On the complete integrability of the periodic quantum Toda lattice

A The case when β is the highest root: The theorem of Etingof

The complete integrability of the periodic quantum Toda lattice corresponding to a Dynkin diagram extended by the highest root was proved by Etingof in [3]. We considered necessary to include the details of his proof for reasons of completeness and clarity. The approach below is slightly different from the original one, in that we preferred to use complex Lie groups and loop spaces rather than formal groups. As background references we indicate [9], [14], and [13]. Let G be a simple, simply connected, complex Lie group of Lie algebra 𝔤 and T⊂G a maximal torus, whose Lie algebra we denote by 𝔥. Let also 〈⋅,⋅〉 be the Killing form on 𝔤 normalized such that 〈α,α〉=2 for any long root α. Pick a simple root system α1,…,αr∈𝔥* and let θ be the corresponding highest root. Consider the differential operator on 𝔥 given by

where Δ is the Laplacian relative to 〈⋅,⋅〉 and K is a parameter. The goal is to construct differential operators on 𝔥 which commute with M. Recall [9] that the corresponding (non-twisted) affine Kac–Moody Lie algebra is 𝔤^=ℒpol⁢(𝔤)⊕ℂ⁢c, where ℒpol⁢(𝔤)=𝔤⊗ℂ⁢[z,z-1] is the space of Laurent polynomials with coefficients in 𝔤, z being on the unit circle S1. The Lie bracket on 𝔤^ is defined by

for any u,v∈ℒpol⁢(𝔤), where Res stands for residue (the coefficient of z-1) and c is a central element.

The Chevalley generators of 𝔤^ are ei,hi,fi, where 0≤i≤r. Here {hi} is the basis of 𝔥^:=𝔥⊕ℂ⁢c consisting of the simple affine coroots, {ei} are root vectors for simple affine roots and {fi} root vectors for the negatives of those roots. Denote by 𝔫+ and 𝔫- the Lie subalgebras of 𝔤^ generated by {ei} and {fi}, respectively. They can be described as

where the first sum runs over all positive affine roots and the second sum over all negative affine roots, 𝔤^α being the corresponding root space. We thus have the triangular decomposition 𝔤^=𝔫+⊕𝔥^⊕𝔫-.

Let us now consider the group ℒ⁢(G) of all smooth maps from the circle S1 to G along with its universal central extension ℒ~⁢(G), see [14, Section 4.4]. Both ℒ⁢(G) and ℒ~⁢(G) are Fréchet–Lie groups, of Lie algebras ℒ⁢(𝔤) (space of all smooth maps S1→𝔤) and ℒ⁢(𝔤)⊕ℂ⁢c, respectively. There is a well-defined exponential map exp:ℒ~⁢(𝔤)→ℒ~⁢(G), which is smooth.

Let U+,U-⊂G be the unipotent radicals of the two standard “opposite” Borel subgroups that contain T. Denote by 𝒰+ the subgroup of ℒ⁢(G) whose elements are smooth boundary values of holomorphic maps γ:{z∈ℂ:|z|<1}→G with γ⁢(0)∈U+. Similarly, 𝒰- is the subgroup of ℒ⁢(G) consisting of smooth boundary values of holomorphic maps γ:{z∈ℂ:|z|>1}∪{∞}→G such that γ⁢(∞)∈U-. There exist canonical embeddings 𝒰±⊂ℒ~⁢(G). Moreover, if 𝒯:=exp⁡(𝔥^)=T×ℂ*, then the map 𝒰+×𝒯×𝒰-→ℒ~⁢(G), (u+,g,u-)↦u+⁢g⁢u- is injective onto 𝒰+⁢𝒯⁢𝒰-, which is an open subspace of ℒ~⁢(G), see [14, Theorem 8.7.2]. Any element of 𝔤^ induces a canonical left-invariant vector field on ℒ~⁢(G), hence also on its open subspace 𝒰+⁢𝒯⁢𝒰-. One obtains in this way a representation of the universal enveloping algebra U⁢(𝔤^) on C∞⁢(𝒰+⁢𝒯⁢𝒰-,ℂ). Concretely, for ξ∈𝔤^, u±∈𝒰±, g∈𝒯, and f:𝒰+⁢𝒯⁢𝒰-→ℂ smooth,

Note that 𝔫+⊕𝔥⊕𝔫-=ℒpol⁢(𝔤), which is a dense subspace of ℒ⁢(𝔤) relative to the Fréchet topology. Moreover, the Lie algebra of 𝒰± is the closure of 𝔫± in ℒ⁢(𝔤) relative to the aforementioned topology. Recall that 𝔫+ is generated as a Lie algebra by ei, 0≤i≤r. Pick some complex numbers χ0+,…,χr+ and consider the Lie algebra homomorphism χ+:𝔫+→ℂ determined by χ+⁢(ei)=χi+ for all 0≤i≤r. Since 𝒰+ is simply connected and its Lie algebra contains 𝔫+ as a dense subspace, there is a unique lift χ+:𝒰+→ℂ* (which is a group homomorphism). In the same way, by picking χ0-,…,χr-∈ℂ, one attaches to them a Lie algebra homomorphism χ-:𝔫-→ℂ such that χ-⁢(fi)=χi-, 0≤i≤r, and then a Lie group homomorphism χ-:𝒰-→ℂ*. A smooth function ϕ:𝒰+⁢𝒯⁢𝒰-→ℂ is called a Whittaker function if

where h∨ is the dual Coxeter number of 𝔤. Such a function ϕ is clearly determined by its restriction to 𝒯. In turn, for any g∈T and any t0∈ℝ,

