Critical points of master functions and mKdV hierarchy of type $A^{(2)}_{2n}$

TL;DR

This paper studies the structure of critical points related to the master function of a twisted affine Lie algebra and shows their invariance under the mKdV hierarchy, revealing a deep connection between algebraic geometry and integrable systems.

Contribution

It constructs a partition of critical points into complex cells and defines rational maps to Miura opers, proving their invariance under the mKdV flows for the first time.

Findings

Critical points form complex cells parametrized by ^m.

Constructed injective rational maps to Miura opers.

Images are invariant under all mKdV flows and fixed by flows with index > 4m.

Abstract

We consider the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra $A^{(2)}_{2n}$. The population is naturally partitioned into an infinite collection of complex cells $\mathbb{C}^m$, where $m$ are some positive integers. For each cell we define an injective rational map $\mathbb{C}^m \to M(A^{(2)}_{2n})$ of the cell to the space $M(A^{(2)}_{2n})$ of Miura opers of type $A^{(2)}_{2n}$. We show that the image of the map is invariant with respect to all mKdV flows on $M(A^{(2)}_{2n})$ and the image is point-wise fixed by all mKdV flows $\frac\partial{\partial t_r}$ with index $r$ greater than $4m$.

Equations

$$u=(u^{(0)},\dots ,u^{(n)}),u^{(j)}=(u_1^{(j)},\dots ,u_{k_j}^{(j)}).$$

$$\Phi (u,z)=\sum _{a<b}({\Lambda }_a,{\Lambda }_b)\ln (z_a-z_b)-\sum _{a,i,j}({\alpha }_j,{\Lambda }_a)\ln (u_i^{(j)}-z_a)+$$

$$+\sum _{j<j^{\prime }}\sum _{i,i^{\prime }}({\alpha }_j,{\alpha }_{j^{\prime }})\ln (u_i^{(j)}-u_{i^{\prime }}^{(j^{\prime })})+\sum _j\sum _{i<i^{\prime }}({\alpha }_j,{\alpha }_j)\ln (u_i^{(j)}-u_{i^{\prime }}^{(j)}),$$

$$I(z,\kappa )=\int e^{\Phi (u,z)/\kappa }A(u,z)du,$$

$$d_u\Phi (u,z)=0.$$

$$\Phi (u_1^{(0)},\dots ,u_{k_0}^{(0)},\dots ,u_1^{(n)},\dots ,u_{k_n}^{(n)})$$

$$=-4\sum _{i,i^{\prime }}\ln (u_i^{(n-1)}-u_{i^{\prime }}^{(n)})-2\sum _{j=0}^{n-2}\sum _{i,i^{\prime }}\ln (u_i^{(j)}-u_{i^{\prime }}^{(j+1)})$$

$$+8\sum _{i<i^{\prime }}\ln (u_i^{(n)}-u_{i^{\prime }}^{(n)})+4\sum _{j=1}^{n-1}\sum _{i<i^{\prime }}\ln (u_i^{(j)}-u_{i^{\prime }}^{(j)})+2\sum _{i<i^{\prime }}\ln (u_i^{(0)}-u_{i^{\prime }}^{(0)}).$$

$$A_{2n}^{(1)}=\left(\begin{array}{cccc}{a}_{0,0}& a_{0,1}& \dots & a_{0,2n}\\ \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots \\ a_{2n,0}& a_{2n,1}& \dots & a_{2n,2n}\end{array}\right)=\left(\begin{array}{cccccc}2& -1& 0& \dots & 0& -1\\ -1& 2& -1& \dots & 0& 0\\ 0& -1& 2& \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots \\ 0& 0& 0& \dots & 2& -1\\ -1& 0& 0& \dots & -1& 2\end{array}\right).$$

$$[E_i,F_j]={\delta }_{i,j}{H}_i,$$

$$[H_i,E_j]=a_{i,j}{E}_j,[H_i,F_j]=-a_{i,j}{F}_j,(\mathrm{ad}{E}_i{)}^{1-a_{i,j}}{E}_j=0,$$

$$(\mathrm{ad}{F}_i{)}^{1-a_{i,j}}{F}_j=0,[H_i,H_j]=0,\sum _{i=0}^{2n}{H}_i=0,$$

$$C(X)=SX{S}^{-1},S=\mathrm{diag}(1,w,\dots ,w^{2n}).$$

$$L({\mathfrak{sl}}_{2n+1},C)={\oplus }_{j\in \mathbb{Z}}{\xi }^j\otimes ({\mathfrak{sl}}_{2n+1}{)}_{j\mathrm{mod}2n+1}.$$

$$E_0\mapsto \xi \otimes e_{1,2n+1},E_i\mapsto \xi \otimes e_{i+1,i},$$

$$F_0\mapsto {\xi }^{-1}\otimes e_{2n+1,1},F_i\mapsto {\xi }^{-1}\otimes e_{i,i+1},$$

$$H_0\mapsto 1\otimes (e_{1,1}-e_{2n+1,2n+1}),H_i\mapsto 1\otimes (-e_{i,i}+e_{i+1,i+1}).$$

$$E_0\mapsto \lambda \otimes e_{1,2n+1},E_i\mapsto 1\otimes e_{i+1,i},$$

$$F_0\mapsto {\lambda }^{-1}\otimes e_{2n+1,1},F_i\mapsto 1\otimes e_{i,i+1},$$

$$H_0\mapsto 1\otimes (e_{1,1}-e_{2n+1,2n+1}),H_i\mapsto 1\otimes (-e_{i,i}+e_{i+1,i+1}).$$

$$A_{(2n+1)m+j}={\xi }^{(2n+1)m+j}\otimes \left(\begin{array}{cc}0& I_j\\ I_{2n+1-j}& 0\end{array}\right)\in L({\mathfrak{sl}}_{2n+1},C),$$

$$B_{(2n+1)m+j}=\left(\begin{array}{cc}0& {\lambda }^{m+1}\otimes I_j\\ {\lambda }^m\otimes I_{2n+1-j}& 0\end{array}\right)\in L({\mathfrak{sl}}_{2n+1},{\sigma }_0).$$

$$E_0={\Lambda }^{(1)}{e}_{2n+1,2n+1},E_i={\Lambda }^{(1)}{e}_{i,i},F_0=e_{2n+1,2n+1}{({\Lambda }^{(1)})}^{-1},F_i=e_{i,i}{({\Lambda }^{(1)})}^{-1},$$

$$e^{g{F}_0}=1+g{e}_{2n+1,2n+1}({\Lambda }^{(1)}{)}^{-1},$$

$$e^{g(F_i+F_{2n+1-i})}=1+g(e_{i,i}+e_{2n+1-i,2n+1-i})({\Lambda }^{(1)}{)}^{-1},$$

$$e^{g2(F_n+F_{n+1})}=1+2g(e_{n,n}+e_{n+1,n+1})({\Lambda }^{(1)}{)}^{-1}+4g^2{e}_{n,n}({\Lambda }^{(1)}{)}^{-2}.$$

$$e_{i+1,i+1}{\Lambda }^{(1)}={\Lambda }^{(1)}{e}_{i,i},e_{i,i}({\Lambda }^{(1)}{)}^{-1}=({\Lambda }^{(1)}{)}^{-1}{e}_{i+1,i+1},$$

$$A_{2n}^{(2)}=\left(\begin{array}{cccc}{a}_{0,0}& a_{0,1}& \dots & a_{0,n}\\ \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots \\ a_{n,0}& a_{2n,1}& \dots & a_{n,n}\end{array}\right)=\left(\begin{array}{cccccccc}2& -1& 0& \dots & \dots & \dots & 0& \\ -2& 2& -1& 0& \dots & \dots & \dots & \\ 0& -1& 2& -1& \dots & \dots & \dots & \\ \dots & 0& -1& \dots & \dots & \dots & \dots & \\ \dots & \dots & \dots & \dots & 2& -1& 0& \\ \dots & \dots & \dots & \dots & -1& 2& -1& \\ 0& \dots & \dots & \dots & 0& -2& 2& \end{array}\right).$$

$$[e_i,f_j]={\delta }_{i,j}{h}_i,[h_i,e_j]=a_{i,j}{e}_j,[h_i,f_j]=-a_{i,j}{f}_j,$$

$$(\mathrm{ad}{e}_i{)}^{1-a_{i,j}}{e}_j=0,(\mathrm{ad}{f}_i{)}^{1-a_{i,j}}{f}_j=0,[h_i,h_j]=0,$$

$$2(h_0+h_1+\cdots +h_{n-1})+h_n=0,$$

$$C(X)=-S{X}^T{S}^{-1},S=\mathrm{diag}(1,-w,w^2,\dots ,w^{2n-2},-w^{2n-1},w^{2n}),$$

$$L({\mathfrak{sl}}_{2n+1},C)={\oplus }_{j\in \mathbb{Z}}{\xi }^j\otimes ({\mathfrak{sl}}_{2n+1}{)}_{j\mathrm{mod}4n+2}.$$

$$e_0\mapsto \xi \otimes e_{1,2n+1},e_n\mapsto \xi \otimes (e_{n+1,n}+e_{n+2,n+1})$$

$$f_0\mapsto {\xi }^{-1}\otimes e_{2n+1,1},f_n\mapsto {\xi }^{-1}\otimes 2(e_{n,n+1}+e_{n+1,n+2})$$

$$e_i\mapsto \xi \otimes (e_{i+1,i}+e_{2n+2-i,2n+1-i}),f_i\mapsto {\xi }^{-1}\otimes (e_{i,i+1}+e_{2n+1-i,2n+2-i}),$$

$$h_0\mapsto 1\otimes (e_{1,1}-e_{2n+1,2n+1}),h_n\mapsto 1\otimes 2(-e_{n,n}+e_{n+2,n+2}),$$

$$h_i\mapsto 1\otimes (-e_{i,i}+e_{i+1,i+1}-e_{2n+1-i,2n+1-i}+e_{2n+2-i,2n+2-i}).$$

$${\sigma }_0(X)=-Q{X}^T{Q}^{-1}=-X^T,Q=\mathrm{diag}(1,-1,\dots ,-1,1).$$

Full text

*Department of Mathematics, University of North Carolina at Chapel Hill

Chapel Hill, NC 27599-3250, USA*

††footnotetext: ⋆ E-mail: anv@email.unc.edu, supported in part by NSF grant DMS-1362924 and Simons Foundation

grant #336826

⋄ E-mail: tykwood@gmail.com

1. 1 Introduction

2. 2 Kac-Moody algebra of type $A_{2n}^{(1)}$

3. 2.1 Definition

4. 2.2 Realizations of ${{\mathfrak{g}}(A_{2n}^{(1)})}$

5. 2.3 Element $\Lambda^{(1)}$

6. 3 Kac-Moody algebra of type $A_{2n}^{(2)}$

7. 3.1 Definition

8. 3.2 Realizations of ${{\mathfrak{g}}(A_{2n}^{(2)})}$

9. 3.3 Element $\Lambda^{(2)}$

10. 3.4 Lie algebra ${{\mathfrak{g}}(A_{2n}^{(2)})}$ as a subalgebra of ${{\mathfrak{g}}(A_{2n}^{(1)})}$

11. 4 mKdV equations

12. 4.1 mKdV equations of type $A^{(1)}_{2n}$

13. 4.2 mKdV equations of type $A^{(2)}_{2n}$

14. 4.3 Comparison of mKdV equations of types $A^{(2)}_{2n}$ and $A^{(1)}_{2n}$

15. 4.4 KdV equations of type $A^{(1)}_{2n}$

16. 4.5 Miura maps

17. 5 Tangent maps to Miura maps

18. 5.1 Tangent spaces

19. 5.2 Formula for the first coefficient

20. 5.3 Intersection of kernels of $d\mathfrak{m}_{i}$

21. 6 Critical points of master functions and generation of tuples of polynomials

22. 6.1 Master function

23. 6.2 Polynomials representing critical points

24. 6.3 Elementary generation

25. 6.4 Degree increasing generation

26. 6.5 Degree-transformations and generation of vectors of integers

27. 6.6 Multistep generation

28. 6.7 Critical points and the population generated from $y^{\emptyset}$

29. 7 Critical points of master functions and Miura opers

30. 7.1 Miura oper associated with a tuple of polynomials, [MV2]

31. 7.2 Deformations of Miura opers of type ${A^{(2)}_{2n}}$, [MV2]

32. 7.3 Miura opers associated with the generation procedure

33. 7.4 Intersection of kernels of $d\mathfrak{m}_{i}$

34. 8 Vector fields

35. 8.1 Statement

36. 8.2 Proof of Theorem 8.1 for $m=1$

37. 8.3 Beginning of proof of Theorem 8.1 for $m>1$

38. 8.4 Proof of Proposition 8.3

39. 8.5 End of proof Theorem 8.1 for $m>1$

1. Introduction

Let ${\mathfrak{g}}$ be a Kac-Moody algebra with invariant scalar product $(\,,\,)$, ${\mathfrak{h}}\subset{\mathfrak{g}}$ Cartan subalgebra, $\alpha_{0},\dots,\alpha_{n}$ simple roots. Let $\Lambda_{1},\dots,\Lambda_{N}$ be dominant integral weights, $k_{0},\dots,k_{n}$ nonnegative integers, $k=k_{0}+\dots+k_{n}$.

