, λ_{−} = 0.

4.4.4. Theorem . Let 𝔤 be one of the Lie algebras 𝔤𝔩(∞; R_{m}) or 𝔤𝔩^{±}(∞; R_{m}) or their hatted versions. The 𝔤-module V with highest weight θ given by its coordinates θ_{ij} = (ε^{j}E_{ii}) is quasifinite if and only if there are only finitely many nonzero coordinates θ_{ij}.

Proof is similar to that of 4.4.1–4.4.3.

Let V be one of the modules M_{R_{m}}^{λ,s} or, if M_{R_{m}}^{λ,s} is reducible, one of its irreducible submodules. We obtain a homomorphism of 𝔤𝔩(λ) into 𝔤 = 𝔤𝔩(∞; R_{m}) or 𝔤𝔩^{±}(∞; R_{m}). Let θ be a linear functional satisfying the conditions of Theorem 4.4.4. We may consider θ be a linear functional on the Cartan subalgebra of 𝔤𝔩(λ). Let us calculate the corresponding generating function.

10 DIMITRY LEITES, ALEXANDER SERGEEV^{1}

4.5. Theorem . The generating function F_{θ}(t) for the functional θ is as follows:

i) For M_{R_{m}}^{λ,s}:

F_{θ}(t) = \frac{∑_{i∈ℤ}e^{(s−2i)t} ∑_{j=0}^{m} \frac{(θ_{ij} − θ_{i−1,j})t^{j}}{j!}}{1 − e^{−2t}} − (\frac{e^{st} − e^{−(λ+1)t}}{1 − e^{−2t}} c + \frac{e^{st}}{1 − e^{−2t}} ∑_{j=1}^{m} \frac{c_{j}}{j!} )  (4.5.1)

ii) For the module with highest weight λ − 1:

F_{θ}(t) = \frac{∑_{i∈ℤ}e^{(λ−2i−1)t} ∑_{j=0}^{m} \frac{(θ_{ij} − θ_{i−1,j})t^{j}}{j!}}{1 − e^{−2t}}  (4.5.2)

iii) For the module with highest weight −λ − 1:

F_{θ}(t) = \frac{∑_{i∈ℤ}e^{(−λ−2i−1)t} ∑_{j=0}^{m} \frac{(θ_{ij} − θ_{i−1,j})t^{j}}{j!}}{1 − e^{−2t}}  (4.5.3)

iv) For the module with lowest weight λ + 1:

F_{θ}(t) = \frac{∑_{i∈ℤ}e^{(λ+2i+1)t} ∑_{j=0}^{m} \frac{(θ_{ij} − θ_{i−1,j})t^{j}}{j!}}{1 − e^{−2t}}  (4.5.4)

v) For the module with lowest weight 1 − λ:

F_{θ}(t) = \frac{∑_{i∈ℤ}e^{(1−λ+2i)t} ∑_{j=0}^{m} \frac{(θ_{ij} − θ_{i−1,j})t^{j}}{j!}}{1 − e^{−2t}}  (4.5.5)

Proof. We will only prove i): the other cases are similar.

Let ϕ be the homomorphism that determines the 𝔤𝔩(λ)-action on M^{λ,s}. Then ϕ(H) = ∑_{i∈ℤ}(s +

ε − 2i)E_{ii}, so ϕ(e^{tH}) = ∑_{i∈ℤ}e^{(s+ε−2i)t}E_{ii} and

\begin{matrix}ϕ((1 − e^{−2t})e^{tH}) = ϕ(e^{tH}) − ϕ(e^{t(H−2)}) = ∑_{i∈ℤ}e^{(s+ε−2i)t}(E_{ii} − E_{i−1,i−1}) = \\ ∑_{i∈ℤ}e^{(s−2i)t} ∑_{j=0}^{m} ε^{j}(E_{ii} − E_{i−1,i−1})\frac{t^{j}}{j!}. \end{matrix}

Therefore,

\begin{matrix}(1 − e^{−2t})F_{θ}(t) = θ(ϕ(e^{tH}) − ϕ(e^{t(H−2)})) = \\ ∑_{i∈ℤ}e^{(s−2i)t} ∑_{j=0}^{m} \frac{θ_{ij} − θ_{i−1,j}}{j!} t^{j}. \end{matrix}

Since only finitely many coordinates θ_{ij} are nonzero, all the sums are, actually, finite.

§5. Quasifinite modules over 𝔤𝔩(λ)

In what follows we will show that not all quasifinite modules over 𝔤𝔩(λ) can be represented in a canonical form similar to that of W_{1+∞} from [KR].

Let 𝔤𝔩^{hol}(λ) be the holomorphic completion of 𝔤𝔩(λ), i.e., 𝔤𝔩_{0}^{hol}(λ) is the algebra of functions in H holomorphic on the whole complex line ℂ and 𝔤𝔩_{i}^{hol}(λ) = {f(H)X^{i}} for i > 0 whereas 𝔤𝔩_{i}^{hol}(λ) = {f(H)Y^{−i}} for i < 0 and f ∈ 𝔤𝔩_{0}^{hol}(λ). The relations in the completed algebra follow from (1.1) and (1.4), namely, they are

Xf(H) = f(H − 2)X, Y f(H) = f(H + 2)Y and XY = \frac{1}{4}(λ^{2} − (H − 1)^{2}).

The following Proposition is proved via the same lines as its counterpart in [KR].

5.1. Proposition . Let V be quasifinite 𝔤𝔩(λ)-module. Then the 𝔤𝔩(λ)-action can be naturally extended to a 𝔤𝔩_{i}^{hol}(λ)-action for i ≠ 0.

5.2. On ℂ, introduce an equivalence relation by setting

[s] = \begin{cases}s + 2ℤ & if s + λ ∉ 2ℤ + 1 \\ s − 2ℤ_{+} & if s = ±λ + 1 \\ s + 2ℤ_{+} & if s = ±λ − 1 \end{cases}

To each class [s] assign an irreducible 𝔰𝔩(2)-module M^{s} with a diagonal action of H whose set of eigenvalues coinsides with [s]; denote by 𝔤𝔩_{s}(∞) the Lie algebra of linear transformations of M^{s} with finitely many nonzero diagonals in the H-diagonal basis.

The following Theorem is similar and is proved via the same lines as its counterpart in [KR].

Theorem . Consider ⊕_{i=1}^{k} M_{R_{m_{i}}}^{s_{i}}, where the [s_{i}] are distinct and the m_{i} are nonegative integers. Let

ϕ : 𝔤𝔩(λ) ⟶ 𝔤 = ⊕_{i=1}^{k} 𝔤𝔩_{s_{i}}(∞; R_{m_{i}}) determined by this module. Then for any quasifinite 𝔤-module V any its 𝔤𝔩(λ)-submodule is a 𝔤-submodule. In particular, if V is irreducible as 𝔤-module, it is irreducible as a 𝔤𝔩(λ)-module.