The initial value problem formed by this equation along with the condition ϕ⁢(exp⁡(t⁢c)⁢g)|t=0=ϕ⁢(g) has a unique solution. Thus, ϕ is uniquely determined by its values on T, hence the space of Whittaker functions can be naturally identified with C∞⁢(T,ℂ).

Equation (A.2) implies readily that for any Whittaker function ϕ and any 0≤i,j≤r one has

There exists a certain completion U~⁢(𝔤^) of U⁢(𝔤^) such that the center of U~⁢(𝔤^)/(c+h∨) is rich, see [4, 5]. First of all, it contains a degree two Casimir element C, which can be expressed as

Here ξ1,…,ξr is an orthonormal basis of 𝔥, and for each positive affine root α∈R^+, the vectors eαk are a basis of 𝔤^α and fαk a basis of 𝔤^-α such that 〈eαk,fαℓ〉=δk⁢ℓ. For α=αi both 𝔤^α and 𝔤^-α are one-dimensional and we take eαi1=ei, fαi1=fi, 0≤i≤r. Finally, ρ denotes the half-sum of all positive roots of (𝔤,𝔥) relative to the basis α1,…,αr and hρ the element of 𝔥 that corresponds to it via the Killing form. Note that even though the second sum in (A.9) is infinite, C belongs to the completion U~⁢(𝔤^).

To any ξ∈𝔱 we attach the directional derivative ∂ξ on C∞⁢(T,ℂ) given by

for all f∈C∞⁢(T,ℂ) and all g∈T. Equations (A.4)–(A.9) above imply that for any Whittaker function ϕ we have (C.ϕ)|T=D(ϕ|T), where D is the differential operator on C∞⁢(T,ℂ) given by

Here ∂ξ1,…,∂ξr,∂hρ are derivatives and eαi is the function T→ℂ given by exp⁡(h)↦eαi⁢(h) for all h∈𝔥, 0≤i≤r. Observe now that hρ=∑i=1rρ⁢(ξi)⁢ξi, hence the composition of D with e-ρ is

Since ∑i=1rρ⁢(ξi)2=〈ρ,ρ〉 and α0=-θ on 𝔥, one obtains

Choose χi+ and χi- such that χi+⁢χi-=-1 for i≠0, χ0+=-1, and χ0-=K. Then

The center of U~(𝔤^/(c+h∨) contains Y1:=C along with Y2,…,Yr, which are of the form Yi=ui+Yi+, where ui∈Sym⁢(𝔥) are fundamental generators of Sym⁢(𝔥)W and Yi+ is a sum of monomials in U⁢(𝔤^)⁢𝔫+ of degree at most equal to deg⁡ui, see [4, 5]. From equations (A.4)–(A.8) one deduces that for any Whittaker function ϕ one has (Yi.ϕ)|T=Di(ϕ|T), 1≤i≤r, where Di is a differential operator on C∞⁢(T,ℂ) which admits a presentation as a polynomial in K⁢e-θ,eα1,…,eαr, ∂ξ1,…,∂ξr (one also uses that (f0.ϕ)|T=χ0-eα0ϕ|T=Ke-θϕ|T). Furthermore, the symbol of Di is ui⁢(∂ξ1,…,∂ξr). The differential operators D=D1,D2,…,Dr commute with each other. But then also

commute with each other. Each of them is a polynomial in K⁢e-θ,eα1,…,eαr, ∂ξ1,…,∂ξr, since

Moreover, the symbol of Di′ is ui⁢(∂ξ1,…,∂ξr) as well.

Let us now consider the “a⁢x+b” algebra 𝔟=𝔞⊕𝔲 corresponding to the Dynkin diagram of 𝔤 extended with the highest root. Its complexification is 𝔟ℂ=𝔞ℂ⊕𝔲ℂ, where one can identify 𝔞ℂ=𝔥. Then 𝔲ℂ has a ℂ-basis X0,…,Xr such that the Lie bracket [⋅,⋅] on 𝔟ℂ is identically zero on both 𝔥 and 𝔲ℂ and for any h∈𝔥 one has

Let Ω1,…,Ωr be the elements of U⁢(𝔟ℂ) which are obtained from the differential operators D1′,…,Dr′ above by making the following replacements:

In this way

turns into 18⁢Ω, see equation (1.1). Moreover, if θ=m1⁢α1+⋯+mr⁢αr, then for any 1≤i,j≤r we have

Thus Ωi is in U⁢(𝔟ℂ)ev and satisfies the three conditions in Theorem 1.1.