Consider ${\mathbb{C}}^{N}$ with coordinates $z=(z_{1},\dots,z_{N})$. Consider ${\mathbb{C}}^{k}$ with coordinates $u$ collected into $n+1$ groups, the $j$-th group consisting of $k_{j}$ variables,

[TABLE]

The master function is the multivalued function on ${\mathbb{C}}^{k}\times{\mathbb{C}}^{N}$ defined by the formula

[TABLE]

with singularities at the places where the arguments of the logarithms are equal to zero.

A point in ${\mathbb{C}}^{k}\times{\mathbb{C}}^{N}$ can be interpreted as a collection of particles in ${\mathbb{C}}$: $z_{a},u^{(j)}_{i}$. A particle $z_{a}$ has weight $\Lambda_{a}$, a particle $u^{(j)}_{i}$ has weight $-\alpha_{j}$. The particles interact pairwise. The interaction of two particles is determined by the scalar product of their weights. The master function is the ”total energy” of the collection of particles.

Notice that all scalar products are integers. So the master function is the logarithm of a rational function. From a ”physical” point of view, all interactions are integer multiples of a certain unit of measurement. This is important for what will follow.

The variables $u$ are the true variables, variables $z$ are parameters. We may think that the positions of $z$-particles are fixed and the $u$-particles can move.

There are ”global” characteristics of this situation,

[TABLE]

where $A(u,z)$ is a suitable density function, $\kappa$ a parameter, and there are ”local” characteristics – critical points of the master function with respect to the $u$-variables,

[TABLE]

A critical point is an equilibrium position of the $u$-particles for fixed positions of the $z$-particles. In this paper we are interested in the equilibrium positions of the $u$-particles.

Examples of master functions associated with ${\mathfrak{g}}=\mathfrak{sl}_{2}$ were considered by Stieltjes and Heine in 19th century, see for example [Sz]. Master functions we introduced in [SV] to construct integral representations for solutions of the KZ equations, see also [V1, V2].

The critical points of master functions with respect to the $u$-variables were used to find eigenvectors in the associated Gaudin models by the Bethe ansatz method, see [BF, RV, V3]. In important cases the algebra of functions on the critical set of a master function is closely related to Schubert calculus, see [MTV].

In [ScV, MV1] it was observed that the critical points of master functions with respect to the $u$-variables can be deformed and form families. Having one critical point, one can construct a family of new critical points. The family is called a population of critical points. A point of the population is a critical point of the same master function or of another master function associated with the same ${\mathfrak{g}},\Lambda_{1},\dots,\Lambda_{N}$ but with different integer parameters $k_{0},\dots,k_{n}$. The population is a variety isomorphic to the flag variety of the Kac-Moody algebra ${\mathfrak{g}}^{t}$ Langlands dual to ${\mathfrak{g}}$, see [MV1, MV2, F].

In [VW], it was discovered that the population originated from the critical point of the master function associated with the affine Lie algebra $\widehat{\mathfrak{sl}}_{n+1}$ and the parameters $N=0,k_{0}=\dots=k_{n}=0$ is connected with the mKdV integrable hierarchy associated with $\widehat{\mathfrak{sl}}_{n+1}$. Namely, that population can be naturally embedded into the space of $\widehat{\mathfrak{sl}}_{n+1}$ Miura opers so that the image of the embedding is invariant with respect to all mKdV flows on the space of Miura opers. For $n=1$, that result follows from the classical paper by M. Adler and J. Moser [AM], which served as a motivation for [VW].

In this paper we prove the analogous statement for the twisted affine Lie algebra ${A^{(2)}_{2n}}$. The special case $A^{(2)}_{2}$ was considered in [VWW].

In Sections 2 - 4 we follow the paper [DS] by V. Drinfled and V. Sokolov. We review the affine Lie algebras $A^{(1)}_{2n}$ and ${A^{(2)}_{2n}}$, the associated mKdV and KdV hierarchies, Miura maps. For example, the ${A^{(2)}_{2n}}$ mKdV hierachy is a sequence of commuting flows on the infinite-dimensional space $\mathcal{M}({A^{(2)}_{2n}})$ of the ${A^{(2)}_{2n}}$ Miura opers.

In Section 5 we study tangent maps to Miura maps. In Section 6 formula (6.1), we introduce our master functions,

[TABLE]

This master function is the special case of the master function in (1.1). The master function in (1.2) is defined by formula (1.1) if ${\mathfrak{g}}$ is the Langlands dual to ${A^{(2)}_{2n}}$ and $N=0$, see a remark in Section 6.1.

Following [MV1, MV2, VW], we describe the generation procedure of new critical points starting from a given one. We define the population of critical points generated from the critical point of the function with no variables, namely, the function corresponding to the parameters $k_{0}=\dots=k_{n}=0$. That population is partitioned into complex cells ${\mathbb{C}}^{m}$ labeled by degree increasing sequences $J=(j_{1},\dots,i_{m})$, see the definition in Section 6.5.

In Theorem 6.5 we deduce from [MV3] that every critical point of the master function in (1.2) with arbitrary parameters $k_{0},\dots,k_{n}$ belongs a cell of our population. Moreover, a function in (1.2) with some parameters $k_{0},\dots,k_{n}$ either does not have critical points at all or its critical points form a cell ${\mathbb{C}}^{m}$ corresponding to a degree increasing sequence.

In Section 7, to every degree increasing sequence $J$ we assign a rational injective map $\mu^{J}:{\mathbb{C}}^{m}\to\mathcal{M}({A^{(2)}_{2n}})$ of the cell corresponding to $J$ to the space $\mathcal{M}({A^{(2)}_{2n}})$ of Miura opers of type ${A^{(2)}_{2n}}$. We describe properties of that map.

In Section 8, we formulate and prove our main result. Theorem 8.1 says that for any degree increasing sequence, the variety $\mu^{J}({\mathbb{C}}^{m})$ is invariant with respect to all mKdV flows on $\mathcal{M}({A^{(2)}_{2n}})$ and that variety is point-wise fixed by all flows $\frac{\partial}{\partial t_{r}}$ with index $r$ greater than $4m$.

Our result shows that there is a deep interrelation between the critical set of the master functions of the form (1.2) and rational finite-dimensional submanifolds of the space $\mathcal{M}({A^{(2)}_{2n}})$, invariant with respect to all flows of the ${A^{(2)}_{2n}}$ mKdV hierarchy.

Initially the critical points of the master functions were related to quantum integrable systems of the Gaudin type through the Bethe ansatz, [SV, BF, RV, V3]. Our result shows that the critical points are also related to the classical integrable systems, namely, the mKdV hierarchies.

In the next papers we plan to extend this result to other affine Lie algebras.

The first author thanks MPI in Bonn for hospitality in 2015-2016.

2. Kac-Moody algebra of type $A_{2n}^{(1)}$

In this section we follow [DS, Section 5].

2.1. Definition

For $n\geqslant 2$, consider the $(2n+1)\times(2n+1)$ Cartan matrix of type $A_{2n}^{(1)}$,

[TABLE]

The Kac-Moody algebra ${{\mathfrak{g}}(A_{2n}^{(1)})}$ of type $A_{2n}^{(1)}$ is the Lie algebra with canonical generators $E_{i},H_{i},F_{i}\in{{\mathfrak{g}}(A_{2n}^{(1)})},i=0,\dots,2n$, subject to the relations:

[TABLE]

see [DS, Section 5]. The Lie algebra ${{\mathfrak{g}}(A_{2n}^{(1)})}$ is graded with respect to the standard grading, $\deg E_{i}=1,\deg F_{i}=-1$, $i=0,\dots,2n$. Let ${{\mathfrak{g}}(A_{2n}^{(1)})}^{j}=\{x\in{\mathfrak{g}}(A_{2n}^{(1)})\ |\ \deg x=j\}$, then ${{\mathfrak{g}}(A_{2n}^{(1)})}=\oplus_{j\in{\mathbb{Z}}}\,{{\mathfrak{g}}(A_{2n}^{(1)})}^{j}$.

Notice that ${\mathfrak{g}}(A_{2n}^{(1)})^{0}$ is the $2n$-dimensional space generated by the $H_{i}$. Denote ${\mathfrak{h}}={\mathfrak{g}}(A_{2n}^{(1)})^{0}$. Introduce elements $\alpha_{j}$ of the dual space ${\mathfrak{h}}^{*}$ by the conditions $\langle\alpha_{j},H_{i}\rangle=a_{i,j}$ for $i,j=0,\dots,2n$. For $j=0,1,\dots 2n$, we denote by ${\mathfrak{n}}_{j}^{-}\subset{{\mathfrak{g}}(A_{2n}^{(1)})}$ the Lie subalgebra generated by $F_{i}$, $i\in\{0,1,\dots,2n\},\,i\neq j$. For example, ${\mathfrak{n}}^{-}_{0}$ is generated by $F_{1},F_{2},\dots,F_{2n}$.

2.2. Realizations of ${{\mathfrak{g}}(A_{2n}^{(1)})}$

Consider the complex Lie algebra $\mathfrak{sl}_{2n+1}$ with standard basis $e_{i,j}$, $i,j=1,\dots,2n+1$.

Let $w=e^{2\pi i/{(2n+1)}}$. Define the Coxeter automorphism $C:\mathfrak{sl}_{2n+1}\to\mathfrak{sl}_{2n+1}$ of order $2n+1$ by the formula

[TABLE]

Denote $(\mathfrak{sl}_{2n+1})_{j}=\{x\in\mathfrak{sl}_{2n+1}\ |\ Cx=w^{j}x\}$. The twisted Lie subalgebra $L(\mathfrak{sl}_{2n+1},C)\subset\mathfrak{sl}_{2n+1}[\xi,\xi^{-1}]$ is the subalgebra

[TABLE]

The isomorphism $\tau_{C}:{{\mathfrak{g}}(A_{2n}^{(1)})}\to L(\mathfrak{sl}_{2n+1},C)$ is defined by the formula, for $i=1,\dots,2n$

[TABLE]

Under this isomorphism we have ${{\mathfrak{g}}(A_{2n}^{(1)})}^{j}=\xi^{j}\otimes({\mathfrak{sl}_{2n+1}})_{j}$.