Following [KR], let us describe now the structure of quasifinite 𝔤𝔩(λ)-modules. Let θ be the highest weight of a quasifinite 𝔤𝔩(λ)-module and F_{θ}(t) = \frac{R(t)}{1 − e^{−2t}}, where R(t) = ∑ r_{i}(t)e^{s_{i}t} the corresponding formal power series, see 3.2. The polynomials r_{i}(t) will be refered to as multiplicities.

5.3. For every s denote by R_{s}(t) the sum of all quasipolynomials with the exponents from class [s]. Then R(t) = ∑ R_{s}(t), where the sum runs over representatives of different equivalency classes. Let R_{s}(t) = ∑ r_{is}(t)e^{(s−2i)t}. The following properties of quasipolynomials follow easily from definitions:

i) R(0) = 0;

ii) the sum of all multiplicities of R_{λ+1}(t) and that of R_{−λ+1}(t)are equal to 0;

iii) the sum of all multiplicities of R_{λ−1}(t) and that of R_{−λ−1}(t)are equal to a constant.

Denote: Λ = [λ − 1] ∩ [λ + 1] ∩ [−λ − 1] ∩ [−λ + 1]. To a quasipolynomial R satisfying i)–iii) assign a 𝔤𝔩(λ)-module V (s) as follows:

1) if s ∉ Λ, then V (s) is the irreducible 𝔤̂𝔩(∞)-module with central charges c_{j} = − ∑_{i∈I} r_{i}^{(j)}(0),

where j = 0, …, max(deg r_{i}), and the other coordinates of the highest weight are θ_{ij} = ∑_{l≤i}(r_{l}^{(j)}(0)+

δ_{l0}c_{j});

2) if s ∉ [λ − 1] ∩ [−λ − 1] and R_{s} = ∑_{i} r_{i}(y)e^{s−2i}, then set c = − ∑_{i∈I} r_{i}(0) and θ_{ij} = ∑_{l≤i}(r_{l}^{(j)}(0) +

δ_{l0}δ_{j0}c_{j}) and let V (s) be the corresponding 𝔤𝔩^{−}(∞)-module.

3) if s ∉ [−λ + 1] ∩ [λ + 1], then θ_{ij} = ∑_{l≤i} r_{l}^{(j)}(0) and V (s) is the corresponding 𝔤𝔩^{+}(∞)-module.

12 DIMITRY LEITES, ALEXANDER SERGEEV^{1}

Theorem . Let the quasipolynomial R = ∑_{i=1}^{k} R_{s_{i}}(t) be decomposed with respect to the distinct

equivalence classes of the exponents; let R satisfy the conditions i)–iii) above. Then the irreducible quasifinite module with highest weight R(t) is isomorphic to

V = V (s_{1}) ⊗ . . . . . . V (s_{k}) ⊗ (αtr),

where αtr is the irreducible 1-dimensional module corresponding to the trace on 𝔤𝔩(λ) and the V (s_{i}) are constructed as above.

Proof. By Theorem 4.5 V is irreducible as 𝔤𝔩(λ)-module. Therefore, it suffices to show that the restriction of the highest weight of this module onto 𝔤𝔩(λ) is equal to \frac{R(t)}{1 − e^{−2t}}. Let s = s_{i} be one of the exponets such that s ∉ Λ. Then Proposition 5.1 implies that there exists a highest weight

θ_{s} for 𝔤̂𝔩(∞; R_{m}) whose generating function is equal to \frac{R_{s}(t) + c_{s}e^{−(λ+1)t}}{1 − e^{−2t}}, where c_{s} = −R_{s}(0).

If s ∈ Λ, then the same Proposition implies that the generating function is equal to \frac{R_{s}(t)}{1 − e^{−2t}}, where the sum of all the exponents of R_{s} vanishes.

Finally, the generating function of the trace is equal to α\frac{e^{(λ−1)t} − e^{−(λ+1)t}}{1 − e^{−2t}}. Therefore, for our

V the generating function is of the form \frac{R(t)}{1 − e^{−2t}}, where

R(t) = ∑_{s∉Λ}(R_{s}(t) + c_{s}e^{−(λ+1)t})+

R_{λ+1}(t) + R_{−λ+1}(t) + R′_{−λ−1}(t) + R′_{λ−1}(t) + α(e^{(λ−1)t} − e^{−(λ+1)t}) =

(by the choice of c_{s}) = ∑_{s∉Λ} R_{s}(t) + R_{λ+1}(t) + R_{−λ+1}(t)+

(R′_{−λ−1}(t) − αe^{−(λ+1)t}) + (R′_{λ−1}(t) + αe^{(λ−1)t}) = ∑_{s∉Λ} R_{s}(t).

§6. Unitary modules over 𝔤𝔩(λ)

Recall that an anti-involution of the Lie algebra 𝔤 over ℂ is a ℝ-linear map ω : 𝔤 ⟶ 𝔤 such that

ω(αx) = ᾱω(x), ω([x, y]) = [ω(x), ω(y)] and ω^{2} = id

for any α ∈ ℂ and x, y ∈ 𝔤.

In presence of an anti-involution ω we can endow V^{∗}, the dual of the 𝔤-module V, with another 𝔤-module structure, namely,

(xl)(v) = l(ω(x)v) for any l ∈ V^{∗}, v ∈ V and x ∈ 𝔤.  (6.0.1)

In particular, if 𝔤 = 𝔤_{−}⊕𝔤_{0}⊕𝔤_{+} and ω interchanges 𝔤_{−} with 𝔤_{+}, then a 𝔤-homomorphism M ⟶ M^{∗} is well-defined, where M is a Verma module with highest weight vector v and the 𝔤-module structure on M^{∗} is given by (6.0.1).

Indeed, let v^{∗} ∈ M^{∗} be such that v^{∗}(v) = 1 and v^{∗}(u) = 0 for all u of lesser weight than that of v. Hence, if x ∈ 𝔤_{+}, then (xv^{∗})(u) = v^{∗}(ω(x)u) = 0, so v^{∗} is also a highest vector and if θ(ω(H)) = \overline{θ(H)}, where θ is the weight of v, then the weight of v^{∗} is also equal to θ. Hence, there exists a 𝔤-isomorphism M ⟶ M^{∗} or an Hermitian 𝔤-invariant form ⟨·, ·⟩ on M. If ⟨·, ·⟩ is positive definite, M is called unitary.

In this section we indicate the conditions on the highest weight of the quasifinite 𝔤𝔩(λ)-module to be unitary. First, let us describe automorphisms of 𝔤𝔩(λ).

As is known, the algebra of functions, F, especially polynomial ones, has undescribably many automorphisms. Hence, so does the Lie algebra 𝔡𝔢𝔯(F) of the differentiations of F. To diminish the amount of automorphisms to a reasonable number, it is natural to consider only the outer automorphisms, i.e., the classes of automorphisms modulo the group of inner ones for which automorphisms induced by the automorphisms of F naturally qualify. Similarly, we say that the automorphism ϕ of 𝔤𝔩(λ) is an inner one if it also serves as an automorphism of the associative algebra 𝔄_{λ}.

Recall that a linear map ϕ : 𝔤 ⟶ 𝔤 is an antiautomorphism of the Lie algebra 𝔤 if ϕ([x, y]) = [ϕ(y), ϕ(x)]. For the associative algebras definition of an antiautomorphism is similar. For example, the map t such that ^{t}|_{𝔤} = −id is called the principal antiautomorphism of the Lie algebra 𝔤. The principal antiautomorphism of 𝔤 can be extended to an antiautomorphism of the associative algebra U(𝔤): ^{t}(x ⊗ y) = ^{t}(y) ⊗^{t}(x). Clearly, t preserves Casimir elements.

6.1. Theorem . The group of outer automorphisms of 𝔤𝔩(λ) is isomorphic to ℤ/2 and is generated by the class of −t (where −t : x ↦ −^{t}(x)).

Proof. Let ψ ∈ Aut(𝔤𝔩(λ)). Then ψ(Y), ψ(H), and ψ(X) span a Lie algebra isomorphic to 𝔰𝔩(2). Moreover, they generate 𝔄_{λ} because the weight of ψ(X)^{n} is equal to 2n with respect to ψ(H). Hence, there exists a surjective homomorphism

τ : U(𝔰𝔩(2)) ⟶ 𝔄_{λ}, X ↦ ψ(X), H ↦ ψ(H), Y ↦ ψ(Y).

Since τ(Ω) ∈ ℂ, there exists a μ ∈ ℂ such that τ(Ω) = \frac{1}{2}(μ^{2} − 1). Therefore, we get a surjective homomorphism, hence, an isomorphism 𝔄_{μ} ⟶ 𝔄_{λ}. In a very difficult technical paper Dixmier proved [Di1] that this may only happen if λ^{2} = μ^{2}. So we may assume that λ = μ and τ is an automorphism. Let ψ_{1} = τ^{−1}ψ. Then ψ_{1}|_{𝔰𝔩(2)} = id.

Therefore, ψ_{1} is an automorphism of 𝔤𝔩(λ), as 𝔰𝔩(2)-module. Therefore, ψ_{1}|_{L^{2i}} = c_{i} ∈ ℂ. Since the Lie algebra 𝔤𝔩(λ) is generated by L^{2} and L^{4}, we deduce that c_{i} = c_{2}^{i−1} for every i ≥ 1. Moreover, since [L^{4}, L^{4}] ⊃ L^{2}, it follows that c_{2}^{2} = 1. Hence, c_{2} = ±1; if c_{2} = 1, then ψ_{1} = id and if c_{2} = −1, then ψ_{1} = ϕ.

6.2. Theorem . If λ^{2} ∉ ℝ, then 𝔤𝔩(λ) has no real forms, i.e., no involutive antilinear automorphisms.

Proof. Let ω be an involutive antilinear automorphism of 𝔤𝔩(λ). Then ω(X), ω(H) and ω(Y) generate 𝔄_{λ}. Hence, there exists a surjective homomorphism

τ : U(𝔰𝔩(2)) ⟶ 𝔄_{λ}, τ = id on X, H, Y;

hence, we obtain an isomorphism τ : 𝔄_{μ} ⟶ 𝔄_{λ}. By [Di1] this may only happen if λ^{2} = μ^{2}. Hence, we may assume that λ = μ and τ is an automorphism. Then ω_{1} = τ^{−1}ω is an antilinear automorphism of 𝔤𝔩(λ) such that

ω_{1}(X) = X, ω_{1}(H) = H, ω_{1}(Y) = Y.

Set z_{i} = (ad X)^{i}(Y^{2}); in particular, z_{0} = z. Then Span(z_{i} : i = 1, …, 4) form a basis of L^{4} and if ω_{1}(Y^{2}) = cY^{2}, then, clearly, ω_{1}(z_{i}) = cz_{i} for i = 1, …, 4. According to [GL] in 𝔤𝔩(λ) the following

14 DIMITRY LEITES, ALEXANDER SERGEEV^{1}

relations hold:

\begin{matrix}3[z_{1}, z_{2}] − 2[z_{1}, z_{3}] = 24(λ^{2} − 4)Y \\ 4[z_{3}, [z_{1}, z_{1}]] − 3[z_{2}, [z, z_{2}]] = 576(λ^{2} − 9)z \end{matrix}

Having applied ω_{1} to both sides of these relations we get

\begin{matrix}c^{2}(λ^{2} − 4) = λ̄^{2} − 4 \\ c^{3}(λ^{2} − 9) = c(λ̄^{2} − 9) \end{matrix}

Therefore, c^{2} = 1 and λ^{2} = λ̄^{2}.

Corollary . If λ^{2} ≠ λ̄^{2}, then 𝔤𝔩(λ) has no involutive antilinear automorphisms.

Proof. Let ω be an involutive antilinear automorphism; clearly τ : X ⟷ Y and τ(H) = H determines an antiautomorphism of 𝔤𝔩(λ). Then τ ∘ω is an involutive antilinear automorphism.

6.3. If λ^{2} = λ̄^{2}, then 𝔄_{λ} possesses an involutive antilinear automorphism ω:

ω(X) = Y, ω(Y) = X, ω(H) = H  (6.3)

In what follows, the unitary modules are considered with respect to this automorphism.

Theorem . Let λ^{2} be real and let F_{θ}(t) = \frac{R(t)}{1 − e^{−2t}}, where R(t) is a quasipolynomial none of whose exponents s_{i} do not belong to Λ = [λ − 1] ∩ [λ + 1] ∩ [−λ − 1] ∩ [−λ + 1]. The 𝔤𝔩(λ)-module with character F_{θ}(t) is unitary if and only if

F_{θ}(t) = ∑ n_{i} \frac{e^{−(λ+1)t} − e^{s_{i}t}}{1 − e^{−2t}} , where n_{i} ∈ ℤ_{+}.