The standard automorphism $\sigma_{0}:\mathfrak{sl}_{2n+1}\to\mathfrak{sl}_{2n+1}$ is the identity: $\sigma_{0}(X)=X$. The isomorphism $\tau_{0}:{{\mathfrak{g}}(A_{2n}^{(1)})}\to L(\mathfrak{sl}_{2n+1},\sigma_{0})$ is defined by the formula, for $i=1,\dots,2n$

[TABLE]

2.3. Element $\Lambda^{(1)}$

Denote by $\Lambda^{(1)}$ the element $\sum_{j=0}^{2n}{E_{j}}\in{{\mathfrak{g}}(A_{2n}^{(1)})}$. Then $\mathfrak{z}(A_{2n}^{(1)})=\{x\in{{\mathfrak{g}}(A_{2n}^{(1)})}\ |\ [\Lambda^{(1)},x]=0\}$ is an Abelian Lie subalgebra of ${\mathfrak{g}}(A_{2n}^{(1)})$. Denote $\mathfrak{z}(A_{2n}^{(1)})^{j}=\mathfrak{z}(A_{2n}^{(1)})\cap{{\mathfrak{g}}(A_{2n}^{(1)})}^{j}$, then $\mathfrak{z}(A_{2n}^{(1)})=\oplus_{j\in{\mathbb{Z}}}\,\mathfrak{z}(A_{2n}^{(1)})^{j}$. We have $\dim\mathfrak{z}(A_{2n}^{(1)})^{j}=1$ if $j\neq 0$ mod $2n+1$ and $\dim\mathfrak{z}(A_{2n}^{(1)})^{j}=0$ otherwise.

Let ${{\mathfrak{g}}(A_{2n}^{(1)})}$ be realized as $L({\mathfrak{sl}_{2n+1}},C)$ and written out as $(2n+1)\times(2n+1)$-matrices. For $m\in{\mathbb{Z}}$ and $1\leqslant j<2n+1$ introduce the element

[TABLE]

where $I_{j}$ is the $j\times j$ identity matrix. We have $A_{(2n+1)m+j}=(A_{1})^{(2n+1)m+j}$.

If ${{\mathfrak{g}}(A_{2n}^{(1)})}$ is realized as $L({\mathfrak{sl}_{2n+1}},\sigma_{0})$, introduce the element

[TABLE]

We have $B_{(2n+1)m+j}=(B_{1})^{(2n+1)m+j}$.

Lemma 2.1**.**

*For any $m\in{\mathbb{Z}}$, $1\leqslant j<2n+1$, the elements $(\tau_{C})^{-1}(A_{(2n+1)m+j})$,

$(\tau_{0})^{-1}(B_{(2n+1)m+j})$ of $\mathfrak{z}(A_{2n}^{(1)})^{(2n+1)m+j}$ are equal. ∎*

Denote by $\Lambda^{(1)}_{(2n+1)m+j}$ the elements $(\tau_{C})^{-1}(A_{(2n+1)m+j})$ and $(\tau_{0})^{-1}(B_{(2n+1)m+j})$ of ${{\mathfrak{g}}(A_{2n}^{(1)})}$. Notice that $\Lambda^{(1)}_{1}=\sum_{i=0}^{2n}{E_{i}}=\Lambda^{(1)}$. For any $m\in{\mathbb{Z}},1\leqslant j<2n+1,$ the element $\Lambda^{(1)}_{(2n+1)m+j}$ generates $\mathfrak{z}(A_{2n}^{(1)})^{(2n+1)m+j}$.

Let $T=\sum_{j=-\infty}^{m}T_{j}$ be a formal series with $T_{j}\in{{\mathfrak{g}}(A_{2n}^{(1)})}^{j}$. Denote $T^{+}=\sum_{j=0}^{n}T_{j},$   $T^{-}=\sum_{j<0}T_{j}$. Let ${{\mathfrak{g}}(A_{2n}^{(1)})}$ be realized as ${\mathfrak{sl}_{2n+1}}[\lambda,\lambda^{-1}]$. Consider $\Lambda^{(1)}=B_{1}$ as a $(2n+1)\times(2n+1)$ matrix depending on the parameter $\lambda$. By [DS, Lemma 3.4], we may represent $T$ uniquely in the form $T=\sum_{j=-\infty}^{k}b_{j}\,(\Lambda^{(1)})^{j}$,  $b_{j}\in{\operatorname{Diag}}$, where ${\operatorname{Diag}}\subset\mathfrak{gl}_{2n+1}$ is the space of diagonal $(2n+1)\times(2n+1)$ matrices. Denote $(T)_{\Lambda^{(1)}}^{+}=\sum_{j=0}^{k}b_{j}\,(\Lambda^{(1)})^{j},$   $(T)_{\Lambda^{(1)}}^{-}=\sum_{j<0}b_{j}\,(\Lambda^{(1)})^{j}$.

Lemma 2.2**.**

We have $(T)_{\Lambda^{(1)}}^{+}=T^{+}$, $(T)_{\Lambda^{(1)}}^{-}=T^{-}$, $b_{0}=T^{0}$.

Proof.

The isomorphism $\iota:{\mathfrak{sl}_{2n+1}}[\lambda,\lambda^{-1}]\to L(\mathfrak{sl}_{2n+1},C^{(1)})$ is given by the formula $\lambda^{m}\otimes e_{k,l}\mapsto\xi^{(2n+1)m+k-l}{\otimes}e_{k,l}$. We have $\iota(b_{0})=\iota(1{\otimes}(b_{0}^{1}e_{1,1}+\dots+b_{0}^{2n+1}e_{2n+1,2n+1}))=1\otimes(b_{0}^{1}e_{1,1}+\dots+b_{0}^{2n+1}e_{2n+1,2n+1})\in{{\mathfrak{g}}(A_{2n}^{(1)})}^{0}$, $\iota(b_{1}\Lambda^{(1)})=\iota((b_{1}^{1}e_{1,1}+b_{1}^{2}e_{2,2}+\dots+b_{1}^{2n+1}e_{2n+1,2n+1})(e_{2,1}+\dots+e_{2n+1,2n}+\lambda e_{1,2n+1}))=\iota(b_{1}^{1}\lambda e_{1,2n+1}+b_{1}^{2}e_{2,1}+\dots+b_{1}^{2n+1}e_{2n+1,2n})=\xi\otimes(b_{1}^{1}e_{1,2n+1}+b_{1}^{2}e_{2,1}+\dots+b_{1}^{2n+1}e_{2n+1,2n})\in{{\mathfrak{g}}(A_{2n}^{(1)})}^{1}$, $\iota(b_{-1}(\Lambda^{(1)})^{-1})=\iota((b_{-1}^{1}e_{1,1}+b_{-1}^{2}e_{2,2}+\dots+b_{-1}^{2n+1}e_{2n+1,2n+1})(e_{1,2}+\dots+e_{2n,2n+1}+\lambda^{-1}e_{2n+1,1}))=\iota(b_{-1}^{1}e_{1,2}+\dots+b_{-1}^{2n}e_{2n,2n+1}+b_{-1}^{2n+1}\lambda^{-1}e_{2n+1,1})=\xi^{-1}\otimes(b_{-1}^{1}e_{1,2}+\dots+b_{-1}^{2n}e_{2n,2n+1}+b_{-1}^{2n+1}e_{2n+1,1})\in{{\mathfrak{g}}(A_{2n}^{(1)})}^{-1}$. Similarly one checks that $\iota(b_{j}\,(\Lambda^{(1)})^{j})\in{{\mathfrak{g}}(A_{2n}^{(1)})}^{j}$ for any $j$. ∎

We have ${(\Lambda^{(1)})}^{-1}=\sum_{i=1}^{2n}e_{i,i+1}+\lambda^{-1}e_{2n+1,1}$,

[TABLE]

for $i=1,\dots,2n$.

Lemma 2.3**.**

Consider the elements $F_{0},F_{i}+F_{2n+1-i},2(F_{n}+F_{n+1})$ for $i=1,\dots,n-1$ as $(2n+1)\times(2n+1)$ matrices. Let $g\in{\mathbb{C}}$. Then

[TABLE]

∎

Lemma 2.4**.**

We have

[TABLE]

for all $i$, where we set $e_{2n+2,2n+2}=e_{1,1}$. ∎

3. Kac-Moody algebra of type $A_{2n}^{(2)}$

In this section we follow [DS, Section 5].

3.1. Definition

For $n\geqslant 2$, consider the $(n+1)\times(n+1)$ Cartan matrix of type ${A^{(2)}_{2n}}$,

[TABLE]

The Kac-Moody algebra ${\mathfrak{g}}({A^{(2)}_{2n}})$ of type ${A^{(2)}_{2n}}$ is the Lie algebra with canonical generators $e_{i},h_{i},f_{i}\in{{\mathfrak{g}}(A_{2n}^{(2)})},i=0,\dots,n$, subject to the relations

[TABLE]

see [DS, Section 5].

The Lie algebra ${{\mathfrak{g}}(A_{2n}^{(2)})}$ is graded with respect to the standard grading, $\deg e_{i}=1,\deg f_{i}=-1$, $i=0,\dots,n$. Let ${{\mathfrak{g}}(A_{2n}^{(2)})}^{j}=\{x\in{{\mathfrak{g}}(A_{2n}^{(2)})}\ |\ \deg x=j\}$, then ${{\mathfrak{g}}(A_{2n}^{(2)})}=\oplus_{j\in{\mathbb{Z}}}\,{{\mathfrak{g}}(A_{2n}^{(2)})}^{j}$.

Notice that ${{\mathfrak{g}}(A_{2n}^{(2)})}^{0}$ is the n-dimensional space generated by the $h_{i}$. Denote ${\mathfrak{h}}={{\mathfrak{g}}(A_{2n}^{(2)})}^{0}$. Introduce elements $\alpha_{j}$ of the dual space ${\mathfrak{h}}^{*}$ by the conditions $\langle\alpha_{j},h_{i}\rangle=a_{i,j}$ for $i,j=0,\dots,n$.

3.2. Realizations of ${{\mathfrak{g}}(A_{2n}^{(2)})}$

Consider the complex Lie algebra $\mathfrak{sl}_{2n+1}$ with standard basis $e_{i,j}$, $i,j=1,2n+1$.

Let $w=e^{2\pi i/(4n+2)}$. Define the Coxeter automorphism $C:\mathfrak{sl}_{2n+1}\to\mathfrak{sl}_{2n+1}$ of order $4n+2$ by the formula

[TABLE]

where the $\{\}^{T}$ denotes transposition across the antidiagonal. Denote $(\mathfrak{sl}_{2n+1})_{j}=\{x\in\mathfrak{sl}_{2n+1}\ |\ Cx=w^{j}x\}$. The twisted Lie subalgebra $L(\mathfrak{sl}_{2n+1},C)\subset{\mathfrak{sl}_{2n+1}}[\xi,\xi^{-1}]$ is the subalgebra

[TABLE]

The isomorphism $\tau_{C}:{{\mathfrak{g}}(A_{2n}^{(2)})}\to L(\mathfrak{sl}_{2n+1},C)$ is defined by the formula

[TABLE]

Under this isomorphism we have ${{\mathfrak{g}}(A_{2n}^{(2)})}^{j}=\xi^{j}\otimes({\mathfrak{sl}_{2n+1}})_{j}$. Define the standard automorphism $\sigma_{0}:\mathfrak{sl}_{2n+1}\to\mathfrak{sl}_{2n+1}$ of order 2 by the formula

[TABLE]

Where the $\{\}^{T}$ is again transposition across the antidiagonal. Let $({\mathfrak{sl}_{2n+1}})_{0,j}=\{x\in{\mathfrak{sl}_{2n+1}}\ |\ \sigma_{0}x=(-1)^{j}x\}$. Then the twisted Lie subalgebra $L(\mathfrak{sl}_{2n+1},\sigma_{0})\subset{\mathfrak{sl}_{2n+1}}[\lambda,\lambda^{-1}]$ is the subalgebra

[TABLE]

The isomorphism $\tau_{0}:{{\mathfrak{g}}(A_{2n}^{(2)})}\to L(\mathfrak{sl}_{2n+1},\sigma_{0})$ is defined by the formula

[TABLE]

3.3. Element $\Lambda^{(2)}$

Denote by $\Lambda^{(2)}$ the element $\sum_{i=0}^{n}{e_{i}}\in{{\mathfrak{g}}(A_{2n}^{(2)})}$. Then $\mathfrak{z}(A_{2n}^{(2)})=\{x\in{{\mathfrak{g}}(A_{2n}^{(2)})}\ |\ [\Lambda^{(2)},x]=0\}$ is an Abelian Lie subalgebra of ${{\mathfrak{g}}(A_{2n}^{(2)})}$. Denote $\mathfrak{z}^{j}(A^{(2)}_{2n})=\mathfrak{z}(A^{(2)}_{2n})\cap{{\mathfrak{g}}(A_{2n}^{(2)})}^{j}$, then $\mathfrak{z}(A_{2n}^{(2)})=\oplus_{j\in{\mathbb{Z}}}\,\mathfrak{z}(A_{2n}^{(2)})^{j}$. We have $\dim\mathfrak{z}(A_{2n}^{(2)})^{j}=0$ if j is even or $j=2n+1\mod 4n+2$ , and $\dim\mathfrak{z}(A_{2n}^{(2)})^{j}=1$ otherwise.