Proof. Let V be the irreducible module with character F_{θ}(t) and P (H) the annihilator of Y v, where v is the highest weight vector. Then P (H) is the characteristic polynomial of the operator H in V_{−1}. But since the unitary form is invariant and ω(H) = H, it follows that H is self-adjoint; hence, all the roots of P are real. Further, if α is a multiple root of multiplicity m > 1, the polynomial P is of the form P = (H − α)^{m}Q(H). For u = (H − α)^{m−1}Q(H)v we have

⟨u, u⟩ = ⟨(H − α)^{m−1}Q(H)v, (H − α)^{m−1}Q(H)v⟩ = ⟨Q(H)v, (H − α)^{2m−2}Q(H)v⟩ = 0.

Thanks to Hermitean property, u = 0. Thus, m = 1.

By Theorem 5.3 V is of the form V (s_{1}) ⊗ ⋯ ⊗ V (s_{k}), where the V (s_{i}) are irreducible 𝔤̂𝔩(∞)- modules. Clearly, V is unitary if and only if each V (s_{i}) is unitary. Now, thanks to [KR] we know that the 𝔤̂𝔩(∞)-module V (s) is unitary if and only if θ_{i} − θ_{i+1} + δ_{i0}c ∈ ℤ_{+} for all i, where the θ_{i} are the coordinates of the highest wieght and c the value of the central charge.

§7. 𝔤𝔩(λ) and the symmetric group

In this section we deduce an explicit realization of certain irreducible 𝔤𝔩(λ)-modules. Namely, we decompose the tensor powers of the Verma module over 𝔰𝔩(2), indicate the corresponding characteristic polynomials and q-characters.

7.1. Lemma . Let A be an associative algebra; let 𝔖_{n} naturally act on A^{⊗n}. Then the algebra (A^{⊗n})^{𝔖_{n}} of 𝔖_{n}-invariants is generated by the elements of the form

a ⊗ 1 ⊗ 1 ⊗ ⋯ ⊗ 1 + … 1 ⊗ 1 ⊗ 1 ⊗ ⋯ ⊗ a.

Proof. Denote: s(a_{1}, …, a_{n}) = ∑_{σ∈𝔖_{n}} a_{σ(1)} ⊗ ⋯ ⊗ a_{σ(n)}. Let B be the algebra generated by the

s(a, 1, …, 1) for all a ∈ A. Let |s(a_{1}, …, a_{n})| be the number of the a_{i} distinct from 1. Let us prove that B ≃ (A^{⊗n})^{𝔖_{n}}.

Let us carry on an induction on |s(a_{1}, …, a_{n})| in order to prove that s(a_{1}, …, a_{n}) ∈ B. Indeed, if |s(a_{1}, …, a_{n})| = 1, by definition s(a_{1}, …, a_{n}) = s(a, 1, …, 1) ∈ B.

Let |s(a_{1}, …, a_{n})| = l > 1. Consider

s(a_{1}, …, a_{l−1}, 1, …, 1)s(a_{l}, 1, …, 1) = αs(a_{1}, …, a_{l}, 1, …, 1) + …,

where α is a nonzero constant and the dots stand for the linear combination of the terms s(b_{1}, …, b_{m}, 1, …, 1) with m < l.

7.2. Theorem . Let V = M^{λ−1} be the Verma module with highest weight λ − 1 over 𝔰𝔩(2) (and 𝔤𝔩(λ)). Then

V^{⊗n} = ⊕_{ν} V^{ν} ⊗ S^{ν},

where V^{ν} is an irreducible 𝔤𝔩(λ)-module and S^{ν} is an irrecible 𝔖_{n}-module and ν runs over the partitions of n.

Proof. Clearly, V is irreducible not only as 𝔤𝔩(λ)-module but also as 𝔄_{λ}-module. Then, obviously, W = V^{⊗n} is irreducible as 𝔄_{λ}^{⊗n}-module.

The image of U(𝔤𝔩(λ)) in End(W) coinsides with the subalgebra generated by s(a, 1, …, 1) for a ∈ 𝔤𝔩(λ), so by Lemma it is isomorphic to (𝔄_{λ}^{⊗n})^{𝔖_{n}}.

Let us decompose W into isotypical 𝔖_{n}-modules: W = ⊕_{ν} W^{ν} and represent each W^{ν} in the form W^{ν} = V^{ν} ⊗ S^{ν}, where V^{ν} is an irreducible 𝔤𝔩(λ)-module. There is no unique way to do this, for example, set V^{ν} = e_{ν}(W) for any minimal idempotent in ℂ[𝔖_{n}] corresponding to the partition ν.

To show that V^{ν} is irreducible as a 𝔤𝔩(λ)-module, consider V_{1}^{ν} = Hom_{𝔖_{n}}(S^{ν}, W). As a 𝔤𝔩(λ)- module, V_{1}^{ν} is isomorphic to V^{ν}. Let ϕ, ψ ∈ V_{1}^{ν} and ϕ ≠ 0. Let us show that there exists an u ∈ U(𝔤𝔩(λ)) such that uϕ = ψ. Indeed, since W is irreducible as 𝔄_{λ}^{⊗n}-module, then the density theorem ([L], Ch. XVII, §3, Th.1) states that there is a w ∈ 𝔄_{λ}^{⊗n} such that wϕ(v_{i}) = ψ(v_{i}) , where the v_{i} form a basis of S^{ν}. (Since S^{ν} is irreducible and ϕ ≠ 0, then the vectors ψ(v_{i}) are linealy independent for i = 1, …, dim S^{ν}.)

Let us average the element wϕ with respect to 𝔖_{n}:

(wϕ)^{♯} = \frac{1}{|𝔖_{n}|} ∑ σ(wϕ)σ^{−1} = \frac{1}{|𝔖_{n}|} ∑ σ(w)σ^{−1}(ϕ) = w^{♯}ϕ.

But, on the other hand, since wϕ = ψ, we deduce that (wϕ)^{♯} = (ψ)^{♯} = ψ, i.e., w^{♯}ϕ = ψ and we may assume that w ∈ (𝔄_{λ}^{⊗n})^{𝔖_{n}}. So, there exists a u ∈ U(𝔤𝔩(λ)) such that uϕ = ψ. In other words, V_{1}^{ν} is irreducible.

7.2.1. Corollary . 1) Let ν = (ν_{1}, …, ν_{n}). The generating function corresponding to the highest weight ν is of the form

∑_{i=1}^{n} ν_{i}e^{(λ−2i+1)t}.  (7.2.1.1)

2) Represent ν in the form ν = (θ_{1}^{α_{1}} … θ_{m}^{α_{m}}), where θ_{1} > ⋯ > θ_{m} > 0 and α_{i} ≠ 0 for all i and where θ^{α} denotes the product of α copies: θ … θ. Then the characteristic polynomial of V^{ν} is equal

16 DIMITRY LEITES, ALEXANDER SERGEEV^{1}

to

P (H) = ∏_{i=1}^{m}(H − λ − 2α_{1} − 2α_{2} − ⋯ − 2α_{i} − 1).

Proof. Heading 1) follows from Theorem 4.5. Multiply (7.2.1.1) by 1−e^{−2t}; we see that the product is of the form

ν_{1}e^{(λ−1)t} + ∑_{i=1}^{n−1}(ν_{i+1} − ν_{i})e^{(λ−2i+1)t} − ν_{n}e^{(λ−2n−1)t}.  (7.2.1.2)

Therefore, sum (7.2.1.2) without the first summand is a solution of an ordinary differential equation with constant coefficients whose characteristic equation is precisely of the form indicated.

7.2.2. Corollary . Set a = e^{λ}, q = e^{−α}, where α is the positive root of 𝔰𝔩(2). Then the q-character of V^{ν} is equal to

χ_{ν} = \frac{a^{|ν|}q^{n(ν)}}{∏_{x∈ν}(1 − q^{h(x)})},  (7.2.2)

where |ν| is the number of cells in the Young tableau corresponding to ν, n(ν) = ∑_{i≥1} ν_{i} and h(x) is

the length of the hook corresponding to the cell x.

Proof. By [Mac], Example 2 in §3, for the S-function s_{ν} we have

χ_{ν} = s_{ν}(a, aq, aq^{2}, …,) = a^{|ν|}s_{ν}(1, q, q^{2}, …,) = rhs of (7.2.2).

§8. Orthogonal polynomials for 𝔤𝔩(n)

For an overview see [NU], [NSU]. (Setting ∏_{i=1}^{l} T_{1}(α_{i}) … T_{l}(α_{i}) = 1 for l = 0 we make formulas

(8.1.1) and the like uniform.)

8.1. Theorem . In 𝔤𝔩(n):

i) Consider the basis e_{kl} = (ad Y)^{k−l}(X^{k}) for 0 ≤ k ≤ n − 1 and −k ≤ l ≤ k. We have

⟨e_{kl}, e_{k′l′}⟩ = δ_{k,k′}δ_{l+l′,0}.

ii) Determine the elements f_{kl} from the equations (0 ≤ l ≤ k)

(ad Y)^{k−l}(X^{k}) = X^{l}f_{kl} and (ad Y)^{k+l}(X^{k}) = f_{k,−l}Y^{l}.

Set T_{0}(H) = \frac{1}{4}(n^{2} − (H − 1)^{2}) and T_{i}(H) = T_{0}(H + 2i) and α_{i} = n − 2i + 1 for i = 1, …, n. For a fixed l ≥ 0 and any k ≥ l the polynomials f_{kl} form an orthonormal basis with respect to the form

⟨f, g⟩ = \begin{cases}∑_{i=l+1}^{n} f(α_{i})g(α_{i})T_{1}(α_{i}) … T_{l}(α_{i}) & for l > 0 \\ ∑_{i=l+1}^{n} f(α_{i})g(α_{i}) & for l = 0. \end{cases}  (8.1.1)

iii) Up to a constant factor the polynomials f_{kl} coincide with the Hahn polynomial of one discrete variable:

f_{kl}(H) = _{3}F_{2} (\begin{matrix}l − k, l + k + 1, \frac{1}{2}(1 − n − H) \\ l + 1, l + 1 − n \end{matrix} | 1) × T_{0}(α_{l+1}) … T_{0}(α_{k}),  (8.1.2)

where

3F_{2} (\begin{matrix}α_{1}, α_{2}, α_{3} \\ β_{1}, β_{2} \end{matrix} | z) = ∑_{i=0}^{∞} \frac{(α_{1})_{i}(α_{2})_{i}(α_{3})_{i}}{(β_{1})_{i}(β_{2})_{i}} \frac{z^{i}}{i!}  (8.1.3)

is a generalized hypergeometric function and (α)_{0} = 1 whereas (α)_{i} = α(α + 1) … (α + i − 1) for i > 0.

Proof. i) First, observe that the subspaces L^{2k} in the decomposition of 𝔤𝔩(n) = L^{0}⊕L^{2}⊕⋯⊕L^{2n−2} are pairwise orthogonal. Indeed, the form A, B ↦ tr AB defines an invariant pairing of L^{k} and L^{l}, hence, an 𝔰𝔩(2)-homomorphism L^{k} ⟶ L^{l} which is only possible if k = l. Moreover, it is clear that ⟨𝔤𝔩(n)_{k}, 𝔤𝔩(n)_{l}⟩ ≠ 0 if and only if k + l = 0. Now observe that 𝔤𝔩(n)_{l} ∩ L^{2k} = Span((adY)^{k−l}(X)). This proves i).

ii) Let X^{l}f(H) ∈ 𝔤𝔩(n)_{l} and g(H)Y^{l} ∈ 𝔤𝔩(n)_{−l}. Then

tr (g(H)Y^{l}X^{l}f(H)) = tr (f(H)g(H)T_{1}(H) … T_{l}(H)).

As is easy to verify

tr f(H) = ∑_{1≤i≤n} f(α_{i}) for any f(H) ∈ 𝔤𝔩(n)_{0}.

This implies (8.1.1).

Moreover, i) implies that for any fixed l lying between −k and k the polynomials f_{kl} and f_{k,−l} form two mutually dual bases, i.e., ⟨f_{kl}, f_{kl′}⟩ = δ_{l+l′,0}. But the degrees of polynomials f_{kl} and f_{k,−l} are equal, which means that they coincide up to a constant factor. This proves ii).

iii) follows from the fact that orthogonal polynomials are uniquely determined by the weight function and the interval over which we consider the scalar product. Eq. (8.1.2) follows from comparison of the coefficients of the leading terms.

Remark . Hahn polynomials are determined for three parameters α, β and N as

h_{p}^{(α,β)}(z, N) = \frac{(−1)^{p}Γ(N)(β)_{p}}{p!Γ(N − p)} _{3}F_{2} (\begin{matrix}−p, α + β + p + 1, −z \\ β + 1, l − N \end{matrix} | 1) .  (8.1.4)

Our f_{kl} coinsides, up to a factor, with h_{p}^{(α,β)}(z, N) at α = β = l, p = k − l, z = \frac{1}{2}(H + n − 1) and N = n − 1.