For example, $\dim\mathfrak{z}^{j}(A^{(2)}_{2})=1$ if $j=1,5\mod 6$ and the dimension equals zero otherwise; $\dim\mathfrak{z}^{j}(A^{(2)}_{4})=1$ if $j=1,3,7,9\mod 10$ and the dimension equals zero otherwise.

If ${{\mathfrak{g}}(A_{2n}^{(2)})}$ is realized as $L({\mathfrak{sl}_{2n+1}},C)$ and written out as $(2n+1)\times(2n+1)$ matrices, then for odd j, $1\leqslant j<2n+1$, introduce the element:

[TABLE]

where $I_{j}$ is the $j\times j$ identity matrix. We have $A_{(4n+2)m\pm j}=A_{1}^{(4n+2)m\pm j}$.

If ${{\mathfrak{g}}(A_{2n}^{(2)})}$ is realized as $L({\mathfrak{sl}_{2n+1}},\sigma_{0})$ and written out as $(2n+1)\times(2n+1)$ matrices, then for odd j, $1\leqslant j<2n+1$, introduce the elements:

[TABLE]

We have $B_{(4n+2)m\pm j}=B_{1}^{(4n+2)m\pm j}$.

Lemma 3.1**.**

For any $m\in{\mathbb{Z}}$, odd j, $1\leqslant j<2n+1$, the elements

[TABLE]

of $\mathfrak{z}(A_{2n}^{(2)})^{(4n+2)m\pm j}$ are equal. ∎

Denote the elements $(\tau_{C})^{-1}(A_{(4n+2)m+j})$, and $(\tau_{0})^{-1}(A_{(4n+2)m-j})$ of ${{\mathfrak{g}}(A_{2n}^{(2)})}$ by $\Lambda^{(2)}_{(4n+2)m+j}$ and $\Lambda^{(2)}_{(4n+2)m-j}$, respectively. Notice that $\Lambda^{(2)}_{1}=\sum_{i=0}^{n}{e_{i}}=\Lambda^{(2)}$. We set $\Lambda^{(2)}_{j}=0$ if j is even or $j=2n+1\mod 4n+2$. The element $\Lambda^{(2)}_{(4n+2)m\pm j}$ generates $\mathfrak{z}(A_{2n}^{(2)})^{(4n+2)m\pm j}$.

3.4. Lie algebra ${{\mathfrak{g}}(A_{2n}^{(2)})}$ as a subalgebra of ${{\mathfrak{g}}(A_{2n}^{(1)})}$

The map $\rho:{{\mathfrak{g}}(A_{2n}^{(2)})}\to{{\mathfrak{g}}(A_{2n}^{(1)})}$,

[TABLE]

where $i=1,\dots,n-1$, realizes the Lie algebra ${{\mathfrak{g}}(A_{2n}^{(2)})}$ as a subalgebra of ${{\mathfrak{g}}(A_{2n}^{(1)})}$. This embedding preserves the standard grading and $\rho(\Lambda^{(2)})=\Lambda^{(1)}$. We have $\rho(\mathfrak{z}(A_{2n}^{(2)})^{j})\subset\mathfrak{z}(A_{2n}^{(1)})^{j}$.

4. mKdV equations

In this section we follow [DS].

4.1. mKdV equations of type $A^{(1)}_{2n}$

Denote by ${\mathcal{B}}$ the space of complex-valued functions of one variable $x$. Given a finite dimensional vector space $W$, denote by ${\mathcal{B}}(W)$ the space of $W$-valued functions of $x$. Denote by $\partial$ the differential operator $\frac{d}{dx}$.

Consider the Lie algebra $\tilde{{\mathfrak{g}}}(A_{2n}^{(1)})$ of the formal differential operators of the form $c\partial+\sum_{i=-\infty}^{k}p_{i}$, $c\in{\mathbb{C}},p_{i}\in{\mathcal{B}}({{\mathfrak{g}}(A_{2n}^{(1)})}^{i})$. Let $U=\sum_{i<0}U_{i}$, $U_{i}\in{\mathcal{B}}({{\mathfrak{g}}(A_{2n}^{(1)})}^{i})$. If ${\mathcal{L}}\in\tilde{\mathfrak{g}}(A_{2n}^{(1)})$, define

[TABLE]

The operator $e^{{\operatorname{ad}}U}({\mathcal{L}})$ belongs to $\tilde{\mathfrak{g}}(A_{2n}^{(1)})$. The map $e^{{\operatorname{ad}}U}$ is an automorphism of the Lie algebra $\tilde{\mathfrak{g}}(A_{2n}^{(1)})$. The automorphisms of this type form a group. If elements of ${{\mathfrak{g}}(A_{2n}^{(1)})}$ are realized as matrices depending on a parameter as in Section 2.2, then $e^{{\operatorname{ad}}U}({\mathcal{L}})=e^{U}{\mathcal{L}}e^{-U}$.

A Miura oper of type $A^{(1)}_{2n}$ is a differential operator of the form

[TABLE]

where $\Lambda^{(1)}=\sum_{i=0}^{2n}E_{i}\in{{\mathfrak{g}}(A_{2n}^{(1)})}$ and $V\in{\mathcal{B}}({{\mathfrak{g}}(A_{2n}^{(1)})}^{0})$. Any Miura oper of type $A^{(1)}_{2n}$ is an element of $\tilde{\mathfrak{g}}(A^{(1)}_{2n})$. Denote by $\mathcal{M}(A^{(1)}_{2n})$ the space of all Miura opers of type $A^{(1)}_{2n}$.

Proposition 4.1** ([DS, Proposition 6.2]).**

For any Miura oper ${\mathcal{L}}$ of type $A^{(1)}_{2n}$ there exists an element $U=\sum_{i<0}U_{i}$, $U_{i}\in{\mathcal{B}}({{\mathfrak{g}}(A_{2n}^{(1)})}^{i})$, such that the operator ${\mathcal{L}}_{0}=e^{{\operatorname{ad}}U}({\mathcal{L}})$ has the form

[TABLE]

where $H=\sum_{j<0}H_{j},H_{j}\in{\mathcal{B}}(\mathfrak{z}(A_{2n}^{(1)})^{j})$. If $U,\tilde{U}$ are two such elements, then $e^{{\operatorname{ad}}U}e^{-{\operatorname{ad}}\tilde{U}}=e^{{\operatorname{ad}}T}$, where $T=\sum_{j<0}T_{j}$, $T_{j}\in\mathfrak{z}(A^{(1)}_{2n})^{j}$. ∎

Let ${\mathcal{L}},U$ be as in Proposition 4.1. Let $r\neq 0$ mod $2n+1$. The element $\phi(\Lambda^{(1)}_{r})=e^{-{\operatorname{ad}}U}(\Lambda^{(1)}_{r})$ does not depend on the choice of $U$ in Proposition 4.1.

The element $\phi(\Lambda^{(1)}_{r})$ is of the form $\sum^{k}_{i=-\infty}\phi(\Lambda^{(1)}_{r})^{i}$, $\phi(\Lambda^{(1)}_{r})^{i}\in{\mathcal{B}}({{\mathfrak{g}}(A_{2n}^{(1)})}^{i})$. We set $\phi(\Lambda^{(1)}_{r})^{+}=\sum^{k}_{i=0}\phi(\Lambda^{(1)}_{r})^{i}$, $\phi(\Lambda^{(1)}_{r})^{-}=\sum_{i<0}\phi(\Lambda^{(1)}_{r})^{i}$.

Let $r\in{\mathbb{Z}}_{>0}$ and $r\neq 0$ mod $2n+1$. The differential equation

[TABLE]

is called the $r$-th mKdV equation of type $A^{(1)}_{2n}$.

Equation (4.2) defines vector fields $\frac{\partial}{\partial t_{r}}$ on the space ${\mathcal{M}}(A^{(1)}_{2n})$ of Miura opers of type $A^{(1)}_{2n}$. For all $r,s$, the vector fields $\frac{\partial}{\partial t_{r}}$, $\frac{\partial}{\partial t_{s}}$ commute, see [DS, Section 6].

Lemma 4.2** ([DS]).**

We have

[TABLE]

∎

4.2. mKdV equations of type $A^{(2)}_{2n}$

A Miura oper of type $A^{(2)}_{2n}$ is a differential operator of the form

[TABLE]

where $\Lambda^{(2)}=\sum_{i=0}^{n}e_{i}\in{{\mathfrak{g}}(A_{2n}^{(2)})}$ and $V\in{\mathcal{B}}({{\mathfrak{g}}(A_{2n}^{(2)})}^{0})$. Denote by $\mathcal{M}(A^{(2)}_{2n})$ the space of all Miura opers of type $A^{(2)}_{2n}$.

Proposition 4.3** ([DS, Proposition 6.2]).**

For any Miura oper ${\mathcal{L}}$ of type $A^{(2)}_{2n}$ there exists an element $U=\sum_{i<0}U_{i}$, $U_{i}\in{\mathcal{B}}({\mathfrak{g}}(A^{(2)}_{2n})^{i})$, such that the operator ${\mathcal{L}}_{0}=e^{{\operatorname{ad}}U}({\mathcal{L}})$ has the form

[TABLE]

where $H=\sum_{j<0}H_{j},H_{j}\in{\mathcal{B}}(\mathfrak{z}(A_{2n}^{(2)})^{j})$. If $U,\tilde{U}$ are two such elements, then $e^{{\operatorname{ad}}U}e^{-{\operatorname{ad}}\tilde{U}}=e^{{\operatorname{ad}}T}$, where $T=\sum_{j<0}T_{j}$, $T_{j}\in\mathfrak{z}(A_{2n}^{(2)})^{j}$. ∎

Let ${\mathcal{L}},U$ be as in Proposition 4.3. Let $r$ be odd, $r\neq 2n+1$ mod $4n+2$. The element $\phi(\Lambda^{(2)}_{r})=e^{-{\operatorname{ad}}U}(\Lambda^{(2)}_{r})$ does not depend on the choice of $U$ in Proposition 4.3.

The element $\phi(\Lambda^{(2)}_{r})$ is of the form $\sum^{k}_{i=-\infty}\phi(\Lambda^{(2)}_{r})^{i}$, $\phi(\Lambda^{(2)}_{r})^{i}\in{\mathcal{B}}({{\mathfrak{g}}(A_{2n}^{(2)})}^{i})$. We set $\phi(\Lambda^{(2)}_{r})^{+}=\sum^{k}_{i=0}\phi(\Lambda^{(2)}_{r})^{i}$, $\phi({\mathcal{L}}^{(2)}a_{r})^{-}=\sum_{i<0}\phi(\Lambda^{(2)}_{r})^{i}$.

Let $r\in{\mathbb{Z}}_{>0}$, $r$ odd and $r\neq 2n+1$ mod $4n+2$. The differential equation

[TABLE]

is called the $r$-th mKdV equation of type $A^{(2)}_{2n}$.