Let us show now how the main properties of the Hahn polynomials follow from the representation theory of 𝔰𝔩(2). For any f(H) ∈ ℂ[H] set

△f(H) = f(H + 2) − f(H), ∇f(H) = f(H) − f(H − 2).  (8.1.5)

8.2. Theorem . Consider 𝔤𝔩(n) as 𝔰𝔩(2)-module with respect to the image of the principal embedding, let Ω be the quadratic Casimir operator (1.2).

i) Ω is self-adjoint with respect to the form ⟨·, ·⟩ and the polynomials X^{l}f_{kl} are eigenfunctions of Ω corresponding to eigenvalue 2k(k + 1). The polynomials f_{kl} satisfy the difference equation

T_{0}(H)∇△(f) − (k + 1)(k + l)△(f) + (k − l)(k + l + 1)f = 0.  (8.2.i))

ii) f_{kl} = \begin{cases}\frac{∇^{k−l}(T_{1} … T_{k})}{T_{1} … T_{l}} & if l > 0 \\ ∇^{k}(T_{1} … T_{k}) & if l = 0. \end{cases}

iii) ⟨f_{kl}, f_{kl}T_{1} … T_{l}⟩ = \frac{(k − l)!}{(k + l)!} \frac{n(n^{2} − 1^{2}) … (n^{2} − k^{2})}{2k + 1}(k!)^{2}.

18 DIMITRY LEITES, ALEXANDER SERGEEV^{1}

Proof. i) Since the form ⟨·, ·⟩ is 𝔰𝔩(2)-invariant, ⟨[w, u], v⟩ = −⟨u, [w, v]⟩ for any u, v ∈ 𝔤𝔩(n) and w ∈ 𝔰𝔩(2). Denote the U(𝔰𝔩(2))-action in 𝔤𝔩(n) by ∗; by induction we easily deduce that ⟨w ∗ u, v⟩ = −⟨u, w^{t} ∗ v⟩ where now w ∈ U(𝔰𝔩(2)) and t is the principal antiinvolution. Since, as is easy to verify, Ω^{t} = Ω, we have ⟨Ω ∗ u, v⟩ = ⟨u, Ω ∗ v⟩, i.e., Ω is self-adjoint. Since 𝔤𝔩(n) = ⊕L^{2i} and X^{l}f_{kl} = (adY)^{k−l}(X^{k}) ∈ L^{2k}, it follows that X^{l}f_{kl} is an eigenfunction of Ω with eigenvalue 2k(k + 1). Applying Ω to X^{l}f_{kl} we obtain eq. (8.2.i).

ii) Recall [Di] the identity

(ady)^{p}(a) = ∑_{j=0}^{p}(−1)^{j}\binom{p}{j}y^{p−j}ay^{j}.

Set a = X^{k}, p = k − l and multiply by Y^{l} from the left. We get

\begin{matrix}Y^{l}X^{l}f_{kl} = Y^{l}(adY)^{k−l}(X^{k}) = Y^{l} ( ∑_{j=0}^{k−l}(−1)^{j}\binom{k−l}{j}Y^{k−l−j}X^{k}Y^{j}) = \\ ∑_{j=0}^{k−l}(−1)^{j}\binom{k−l}{j}y^{k−j}X^{k}Y^{j} = ∑_{j=0}^{k}(−1)^{j}\binom{k}{j}T_{j−1} … T_{0}T_{1} … T_{k−j}. \end{matrix}

But ∇^{k−l}(f)(H) = ∑_{j=0}^{k}(−1)^{j}\binom{k}{j}f(H − 2j) for any function f, so

Y^{l}X^{l}f_{kl} = ∇^{k−l}(T_{1} … T_{k}).

Since Y^{l}X^{l} = T_{1} … T_{l}, we are done.

iii) The module L^{2k} is the linear span of the vectors v_{l} = (adY)^{k−l}(X^{k}) for −k ≤ l ≤ k; hence,

⟨v_{l}, v_{m}⟩ = −⟨v_{l−1}, v_{m+1}⟩ = ⋯ = (−1)^{l}⟨v_{0}, v_{l+m}⟩

by invariance of the form. Hence, ⟨v_{l}, v_{−l}⟩ = (−1)^{l}⟨v_{0}, v_{0}⟩. But v_{l} = X^{l}f_{kl} = (ad Y)^{k−l}(X^{k}) and v_{−l} = f_{k,−l}Y^{l} = (ad Y)^{k+l}(X^{k}).

Now recall that f_{kl} and f_{k,−l} are identical up to a constant factor and compare the leading coefficients. We see that

f_{k,−l} = (−1)^{l} \frac{(k + l)!}{(k − l)!}f_{k,l}.

Hence,

⟨f_{k,l}, f_{k,l}T_{1} … T_{l}⟩ = tr (f_{k,l}Y^{l} · X^{l}f_{k,l}) = (−1)^{l} \frac{(k + l)!}{(k − l)!}⟨v_{l}, v_{−l}⟩ = \frac{(k + l)!}{(k − l)!}⟨v_{0}, v_{0}⟩.

It remains to demonstrate that

⟨v_{0}, v_{0}⟩ = \frac{n(n^{2} − 1^{2}) … (n^{2} − k^{2})}{2k + 1}(k!)^{2}.

Obviously, f_{k} = ((ad Y)^{k}(X^{k}))^{2} ∈ U(𝔰𝔩(2)). So ⟨v_{0}, v_{0}⟩ = tr_{n}ϕ_{n}(f_{k}), where tr_{n} is the trace on 𝔤𝔩(n) and ϕ_{n} is the homomorphism of U(𝔰𝔩(2)) into 𝔤𝔩(n) induced by the principle embedding.

It is clear now that P_{k}(n) = tr_{n}(ϕ_{n}(f_{k})) is a polynomial of degree 2k + 1; moreover, this polynomial is an odd one. But ϕ_{n}(X^{k}) = 0 for n ≤ k and ϕ_{n}(f_{k}) = 0; hence, P_{k}(n) = 0 if n ≤ k and P_{k}(−n) = 0 because P_{k} is odd. Therefore,

P_{k}(n) = c_{k}n(n^{2} − 1^{2}) … (n^{2} − k^{2}).

ORTHOGONAL POLYNOMIALS AND COMPLEX SIZE MATRICES 19

To calculate the constant c_{k}, it suffices to compute P_{k}(k + 1) = tr_{k+1}(ϕ_{k+1}(f_{k})). But in this case X^{k} = (k!)^{2}e_{1,k+1} and

(ad Y)^{k}(X^{k}) = (k!)^{2}(ad Y)^{k}(e_{1,k+1}) = (k!)^{2} ∑_{j=0}^{k}(−1)^{j}\binom{k}{j}e_{k+1−j,k+1−j}.

Thus,

tr_{k+1}(ϕ_{k+1}(f_{k})) = (k!)^{4} ∑_{j=0}^{k} \binom{k}{j}^{2} = (k!)^{4}\binom{2k}{k}

implying c_{k} = \frac{(k!)^{2}}{2k + 1}.

8.3.1. Proposition . The following orthogonality relations (i) and (ii) hold:

(i)  ∑_{i} f_{kl}(α_{i})f_{k_{1},l}(α_{i})T_{1}(α_{i}) … T_{l}(α_{i}) = δ_{k,k_{1}}c_{k,l},

where c_{k,l} = \frac{(k − l)!}{(k + l)!} \frac{n(n^{2} − 1^{2}) … (n^{2} − k^{2})}{2k + 1}(k!)^{2}.