Equation (4.5) defines vector fields $\frac{\partial}{\partial t_{r}}$ on the space ${\mathcal{M}}(A^{(2)}_{2})$ of Miura opers. For all $r,s$, the vector fields $\frac{\partial}{\partial t_{r}}$, $\frac{\partial}{\partial t_{s}}$ commute, see [DS, Section 6].

Lemma 4.4** ([DS]).**

We have

[TABLE]

∎

4.3. Comparison of mKdV equations of types $A^{(2)}_{2n}$ and $A^{(1)}_{2n}$

Consider ${\mathfrak{g}}(A^{(2)}_{2n})$ as a Lie subalgebra of ${{\mathfrak{g}}(A_{2n}^{(1)})}$, see Section 3.4. If ${\mathcal{L}}$ is a Miura oper of type $A^{(2)}_{2n}$, then it is also a Miura oper of type $A^{(1)}_{2n}$. We have $\mathcal{M}(A^{(2)}_{2n})\subset\mathcal{M}(A^{(1)}_{2n})$,

[TABLE]

Lemma 4.5**.**

Let $r$ be odd, $r\neq 2n+1$ mod $4n+2$, $r>0$. Let ${\mathcal{L}}^{A^{(2)}_{2n}}(t_{r})$ be the solution of the $r$-th mKdV equation of type $A^{(2)}_{2n}$ with initial condition ${\mathcal{L}}^{A^{(2)}_{2n}}(0)={\mathcal{L}}$. Let ${\mathcal{L}}^{A^{(1)}_{2n}}(t_{r})$ be the solution of the $r$-th mKdV equation of type $A^{(1)}_{2n}$ with initial condition ${\mathcal{L}}^{A^{(1)}_{2n}}(0)={\mathcal{L}}$. Then ${\mathcal{L}}^{A^{(2)}_{2n}}(t_{r})={\mathcal{L}}^{A^{(1)}_{2n}}(t_{r})$ for all values of $t_{r}$. ∎

Proof.

The element $U$ in Proposition 4.3 which is used to construct the mKdV equation of type $A^{(2)}_{2n}$ can be used also to construct the mKdV equation of type $A^{(1)}_{2n}$. ∎

4.4. KdV equations of type $A^{(1)}_{2n}$

Let ${\mathcal{B}}((\partial^{-1}))$ be the algebra of formal pseudodifferential operators of the form $a=\sum_{i\in{\mathbb{Z}}}a_{i}\partial^{i}$, with $a_{i}\in{\mathcal{B}}$ and finitely many terms with $i>0$. The relations in this algebra are

[TABLE]

for any $k\in{\mathbb{Z}}$ and $u\in{\mathcal{B}}$. For $a=\sum_{i\in{\mathbb{Z}}}a_{i}\partial^{i}\in{\mathcal{B}}((\partial^{-1}))$, define $a^{+}=\sum_{i\geqslant 0}a_{i}\partial^{i}$.

Denote ${\mathcal{B}}[\partial]\subset{\mathcal{B}}((\partial^{-1}))$ the subalgebra of differential operators $a=\sum_{i=0}^{m}a_{i}\partial^{i}$ with $m\in{\mathbb{Z}}_{\geqslant 0}$. Denote ${\mathcal{D}}\subset{\mathcal{B}}[\partial]$ the affine subspace of differential operators of the form

$L=\partial^{2n+1}+\sum\limits_{i=0}^{2n-1}u_{i}\partial^{i}$.

For $L\in{\mathcal{D}}$, there exists a unique $L^{\frac{1}{2n+1}}=\partial+\sum_{i\leqslant 0}a_{i}\partial^{i}\in{\mathcal{B}}((\partial^{-1}))$ such that $(L^{\frac{1}{2n+1}})^{2n+1}=L$. For $r\in{\mathbb{N}}$, we have $L^{\frac{r}{2n+1}}=\partial^{r}+\sum^{r-1}_{i=-\infty}b_{i}\partial^{i}$, $b_{i}\in{\mathcal{B}}$. We set $(L^{\frac{r}{2n+1}})^{+}=\partial^{r}+\sum^{r-1}_{i=0}b_{i}\partial^{i}$.

For $r\in{\mathbb{N}}$, the differential equation

[TABLE]

is called the $r$-th KdV equation of type $A^{(1)}_{2n}$.

Equation (4.8) defines flows $\frac{\partial}{\partial t_{r}}$ on the space ${\mathcal{D}}$. For all $r,s\in{\mathbb{N}}$ the flows $\frac{\partial}{\partial t_{r}}$ and $\frac{\partial}{\partial t_{s}}$ commute, see [DS].

4.5. Miura maps

Let ${\mathcal{L}}=\partial+\Lambda^{(1)}+V$ be a Miura oper of type $A^{(1)}_{2n}$ with $V=\sum_{k=1}^{2n+1}v_{k}e_{k,k}$, $\sum_{k=1}^{2n+1}v_{k}=0$. For $i=0,\dots,2n+1$, define the scalar differential operator $L_{i}=\partial^{2n+1}+\sum_{j=0}^{2n-1}u_{j,i}\partial^{j}\in{\mathcal{D}}$ by the formula:

[TABLE]

for $i=1,\dots,2n$.

Theorem 4.6** ([DS, Proposition 3.18]).**

Let a Miura oper ${\mathcal{L}}$ satisfy the mKdV equation (4.2) for some $r$. Then for every $i=0,\dots,2n$ the differential operator $L_{i}$ satisfies the KdV equation (4.8).

For $i=0,\dots,2n+1$, we define the $i$-th Miura map by the formula

[TABLE]

see (4.9).

For $i=0,1,\dots,2n$, an $i$*-oper * is a differential operator of the form

[TABLE]

with $V\in{\mathcal{B}}({{\mathfrak{g}}(A_{2n}^{(1)})}^{0})$ and $W\in{\mathcal{B}}({\mathfrak{n}}^{-}_{i})$. For $w\in{\mathcal{B}}({\mathfrak{n}}^{-}_{i})$ and an $i$-oper ${\mathcal{L}}$, the differential operator $e^{\text{ad}\,w}({\mathcal{L}})$ is an $i$-oper. The $i$-opers ${\mathcal{L}}$ and $e^{\text{ad}\,w}({\mathcal{L}})$ are called $i$-gauge equivalent. A Miura oper is an $i$-oper for any $i$.

Theorem 4.7** ([DS, Proposition 3.10]).**

If Miura opers ${\mathcal{L}}$ and $\tilde{\mathcal{L}}$ are $i$-gauge equivalent, then $\mathfrak{m}_{i}({\mathcal{L}})=\mathfrak{m}_{i}(\tilde{\mathcal{L}})$. ∎

5. Tangent maps to Miura maps

5.1. Tangent spaces

Consider the spaces of Miura opers $\mathcal{M}(A^{(2)}_{2n})\subset\mathcal{M}(A^{(1)}_{2n})$. The tangent space to $\mathcal{M}(A^{(2)}_{2n})$ at a point ${\mathcal{L}}$ is

[TABLE]

where $X_{i}$ are functions of variable $x$. Recall ${\mathcal{D}}=\{L=\partial^{2n+1}+\sum\limits_{i=0}^{2n-1}u_{i}\partial^{i}\}$. The tangent space to ${\mathcal{D}}$ at a point $D$ is $T_{D}{\mathcal{D}}=\{Z=\sum_{i=0}^{2n-1}Z_{i}\partial^{i}\}$, where $Z_{i}$ are functions of $x$.

Consider the restrictions of Miura maps to $\mathcal{M}(A^{(2)}_{2n})$ and the corresponding tangent maps

[TABLE]

By definition, if ${\mathcal{L}}=\partial+\Lambda^{(1)}+\sum_{i=1}^{2n+1}v_{i}e_{i,i}\in\mathcal{M}(A^{(2)}_{2n})$, $X=\sum_{i=1}^{2n+1}X_{i}e_{i,i}\in T_{\mathcal{L}}\mathcal{M}(A^{(2)}_{2n})$, $d\mathfrak{m}_{i}(X)=Z^{i}=\sum_{j=0}^{2n-1}Z_{j}^{i}\partial^{j}$, then

[TABLE]

In what follows we study the intersection of kernels of these tangent maps when $i$ runs through certain subsets of $\{1,\dots,2n+1\}$.

5.2. Formula for the first coefficient

Proposition 5.1**.**

Let ${\mathcal{L}}=\partial+\Lambda^{(1)}+\sum_{i=1}^{2n+1}v_{i}e_{i,i}\in\mathcal{M}(A^{(1)}_{2n})$, $X=\sum_{i=1}^{2n+1}X_{i}e_{i,i}$ $\in T_{\mathcal{L}}\mathcal{M}(A^{(2)}_{2n})$, $d\mathfrak{m}_{i}(X)=Z^{i}=\sum_{j=0}^{2n-1}Z_{j}^{i}\partial^{j}$. Then

[TABLE]

Proof.

The proof uses only the identity $\sum_{j=1}^{2n+1}v_{j}=0$ and is straightforward. ∎

5.3. Intersection of kernels of $d\mathfrak{m}_{i}$

Lemma 5.2**.**

Let ${\mathcal{L}}=\partial+\Lambda^{(1)}+\sum_{k=1}^{2n+1}v_{k}e_{k,k}\in\mathcal{M}(A^{(2)}_{2n})$, $X=\sum_{k=1}^{2n+1}X_{k}e_{k,k}$ $\in T_{\mathcal{L}}\mathcal{M}(A^{(2)}_{2n})$, $d\mathfrak{m}_{i}(X)=Z^{i}=\sum_{j=0}^{2n-1}Z_{j}^{i}\partial^{j}$. Assume that $Z^{i}_{2n-1}=0$ for $i=1,\dots,2n$, then

[TABLE]

Proof.

By assumption we have the system of equations

[TABLE]

By subtracting the first equation from the second we get $2nX^{\prime}_{2n}-X^{\prime}_{2n-1}-X^{\prime}_{2n-2}-\dots-X^{\prime}_{1}-X^{\prime}_{2n+1}=0$, equivalently $(2n+1)X^{\prime}_{2n}-\sum_{k=1}^{2n+1}X^{\prime}_{k}=0$. Since $\sum_{k=1}^{2n+1}X_{k}=0$, we get $X^{\prime}_{2n}=0$. By subtracting the second from the third we get $X^{\prime}_{2n-1}=0$. Similarly we obtain

[TABLE]

Applying (5.7) to the last equation in (5.6) yields $X^{\prime}_{2n+1}+\sum_{k=1}^{2n+1}v_{k}X_{k}=0$. By pulling out the terms for $k=1,2n+1$ we obtain $X^{\prime}_{2n+1}+v_{1}X_{1}+v_{2n+1}X_{2n+1}=-(X^{\prime}_{1}-2v_{1}X_{1})=-\sum_{k=2}^{2n}v_{k}X_{k}$. ∎

Lemma 5.3**.**

Let $j\in\{1,\dots,n-1\}$. Let ${\mathcal{L}}=\partial+\Lambda^{(1)}+\sum_{k=1}^{2n+1}v_{k}e_{k,k}\in\mathcal{M}(A^{(2)}_{2n})$, $X=\sum_{k=1}^{2n+1}X_{k}e_{k,k}$ $\in T_{\mathcal{L}}\mathcal{M}(A^{(2)}_{2n})$, $d\mathfrak{m}_{i}(X)=Z^{i}=\sum_{j=0}^{2n-1}Z_{j}^{i}\partial^{j}$. Assume that $Z^{i}_{2n-1}=0$ for all $i\notin\{j,2n+1-j\}$, then

[TABLE]

for $i\notin\{j,j+1,2n+1-j,2n+2-j\}$.

Proof.

By assumption we have the system of equations

[TABLE]

Subtracting the second line from the first gives $X^{\prime}_{2n+1}=0$, cf. the proof of Lemma 5.2. Similarly, for $i\notin\{j,j+1,2n+1-j,2n+2-j\}$ considering the difference $Z^{i-1}_{2n-1}-Z^{i}_{2n-1}=0$ we obtain $X^{\prime}_{i}=0$.

Considering the difference $Z^{2n+2-j}_{2n-1}-Z^{2n-j}_{2n-1}=0$ we obtain

[TABLE]

Hence $X^{\prime}_{2n+1-j}+X^{\prime}_{2n+2-j}=0$ and $X^{\prime}_{j}+X^{\prime}_{j+1}=0$. Now we can rewrite equation $Z^{2n+1}_{2n-1}=0$ as

[TABLE]

Or equivalently

[TABLE]

∎

Lemma 5.4**.**

Let ${\mathcal{L}}=\partial+\Lambda^{(1)}+\sum_{k=1}^{2n+1}v_{k}e_{k,k}\in\mathcal{M}(A^{(2)}_{2n})$, $X=\sum_{k=1}^{2n+1}X_{k}e_{k,k}$ $\in T_{\mathcal{L}}\mathcal{M}(A^{(2)}_{2n})$, $d\mathfrak{m}_{i}(X)=Z^{i}=\sum_{j=0}^{2n-1}Z_{j}^{i}\partial^{j}$. Assume that $Z^{i}_{2n-1}=0$ for all $i\notin\{n,n+1\}$, then

[TABLE]

Proof.

By assumption we have the system of equations

[TABLE]

Subtracting the second line from the first gives $X^{\prime}_{2n+1}=0$, cf. the proof of Lemma 5.2. Similarly, for $i\notin\{n,n+1,n+2\}$ considering the difference $Z^{i-1}_{2n-1}-Z^{i}_{2n-1}=0$ we obtain $X^{\prime}_{i}=0$. Notice that $X_{n+1}=0$ by assumption.

Now we can rewrite equation $Z^{2n+1}_{2n-1}=0$ as

[TABLE]

∎

6. Critical points of master functions and generation of tuples of polynomials

In this section we follow [MV1]. For functions $f(x),g(x)$, we denote

[TABLE]

the Wronskian determinant, and $f^{\prime}(x):=\frac{df}{dx}(x)$.

6.1. Master function

Choose nonnegative integers $\boldsymbol{k}=(k_{0},k_{1},\dots,k_{n})$. Consider variables $u=(u_{i}^{(j)})$, where $j=0,1,\dots,n$ and $i=1,\dots,k_{j}$. The master function $\Phi(u;\boldsymbol{k})$ is defined by the formula:

[TABLE]

The product of symmetric groups $\Sigma_{\boldsymbol{k}}=\Sigma_{k_{0}}\times\Sigma_{k_{1}}\times\dots\times\Sigma_{k_{n}}$ acts on the set of variables by permuting the coordinates with the same upper index. The function $\Phi$ is symmetric with respect to the $\Sigma_{\boldsymbol{k}}$-action. A point $u$ is a critical point if $d\Phi=0$ at $u$. In other words, $u$ is a critical point if and only if the following equations equal zero:

[TABLE]

All the orbits have the same cardinality $\prod_{i=0}^{n}k_{i}!$ . We do not make distinction between critical points in the same orbit.

Remark**.**

The definition of master functions can be found in [SV], see also [MV1, MV2]. The master functions $\Phi(u,\boldsymbol{k})$ in (6.1) are associated with the Kac-Moody algebra with Cartan matrix of type

[TABLE]

which is dual to the Cartan matrix ${A^{(2)}_{2n}}$, see this Langlands duality in [MV1, MV2, VWW].

6.2. Polynomials representing critical points

Let $u=(u_{i}^{(j)})$ be a critical point of the master function $\Phi$. Introduce the $(n+1)$-tuple of polynomials $y=(y_{0}(x),\dots,y_{n}(x))$,

[TABLE]

This tuple of polynomials defines a point in the direct product $({\mathbb{C}}[x])^{n+1}$. We say that the tuple represents the critical point.

Each polynomial of the tuple will be considered up to multiplication by a nonzero number.

It is convenient to think that the $(n+1)$-tuple $\boldsymbol{y}^{\emptyset}=(1,\dots,1)$ of constant polynomials represents in $({\mathbb{C}}[x])^{n+1}$, the critical point of the master function with no variables. This corresponds to the case $\boldsymbol{k}=(0,\dots,0)$.

We say that a given tuple $y\in({\mathbb{C}}[x])^{n+1}$ is generic if each polynomial $y_{i}(x)$ has no multiple roots and for $i=0,\dots,n-1$ the polynomials $y_{i}(x)$ and $y_{i+1}(x)$ have no common roots. If a tuple represents a critical point, then it is generic, see (6.2). For example, the tuple ${y}^{\emptyset}$ is generic.

6.3. Elementary generation

An $(n+1)$-tuple is called fertile if there exist polynomials $\tilde{y}_{0},\dots,\tilde{y}_{n}\in({\mathbb{C}}[x])^{n+1}$ such that

[TABLE]

where $a_{i,j}$ are the entries of the Cartan matrix of type $A^{(2)}_{2n}$, that is,

[TABLE]

For example, $y^{\emptyset}$ is fertile and $\tilde{y}_{j}=x+c_{j},$ where the $c_{j}$ are arbitrary numbers.

Assume that an $(n+1)$-tuple of polynomials $y=(y_{0},\dots,y_{n})$ is fertile. Equations (6.5) give us first order inhomogeneous differential equations with respect to $\tilde{y}_{i}$. The solutions are

[TABLE]

where $c_{0},\dots,c_{n}$ are arbitrary numbers. For each $i=0,\dots,n$, the tuple

[TABLE]

forms a one-parameter family. This family is called the generation of tuples from $y$ in the $i$-th direction. A tuple of this family is called an immediate descendant of $y$ in the $i$-th direction.

Theorem 6.1** ([MV1]).**

**

1. (i)

A generic tuple $y=(y_{0},\dots,y_{n})$, $\deg y_{i}=k_{i}$, represents a critical point of the master function $\Phi(u;\boldsymbol{k})$ if and only if $y$ is fertile. 2. (ii)

If $y$ represents a critical point, then for any $c\in{\mathbb{C}}$ the tuple $y^{(j)}(x,c)$ $j=0,\dots,n$ is fertile. 3. (iii)

If $y$ is generic and fertile, then for almost all values of the parameter $c\in{\mathbb{C}}$ the tuple $y^{(j)}(x,c)$ is generic. The exceptions form a finite set in ${\mathbb{C}}$. 4. (iv)

Assume that a sequence ${y}_{i},i=1,2,\dots$, of fertile tuples has a limit ${y}_{\infty}$ in $({\mathbb{C}}[x])^{n+1}$ as $i$ tends to infinity.

1. (a)

Then the limiting tuple ${y}_{\infty}$ is fertile. 2. (b)

For $j=0,\dots,n$, let ${y}_{\infty}^{(j)}$ be an immediate descendant of ${y}_{\infty}$. Then for all $j$ there exist immediate descendants ${y}_{i}^{(j)}$ of ${y}_{i}$ such that ${y}_{\infty}^{(j)}$ is the limit of ${y}_{i}^{(j)}$ as $i$ tends to infinity.

∎

6.4. Degree increasing generation

Let $y=(y_{0},\dots,y_{n})$ be a generic fertile $(n+1)$-tuple of polynomials. Define $k_{j}=\deg y_{j}$ for $j=0,\dots,n$.

The polynomial $\tilde{y}_{0}$ in (6.7) is of degree $k_{0}$ or $\tilde{k}_{0}=2k_{1}+1-k_{0}$. We say that the generation $(y_{0},\dots,y_{n})\to(\tilde{y}_{0},\dots,y_{n})$ is *degree increasing * in the [math]-th direction if $\tilde{k}_{0}>k_{0}$. In that case $\deg\tilde{y}_{0}=\tilde{k}_{0}$ for all $c$.

For $i=1,\dots,n-2$, the polynomial $\tilde{y}_{i}$ in (6.8) is of degree $k_{i}$ or $\tilde{k}_{i}=k_{i-1}+k_{i+1}+1-k_{i}$. We say that the generation $(y_{0},\dots,y_{i},\dots,y_{n})\to(y_{0},\dots,\tilde{y}_{i},\dots,y_{n})$ is *degree increasing * in the $i$-th direction if $\tilde{k}_{i}>k_{i}$. In that case $\deg\tilde{y}_{i}=\tilde{k}_{i}$ for all $c$.

The polynomial $\tilde{y}_{n-1}$ in (6.9) is of degree $k_{n-1}$ or $\tilde{k}_{n-1}=k_{n-2}+2k_{n}+1-k_{n-1}$. We say that the generation $(y_{0},\dots,y_{n-1},y_{n})\to(y_{0},\dots,\tilde{y}_{n-1},y_{n})$ is *degree increasing * in the $n-1$-st direction if $\tilde{k}_{n-1}>k_{n-1}$. In that case $\deg\tilde{y}_{n-1}=\tilde{k}_{n-1}$ for all $c$.

The polynomial $\tilde{y}_{n}$ in (6.10) is of degree $k_{n}$ or $\tilde{k}_{n}=k_{n-1}+1-k_{n}$. We say that the generation $(y_{0},\dots,y_{n-1},y_{n})\to(y_{0},\dots,y_{n-1},\tilde{y}_{n})$ is *degree increasing * in the $n$-th direction if $\tilde{k}_{n}>k_{n}$. In that case $\deg\tilde{y}_{n}=\tilde{k}_{n}$ for all $c$.

For $i=0,\dots,n$, if the generation is degree increasing in the $i$-th direction we normalize family (6.11) and construct a map $Y_{y,i}:{\mathbb{C}}\to({\mathbb{C}}[x])^{n+1}$ as follows. First we multiply the polynomials $y_{0},\dots,y_{n}$ by numbers to make them monic. Then we choose a monic polynomial $y_{i,0}$ satisfying the equation $\operatorname{Wr}({y_{i,0}},y_{i})=\epsilon\,\prod_{j\neq i}y_{j}^{-a_{j,i}},$ for some nonzero interger $\epsilon$, and such that the coefficient of $x^{k_{i}}$ in $y_{i,0}$ equals zero. Set

[TABLE]

and define

[TABLE]

The polynomials of this $(n+1)$-tuple are monic.

6.5. Degree-transformations and generation of vectors of integers

The degree-transformations

[TABLE]

correspond to the shifted action of reflections $w_{0},\dots,w_{n}\in W$, where $W$ is the Weyl group associated with the Cartan matrix $A$ in (6.3) and $w_{0},\dots,w_{n}$ are the standard generators, see [MV1, Lemma 3.11] for more detail.

We take formula (6.5) as the definition of degree-transformations:

[TABLE]

acting on arbitrary vectors $\boldsymbol{k}=(k_{0},\dots,k_{n})$.

We start with the vector $\boldsymbol{k}^{\emptyset}=(0,\dots,0)$ and a sequence $J=(j_{1},j_{2},\dots,j_{m})$ of integers such that $j_{i}\in\{0,\dots,n\}$ for all $i$. We apply the corresponding degree transformations to $\boldsymbol{k}^{\emptyset}$ and obtain a sequence of vectors $\boldsymbol{k}^{\emptyset},$ $\boldsymbol{k}^{(j_{1})}=w_{j_{1}}\boldsymbol{k}^{\emptyset},$ $\boldsymbol{k}^{(j_{1},j_{2})}=w_{j_{2}}w_{j_{1}}\boldsymbol{k}^{\emptyset}$,…,

[TABLE]

We say that the vector $\boldsymbol{k}^{J}$ is generated from $(0,\dots,0)$ in the direction of $J$.

We call a sequence $J$ degree increasing if for every $i$ the transformation $w_{j_{i}}$ applied to $w_{j_{i-1}}\dots w_{j_{1}}\boldsymbol{k}^{\emptyset}$ increases the $j_{i}$-th coordinate.