(ii)  ∑_{k=l}^{n−1} \frac{1}{c_{kl}} f_{kl}(α_{i})f_{kl}(α_{j})T_{1}(α_{i}) … T_{l}(α_{i}) = δ_{ij} for any i, j > l.

Proof. (i) follows fromTheorem 8.2, iii). To prove ii), express e_{ij} via X^{l}f_{kl} and f_{k,−l}Y^{l}. We have, in particular, e_{i,i+l} = ∑_{k=l}^{n−1} α_{ik}X^{l}f_{kl} for any k between l and n − 1. But, as is easy to verify,

\begin{matrix}f_{kl}Y^{l} = (∑_{i=1}^{n} f_{kl}(α_{i})e_{ii})( ∑_{i=1}^{n−1} e_{i+1,i})^{l} = \\ (∑_{i=1}^{n} f_{kl}(α_{i})e_{ii})(∑_{i=1}^{n−l} e_{i+l,i}) = ∑_{i=1}^{n−l} f_{kl}(α_{i+l})e_{i+l,i}. \end{matrix}

Hence, by Theorem 8.2 we have

f_{kl}(α_{i+l}) = ⟨e_{i,i+l}, f_{kl}Y^{l}⟩ = α_{ik}⟨f_{kl}Y^{l}, X^{l}f_{kl}⟩ = α_{ik}c_{kl}.

In other words,

e_{i,i+l} = ∑_{k=l}^{n−1} \frac{f_{kl}(α_{i+l})}{c_{kl}}X^{l}f_{kl}.

Similarly,

e_{i+l,i} = ∑_{k=l}^{n−1} \frac{f_{kl}(α_{i+l})T_{1}(α_{i+1}) … T_{l}(α_{i+l−1})}{c_{kl}}f_{kl}Y^{l}.

Therefore, by setting i′ = i + l, j′ = j + l we deduce that

⟨e_{i+l,i}, e_{j,j+l}⟩ = ∑_{i=l}^{n−1} \frac{1}{c_{kl}} f_{kl}(α_{i+l})f_{kl}(α_{j+l})T_{1}(α_{i+1}) … T_{l}(α_{i+l−1}) =

∑_{i=l}^{n−1} \frac{1}{c_{kl}} f_{kl}(α_{i′})f_{kl}(α_{j′})T_{1}(α_{i′−1}) … T_{l}(α_{i′−l+1}) =

∑_{i=l}^{n−1} \frac{1}{c_{kl}} f_{kl}(α_{i′})f_{kl}(α_{j′})T_{1}(α_{i′}) … T_{l}(α_{i′}).

20 DIMITRY LEITES, ALEXANDER SERGEEV^{1}

Expressing Casimir elements Ω_{2}, Ω_{3}, …, Ω_{n} for 𝔤𝔩(n) in terms of the orthogonal polynomials rather than matrix units, we can derive various identities relating the polynomials. Some of these identities might be even new, cf. 9.3. For example, for Ω_{2} we have:

8.3.2. Proposition . In 𝔤𝔩(n), we have

∑_{k=0}^{n−1} (\frac{1}{c_{k0}} f_{k0}^{2} + 2 ∑_{l=1}^{k} \frac{1}{c_{kl}} f_{kl}^{2}T_{1} … T_{l} − \frac{2k + 1}{n}) = −H.

Proof. Let us rewrite the Casimir operator for 𝔤𝔩(n) expressed in terms of matrix units

\begin{matrix}Ω_{2} = ∑_{k=1}^{n} e_{kk} ⊗ e_{kk} + ∑_{i<j}(e_{ij} ⊗ e_{ji} + e_{ji} ⊗ e_{ij}) = \\ ∑_{k=1}^{n} e_{kk} ⊗ e_{kk} + ∑_{k=1}^{n}(n − 2k + 1)e_{kk}2 ∑_{i<j} e_{ji} ⊗ e_{ij}) \end{matrix}

via orthogonal polynomials:

\begin{matrix}Ω_{2} = ∑_{k=0}^{n−1} \frac{1}{c_{k0}} f_{k0} ⊗ f_{k0} + ∑_{l=1}^{n−1} ∑_{k=l}^{n−1} \frac{1}{c_{kl}} (X^{l}f_{kl} ⊗ f_{kl}Y^{l} + f_{kl}Y^{l} ⊗ X^{l}f_{kl}) = \\ ∑_{k=0}^{n−1} \frac{1}{c_{k0}} f_{k0} ⊗ f_{k0} + 2 ∑_{l=1}^{n−1} ∑_{k=l}^{n−1} \frac{1}{c_{kl}} f_{kl}Y^{l} ⊗ X^{l}f_{kl} + ∑_{k>l} \frac{1}{c_{kl}} [X^{l}f_{kl}, f_{kl}Y^{l}]. \end{matrix}

Let us now transform the last sum. For this, consider a homomorphism φ : U(𝔤𝔩(n)) ⟶ 𝔤𝔩(n) induced by the identity representation. Since

\frac{1}{c_{k0}} f_{k0} ⊗ f_{k0} + ∑_{l=1}^{k} \frac{1}{c_{kl}} (X^{l}f_{kl} ⊗ f_{kl}Y^{l} + f_{kl}Y^{l} ⊗ X^{l}f_{kl})

is 𝔰𝔩(2)-invariant, its image in 𝔤𝔩(n) is also an 𝔰𝔩(2)-invariant, i.e., is a constant, D(k):

\frac{1}{c_{k0}} f_{k0} · f_{k0} + 2 ∑_{l=1}^{k} \frac{1}{c_{kl}} f_{kl}Y^{l} · X^{l}f_{kl} + ∑_{l=1}^{k} \frac{1}{c_{kl}} [X^{l}f_{kl}, f_{kl}Y^{l}] = D(k).

Let us calculate the trace of both sides. We see that

nD(k) = \frac{1}{c_{k0}} tr(f_{k0}^{2}) + 2 ∑_{l=1}^{k} \frac{1}{c_{kl}} tr(f_{kl}^{2}T_{1} … T_{l}) = 2k + 1

implying D(k) = \frac{2k + 1}{n}. Therefore,

∑_{l=1}^{k} \frac{1}{c_{kl}} [X^{l}f_{kl}, f_{kl}Y^{l}] = \frac{2k + 1}{n} − (\frac{1}{c_{k0}} f_{k0} · f_{k0} + 2 ∑_{l=1}^{k} \frac{1}{c_{kl}} f_{kl}Y^{l} · X^{l}f_{kl}) .