6.6. Multistep generation

Let $J=(j_{1},\dots,j_{m})$ be a degree increasing sequence. Starting from $y^{\emptyset}=(1,\dots,1)$ and $J$, we construct a map

[TABLE]

by induction on $m$. If $J=\emptyset$, the map $Y^{\emptyset}$ is the map ${\mathbb{C}}^{0}=(pt)\ \mapsto y^{\emptyset}$. If $m=1$ and $J=(j_{1})$, the map $Y^{(j_{1})}:{\mathbb{C}}\to({\mathbb{C}}[x])^{n+1}$ is given by formula (6.13) for $y=y^{\emptyset}$ and $j=j_{1}$. More precisely, equation (6.5) takes the form $\operatorname{Wr}(\tilde{y}_{j_{1}},1)=1$. Then $\tilde{y}_{j_{1},0}=x$ and

[TABLE]

By Theorem 6.1 all tuples in the image are fertile and almost all tuples are generic (in this example all tuples are generic). Assume that for ${\tilde{J}}=(j_{1},\dots,j_{m-1})$, the map $Y^{{\tilde{J}}}$ is constructed. To obtain $Y^{J}$ we apply the generation procedure in the $j_{m}$-th direction to every tuple of the image of $Y^{{\tilde{J}}}$. More precisely, if

[TABLE]

then

[TABLE]

The map $Y^{J}$ is called the generation of tuples from $y^{\emptyset}$ in the $J$-th direction.

Lemma 6.2**.**

All tuples in the image of $Y^{J}$ are fertile and almost all tuples are generic. For any $c\in{\mathbb{C}}^{m}$ the $(n+1)$-tuple $Y^{J}(c)$ consists of monic polynomials. The degree vector of this tuple equals $\boldsymbol{k}^{J}$. ∎

Lemma 6.3**.**

The map $Y^{J}$ sends distinct points of ${\mathbb{C}}^{m}$ to distinct points of $({\mathbb{C}}[x])^{n+1}$.

Proof.

The lemma is easily proved by induction on $m$. ∎

6.7. Critical points and the population generated from $y^{\emptyset}$

The set of all tuples $(y_{0},\dots,y_{n})\in({\mathbb{C}}[x])^{n+1}$ obtained from $y^{\emptyset}=(1,\dots,1)$ by generations in all directions $J=(j_{1},\dots,j_{m})$, $m\geqslant 0$, (not necessarily degree increasing) is called the population of tuples generated from $y^{\emptyset}$, see [MV1, MV2].

Theorem 6.4** ([MV3]).**

If a tuple of polynomials $(y_{0},\dots,y_{n})$ represents a critical point of the master function $\Phi(u,\boldsymbol{k})$ defined in (6.1) for some parameters $\boldsymbol{k}=(k_{0},\dots,k_{n})$, then $(y_{0},\dots,y_{n})$ is a point of the population generated from $y^{\emptyset}$ by a degree increasing generation, that is, there exist a degree increasing sequence $J=(j_{1},\dots,j_{m})$ and a point $c\in{\mathbb{C}}^{m}$ such that $(y_{0}(x),\dots,y_{n}(x))=Y^{J}(x,c)$. Moreover, for any other critical point of that function $\Phi(u,\boldsymbol{k})$ there is a point $c^{\prime}{}\in{\mathbb{C}}^{m}$ such that the tuple $Y^{J}(x,c^{\prime}{})$ represents that other critical point.

By Theorem 6.4 a function $\Phi(u,\boldsymbol{k})$ either does not have critical points at all or all of its critical points form one cell ${\mathbb{C}}^{m}$.

Proof.

Theorem 3.8 in [MV2] says that $(y_{0},\dots,y_{n})$ is a point of the population generated from $y^{\emptyset}$. The fact that $(y_{0},\dots,y_{n})$ can be generated from $y^{\emptyset}$ by a degree increasing generation is a corollary of Lemmas 3.5 and 3.7 in [MV2]. The same lemmas show that any other critical point of the master function $\Phi(u,\boldsymbol{k})$ is represented by the tuple $Y^{J}(x,c^{\prime}{})$ for a suitable $c^{\prime}{}\in{\mathbb{C}}^{m}$. ∎

7. Critical points of master functions and Miura opers

7.1. Miura oper associated with a tuple of polynomials, [MV2]

We say that a Miura oper of type ${A^{(2)}_{2n}}$, ${\mathcal{L}}=\partial+\Lambda^{(2)}+V$, is associated to the $(n+1)$-tuple of polynomials $y$ if $V=-\sum_{i=0}^{n}\ln^{\prime}(y_{i})\,h_{i}$, where $\ln^{\prime}(f(x))=\frac{f^{\prime}(x)}{f(x)}$. If ${\mathcal{L}}$ is associated to $y$ and $V=\sum_{i=1}^{2n+1}v_{i}e_{i,i}$, then

[TABLE]

We also have

[TABLE]

where $a_{i,j}$ are entries of the Cartan matrix of type $A^{(2)}_{2n}$. More precisely,

[TABLE]

For example,

[TABLE]

is associated to the tuple $y^{\emptyset}=(1,\dots,1)$.

Define the map

[TABLE]

which sends a tuple $y=(y_{0},\dots,y_{n})$ to the Miura oper ${\mathcal{L}}=\partial+\Lambda^{(2)}+V$ associated to $y$.

7.2. Deformations of Miura opers of type ${A^{(2)}_{2n}}$, [MV2]

Lemma 7.1** ([MV2]).**

Let ${\mathcal{L}}$ $=\partial+\Lambda^{(2)}+V$ be a Miura oper of type ${A^{(2)}_{2n}}$. Let $g\in{\mathcal{B}}$ and $j\in\{0,\dots,n\}$. Then

[TABLE]

∎

Corollary 7.2** ([MV2]).**

Let ${\mathcal{L}}=\partial+\Lambda^{(2)}+V$ be a Miura oper of type ${A^{(2)}_{2n}}$. Then $e^{\operatorname{ad}\,gf_{j}}{\mathcal{L}}$ is a Miura oper if and only if the scalar function $g$ satisfies the Riccati equation

[TABLE]

∎

Let ${\mathcal{L}}=\partial+\Lambda^{(2)}+V$ be a Miura oper. For $j\in\{0,\dots,n\}$, we say that ${\mathcal{L}}$ is deformable in the $j$-th direction if equation (7.6) has a nonzero solution $g$, which is a rational function.

Theorem 7.3** ([MV2]).**

Let ${\mathcal{L}}=\partial+\Lambda^{(2)}+V$ be the Miura oper associated to the tuple of polynomials $y=(y_{0},\dots,y_{n})$. Let $j\in\{0,\dots,n\}$. Then ${\mathcal{L}}$ is deformable in the $j$-th direction if and only if there exists a polynomial $\tilde{y}_{j}$ satisfying equation (6.5). Moreover, in that case any nonzero rational solution $g$ of the Riccati equation (7.6) has the form $g=\mathrm{ln}^{\prime}(\tilde{y}_{j}/y_{j})$ where $\tilde{y}_{j}$ is a solution of equation (6.5). If $g=\mathrm{ln}^{\prime}(\tilde{y}_{j}/y_{j})$, then the Miura oper

[TABLE]

is associated to the tuple $y^{(j)}$, which is obtained from the tuple $y$ by replacing $y_{j}$ with $\tilde{y}_{j}$.

7.3. Miura opers associated with the generation procedure

Let $J=(j_{1},\dots,j_{m})$ be a degree increasing sequence, see Section 6.5. Let $Y^{J}:{\mathbb{C}}^{m}\to({\mathbb{C}}[x])^{n+1}$ be the generation of tuples from $y^{\emptyset}$ in the $J$-th direction. We define the associated family of Miura opers by the formula:

[TABLE]

The map $\mu^{J}$ is called the generation of Miura opers from ${\mathcal{L}}^{\emptyset}$ in the $J$-th direction, see ${\mathcal{L}}^{\emptyset}$ in (7.4).

For $\ell=1,\dots,m$, denote $J_{\ell}=(j_{1},\dots,j_{\ell})$ the beginning $\ell$-interval of the sequence $J$. Consider the associated map $Y^{J_{\ell}}:{\mathbb{C}}^{\ell}\to({\mathbb{C}}[x])^{n+1}$. Denote

[TABLE]

Introduce

[TABLE]

for $\ell=2,\dots,m$. For $c\in{\mathbb{C}}^{m}$, define $U^{J}(c)=\sum_{i<0}(U^{J}(c))_{i}$, $(U^{J}(c))_{i}\in{\mathcal{B}}({\mathfrak{g}}(A^{(2)}_{2n})^{i})$, depending on $c\in{\mathbb{C}}^{m}$, by the formula

[TABLE]

Lemma 7.4**.**

For $c\in{\mathbb{C}}^{m}$, we have

[TABLE]

Proof.

The lemma follows from Theorem 7.3. ∎

Corollary 7.5**.**

Let $r>0$, odd and $r\neq n+1$ mod $4n+2$. Let $c\in{\mathbb{C}}^{m}$. Let \frac{\partial\phantom{a}}{\partial t_{r}}\big{|}_{\mu^{J}(c)} be the value at $\mu^{J}(c)$ of the vector field of the $r$-th mKdV flow on the space $\mathcal{M}(A^{(2)}_{2n})$, see (4.5). Then

[TABLE]

Proof.

The corollary follows from (4.6) and (7.10). ∎

We have the natural embedding $\mathcal{M}({A^{(2)}_{2n}})\hookrightarrow\mathcal{M}({A_{2n}^{(1)}})$, see Section 3.4. Let $J=(j_{1},j_{2}$, …, $j_{m})$. Denote ${\tilde{J}}=(j_{1},\dots,j_{m-1})$. Consider the associated family $\mu^{{\tilde{J}}}:{\mathbb{C}}^{m-1}\to\mathcal{M}({A^{(2)}_{2n}})$. Denote ${\tilde{c}}=(c_{1},\dots,c_{m-1})$.

Proposition 7.6**.**

For any $r>0$ the difference \frac{\partial}{\partial t_{r}}\big{|}_{\mu^{J}(c)}-\frac{\partial}{\partial t_{r}}\big{|}_{\mu^{\tilde{J}}(\tilde{c})} has the following form for some scalar functions $u_{1}(x,c)$, $u_{2}(x,c)$, $u_{3}(x,c)$:

1. (i)

if $j_{m}\in\{1,2,\dots,n-1\}$, then

[TABLE] 2. (ii)

if $j_{m}=0$, then

[TABLE] 3. (iii)

if $j_{m}=n$, then

[TABLE]

Proof.

We will write $\Lambda$ for $\Lambda^{(2)}=\Lambda^{(1)}$. Denote

[TABLE]

Expand $A_{r}=\sum_{i}A_{r}^{i}\Lambda^{i}$ where $A_{r}^{i}=\sum_{l=1}^{2n+1}A_{r}^{i,l}e_{l,l}$ with scalar coefficients $A_{r}^{i,l}$. Then \frac{\partial}{\partial t_{r}}\big{|}_{\mu^{\tilde{J}}(\tilde{c})}=-\frac{\partial}{\partial x}A_{r}^{0}. Assume that $j_{m}\in\{1,\dots,n-1\}$. Then

[TABLE]

The last term is zero since

[TABLE]

and we get

[TABLE]

The cases $j_{m}=0,n$ are proved similarly. ∎

Let $\mathfrak{m}_{i}:\mathcal{M}({A_{2n}^{(1)}})\to{\mathcal{D}},\ {\mathcal{L}}\mapsto L_{i},$ be the Miura maps defined in Section 4.5 for $i=0,\dots,2n$. Below we consider the composition of the embedding $\mathcal{M}({A^{(2)}_{2n}})\hookrightarrow\mathcal{M}({A_{2n}^{(1)}})$ and a Miura map.