Summing over k we obtain

∑_{k=0}^{n−1} (\frac{1}{c_{k0}} f_{k0} · f_{k0} + 2 ∑_{l=1}^{k} \frac{1}{c_{kl}} f_{kl}Y^{l} · X^{l}f_{kl} − \frac{2k + 1}{n}) = − ∑_{k=0}^{n−1} ∑_{l=1}^{k} \frac{1}{c_{kl}} [X^{l}f_{kl}, f_{kl}Y^{l}].

The linear parts of the Casimir operator coincide, so the last sum is equal to − ∑_{i=1}^{n}(n − 2i + 1)e_{ii} = −H.

§9. Orthogonal polynomials and 𝔤𝔩(λ)

The above results for 𝔤𝔩(n), n = 1, 2, etc. also hold mutatis mutandis for 𝔤𝔩(λ) with any complex λ ∉ ℤ \ {0}. In this section we only consider such values of λ.

Observe that 𝔤𝔩(λ), unlike 𝔤𝔩(∞) or 𝔤𝔩^{±}(∞), has no basis consisting of matrix units. But we can represent it in the form

𝔤𝔩(λ) = ⊕_{k=0}^{∞}L^{2k} and 𝔤𝔩(λ) = ⊕_{i=−∞}^{∞}𝔤𝔩(λ)_{i},

where 𝔤𝔩(λ)_{0} = ℂ[H] while 𝔤𝔩(λ)_{i} = X^{i}𝔤𝔩(λ)_{0} and 𝔤𝔩(λ)_{−i} = 𝔤𝔩(λ)_{0}Y^{i} for i > 0.

As was shown above, there exists a trace on 𝔤𝔩(λ) whose restriction onto 𝔤𝔩(λ)_{0} has for generating function

\frac{e^{λt} − e^{−λt}}{e^{t} − e^{−t}} if λ ≠ 0 and \frac{2t}{e^{t} − e^{−t}} if λ = 0.

In the first case we have normed the trace so that tr(1) = λ by anology with the finite dimensional case when the scalars are naturally represented by scalar matrices and tr(1_{n}) = n; in the second case, λ = 0, we assume that tr(1) = 1.

Observe that for an integer λ = n we have

\frac{e^{nt} − e^{−nt}}{e^{t} − e^{−t}} = e^{(1−n)t} + e^{(3−n)t} + ⋯ + e^{(n−3)t} + e^{(n−1)t}

and the respective functional is of the form

tr(f(H)) = ∑_{i=1}^{n} f(α_{i}), α_{i} = n − 2i + 1.

The functional tr on 𝔤𝔩(λ) gives rise to a nondegenerate symmetric invariant bilinear form ⟨u, v⟩ = tr uv.

9.1. Theorem . In notations of §8: i) ⟨e_{k,l}, e_{k′,l′}⟩ = δ_{k,k′}δ_{l+l′,0}.

ii) The polynomials f_{k,l} for a fixed l ≥ 0 and k ≥ l form an orthogonal basis with respect to the scalar product (8.1.1).

iii) The polynomials f_{k,l} coinside, up to a constant factor, with continuous Hahn polynomials of discrete variable given by (8.1.2) for n = λ.

Proof. The proof of headings i) and ii) is similar to that of respective stements of Theorem 8.1 for integer λ. To prove iii) observe that for a nonnegative integer eq. (9.1) holds thanks to Theorem 8.1. But both sides of (9.1) are polynomials in H whose coefficients are polynomials in λ. Due to continuity of these expressions in Zariski topology we are done.

9.2. Theorem . Let Ω be the quadratic Casimir operator for 𝔰𝔩(2) ⊂ 𝔤𝔩(λ).

i) Ω is self-adjoint with respect to the form (8.1.1) and the polynomials X^{l}f_{kl} for k ∈ ℤ_{+} and l ∈ [−k, k] ∩ ℤ are eigenfunctions of Ω corresponding to eigenvalue 2k(k + 1). The polynomials f_{kl} satisfy the difference equation (8.2.i) and are of the form (8.2.ii)

ii) ⟨f_{kl}, f_{kl}T_{1} … T_{l}⟩ = \begin{cases}\frac{(k − l)!}{(k + l)!} \frac{(k!)^{2}}{2k + 1}λ(λ^{2} − 1^{2}) … (λ^{2} − k^{2}) & if λ ≠ 0 \\ (−1)^{k} \frac{(k − l)!}{(k + l)!} \frac{(k!)^{4}}{2k + 1} & if λ = 0. \end{cases}

Proof. Proof of heading i) is the same as that of the corresponding statements of Theorem 8.2. If λ ≠ 0, then both sides of eq. ii) are polynomials in λ which coinside at integer values of λ and, therefore, are equal by continuity in Zariski topology. To embrace λ = 0, consider lim_{λ⟶0} \frac{tr}{λ}; as

before, both sides become polynomials in λ implying ii).

It is somewhat unexpected that the dual orthogonality relations hold for sufficiently large |λ| . Indeed, set

c_{kl} = \begin{cases}\frac{(k − l)!}{(k + l)!} \frac{(k!)^{2}}{2k + 1}λ(λ^{2} − 1^{2}) … (λ^{2} − k^{2}) & if λ ≠ 0 \\ (−1)^{k} \frac{(k − l)!}{(k + l)!} \frac{(k!)^{4}}{2k + 1} & if λ = 0. \end{cases}  (9.2.3)

9.3. Conjecture . For sufficiently large |λ| we have (for α_{i} = λ − 2i + 1):

i)  ∑_{k=l}^{∞} \frac{1}{c_{kl}} f_{kl}(α_{i})f_{kl}(α_{j})T_{1}(α_{i}) … T_{l}(α_{i}) = δ_{ij} for i, j > l.  (9.3.1)

ii) ∑_{k=0}^{∞} (\frac{1}{c_{k0}} f_{k0}(α_{i})^{2} + 2 ∑_{l=1}^{k} \frac{1}{c_{kl}} f_{kl}(α_{i})^{2}T_{1}(α_{i}) … T_{l}(α_{i}) − \frac{2k + 1}{λ}) = −α_{i} for i = 1, 2, ….

(9.3.2)

A priori, |λ| depends on l. We do not know a uniform proof and only checked Conjecture in certain particular cases. We are thankful to V. Gerdt and P. Grozman who helped us with numerical experiments on Maple and MATHEMATICA, respectively. (It seems that Maple is much more convenient and suitable for the task.)

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Correspondence: Dept. of Math., Univ. of Stockholm, Roslagsv. 101, Kräftriket hus 6, S-106 91, Stockholm, Sweden, mleites@matematik.su.se (^{1}On leave of absence from Balakovo University of Technique, Technology and Information, Balakovo, Saratov region, Russia)