Lemma 7.7**.**

If $j_{m}=0$, we have $\mathfrak{m}_{i}\circ\mu^{J}({\tilde{c}},c_{m})=\mathfrak{m}_{i}\circ\mu^{{\tilde{J}}}({\tilde{c}})$ for all $i\neq 0$. If $j_{m}=1,\dots,n$, we have $\mathfrak{m}_{i}\circ\mu^{J}({\tilde{c}},c_{m})=\mathfrak{m}_{i}\circ\mu^{{\tilde{J}}}({\tilde{c}})$ for all $i\neq j_{m},2n+1-j_{m}$.

Proof.

The lemma follows from formula (7.10) and Theorem 4.7. ∎

Lemma 7.8**.**

If $j_{m}=0$, then

[TABLE]

for some positive integer $a$. If $j_{m}=1,\dots,n-2$, then

[TABLE]

for some positive integer $a$. If $j_{m}=n-1$, then

[TABLE]

for some positive integer $a$. If $j_{m}=n$, then

[TABLE]

for some positive integer $a$.

Notice that the right-hand side of these formulas can be written as

[TABLE]

Proof.

Let $j_{m}=0$. Then $y_{0}(x,{\tilde{c}},c_{m},m)=y_{0,0}(x,{\tilde{c}})+c_{m}y_{0}(x,{\tilde{c}},m-1),$ where $y_{0,0}(x,{\tilde{c}})$ is such that

[TABLE]

for some positive integer $a$, see (6.12). We have $g_{m}=\ln^{\prime}(y_{0}(x,{\tilde{c}},c_{m},m))-\ln^{\prime}(y_{0}(x,{\tilde{c}},m-1))$.

By formula (7.11), we have

[TABLE]

This proves formula (7.16). The other formulas are proved similarly. ∎

7.4. Intersection of kernels of $d\mathfrak{m}_{i}$

Let $J=(j_{1},\dots,j_{m})$ be a degree increasing sequence and $\mu^{J}:{\mathbb{C}}^{m}\to\mathcal{M}({A^{(2)}_{2n}})$ the generation of Miura opers from ${\mathcal{L}}^{\emptyset}$ in the $J$-th direction. We have $\mu^{J}(c)=\partial+\Lambda^{(1)}+\sum_{k=1}^{2n+1}v_{k}(x,c)e_{k,k},$ where

[TABLE]

Let $X(c)=\sum_{k=1}^{2n+1}X_{k}(x,c)e_{k,k}\in T_{\mu^{J}(c)}\mathcal{M}({A^{(2)}_{2n}})$ be a field of tangent vectors to $\mathcal{M}({A^{(2)}_{2n}})$ at the points of the image of $\mu^{J}$,

[TABLE]

Our goal is to show that under certain conditions we have

[TABLE]

for some scalar function $A(c)$ on ${\mathbb{C}}^{m}$.

Proposition 7.9**.**

Let $j_{m}=0$ and $X(c)\in T_{\mu^{J}(c)}\mathcal{M}({A^{(2)}_{2n}})$. Assume that d\mathfrak{m}_{i}\big{|}_{\mu^{J}(c)}(X(c))=0 for all $i=1,\dots,2n$ and all $c\in{\mathbb{C}}^{m}$. Assume that $X(c)$ has the form indicated in the right-hand side of formula (7.14). Then equation (7.21) holds.

Proof.

Since $X_{k}(x,c)=0$ for $k=2,\dots,2n$, equation (5.5) takes the form $X^{\prime}_{1}-2v_{1}X_{1}=0$, or more precisely, X^{\prime}_{1}=2\ln^{\prime}\big{(}\frac{y_{1}(x,{\tilde{c}},m-1)}{y_{0}(x,{\tilde{c}},c_{m},m)}\big{)}X_{1}. Hence $X_{1}(x,c)=-X_{2n+1}=A(c)\frac{y_{1}(x,{\tilde{c}},m-1)^{2}}{y_{0}(x,{\tilde{c}},c_{m},m)^{2}}$ for some scalar $A(c)$. Lemma 7.8 implies equation (7.21). ∎

Proposition 7.10**.**

Let $j_{m}\in\{1,\dots,n-1\}$ and $X(c)\in T_{\mu^{J}(c)}\mathcal{M}({A^{(2)}_{2n}})$. Assume that d\mathfrak{m}_{i}\big{|}_{\mu^{J}(c)}(X(c))=0 for all $i\notin\{j_{m},2n+1-j_{m}\}$ and all $c\in{\mathbb{C}}^{m}$. Assume that $X(c)$ has the form indicated in the right-hand side of formula ((i)). Then equation (7.21) holds.

Proof.

By Lemma 5.3 we have $X^{\prime}_{j_{m}}+(v_{j_{m}}-v_{j_{m}+1})X_{j_{m}}=0$. Then for $j_{m}=1,\dots,n-2$, we have

[TABLE]

and for $j_{m}=n-1$, we have

[TABLE]

Lemma 7.8 yields equation (7.21). ∎

Proposition 7.11**.**

Let $j_{m}=n$ and $X(c)\in T_{\mu^{J}(c)}\mathcal{M}({A^{(2)}_{2n}})$. Assume that d\mathfrak{m}_{i}\big{|}_{\mu^{J}(c)}(X(c))=0 for all $i\neq n,n+1$, and $c\in{\mathbb{C}}^{m}$. Assume that $X(c)$ has the form indicated in the right-hand side of formula (7.15). Then equation (7.21) holds.

Proof.

By assumptions we have $X_{n+1}=0$ as well as $X_{i}=0$ for $i\neq n,n+1,n+2$. By Lemma 5.4 we have $X_{n}^{\prime}{}+v_{n}X_{n}=0$ where $v_{n}=\ln^{\prime}\frac{y_{n}^{2}}{y_{n-1}}$. Hence $X_{n}=-X_{n+2}=A(c)\frac{y_{n-1}(x,{\tilde{c}},m-1)}{y_{n}(x,{\tilde{c}},c_{m},m)^{2}}$ for some scalar function $A(c)$. Lemma 7.8 yields equation (7.21). ∎

8. Vector fields

8.1. Statement

Let $r>0$ be odd, and $r\neq 2n+1$ mod $4n+2$. Recall that we denote by $\frac{\partial}{\partial t_{r}}$ the $r$-th mKdV vector field on the space $\mathcal{M}({A^{(2)}_{2n}})$ of Miura opers of type ${A^{(2)}_{2n}}$. We also denote by $\frac{\partial}{\partial t_{r}}$ the $r$-th mKdV vector field of type ${A_{2n}^{(1)}}$ on the space $\mathcal{M}({A_{2n}^{(1)}})$ of Miura opers of type ${A_{2n}^{(1)}}$. We have a natural embedding $\mathcal{M}({A^{(2)}_{2n}})\hookrightarrow\mathcal{M}({A_{2n}^{(1)}})$. Under this embedding the vector $\frac{\partial}{\partial t_{r}}$ on $\mathcal{M}({A^{(2)}_{2n}})$ equals the vector filed $\frac{\partial}{\partial t_{r}}$ on $\mathcal{M}({A_{2n}^{(1)}})$ restricted to $\mathcal{M}({A^{(2)}_{2n}})$, see Section 4.3. We also denote by $\frac{\partial}{\partial t_{r}}$ the $r$-th KdV vector field on the space ${\mathcal{D}}$, see Section 4.4.

For a Miura map $\mathfrak{m}_{i}:\mathcal{M}\to{\mathcal{D}},\ {\mathcal{L}}\mapsto L_{i},$ denote by $d\mathfrak{m}_{i}$ the associated derivative map $T\mathcal{M}({A_{2n}^{(1)}})\to T{\mathcal{D}}$ of tangent spaces. By Theorem 4.6 we have d\mathfrak{m}_{i}:\frac{\partial}{\partial t_{r}}\big{|}_{{\mathcal{L}}}\mapsto\frac{\partial}{\partial t_{r}}\big{|}_{L_{i}}.

Fix a degree increasing sequence $J=(j_{1},\dots,j_{m})$. Consider the associated family $\mu^{J}:{\mathbb{C}}^{m}\to\mathcal{M}({A^{(2)}_{2n}})$ of Miura opers. For a vector field $\Gamma$ on ${\mathbb{C}}^{m}$, we denote by $\mathfrak{L}_{\Gamma}\mu^{J}$ the derivative of $\mu^{J}$ along the vector field. The derivative is well-defined since $\mathcal{M}({A^{(2)}_{2n}})$ is an affine space.

Theorem 8.1**.**

Let $r>0$ be odd, $r\neq 2n+1$ mod $4n+2$. Then there exists a polynomial vector field $\Gamma_{r}$ on ${\mathbb{C}}^{m}$ such that

[TABLE]

for all $c\in{\mathbb{C}}^{m}$. If $r>4m$, then \frac{\partial}{\partial t_{r}}\big{|}_{\mu^{J}(c)}=0.

Corollary 8.2**.**

The family $\mu^{J}$ of Miura opers is invariant with respect to all mKdV flows of type ${A^{(2)}_{2n}}$ and is point-wise fixed by flows with $r>4m$.

In other words, every mKdV flow corresponds to a flow on the space of integration parameters $c\in{\mathbb{C}}^{m}$. Informally speaking, we may say, that the integration parameters $c=(c_{1},\dots,c_{m})$ are times of the mKdV flows.

8.2. Proof of Theorem 8.1 for $m=1$

Let $J=(j_{1})$. Then $\mu^{J}(c_{1})=e^{g_{1}f_{j_{1}}}{\mathcal{L}}^{\emptyset}e^{-g_{1}f_{j_{1}}}=\partial+\Lambda-g_{1}h_{j_{1}}$, where $g_{1}=\frac{1}{x+c_{1}}$, see formula (7.9). We have

[TABLE]

Assume $j_{1}\in\{1,\dots,n-1\}$. Then $e^{g_{1}f_{j_{1}}}=1+g_{1}(e_{j_{1},j_{1}}+e_{2n+1-j_{1},2n+1-j_{1}})\Lambda^{-1}$. If $r$ is odd and $r>1$, then the right-hand side of (8.2) is zero, hence $\frac{\partial}{\partial t_{r}}|_{\mu^{J}(c_{1})}=\Gamma_{r}=0$. Let $r=1$, then

[TABLE]

Hence $\Gamma_{1}=-\frac{\partial}{\partial c_{1}}$. Assume $j_{1}=n$, then by formula (7.12),

[TABLE]

It follows from (8.2) and Lemma 2.4 that \frac{\partial}{\partial t_{r}}\big{|}_{\mu^{J}(c_{1})}=0 for $r$ odd and $r>1$, hence $\Gamma_{r}=0$. For $r=1$ we have \frac{\partial}{\partial t_{r}}\big{|}_{\mu^{J}(c_{1})}=-2\frac{dg_{1}}{dx}(e_{n,n}-e_{n+2,n+2})=\frac{dg_{1}}{dx}h_{n}=-\frac{1}{(x+c_{1})^{2}}h_{n}=-\frac{\partial\mu^{J}}{\partial c_{1}}(c_{1}). Hence $\Gamma_{1}=-\frac{\partial}{\partial c_{1}}$. Similarly, if $j_{1}=0$, then $\Gamma_{1}=-\frac{\partial}{\partial c_{1}}$ and $\Gamma_{r}=0$ for $r>1$. Theorem 8.1 is proved for $m=1$.

8.3. Beginning of proof of Theorem 8.1 for $m>1$

We prove the first statement of Theorem 8.1 by induction on $m$. Let $J=(j_{1},\dots,j_{m})$. Assume that the statement is proved for ${\tilde{J}}=(j_{1},\dots,j_{m-1})$. Let

[TABLE]

be the generation of tuples in the ${\tilde{J}}$-th direction. Then the generation of tuples in the $J$-th direction is

[TABLE]

see (6.17) and (6.18). We have $g_{m}=\ln^{\prime}(y_{j_{m},0}(x,{\tilde{c}})+c_{m}y_{j_{m}}(x,{\tilde{c}}))-\ln^{\prime}(y_{j_{m}}(x,{\tilde{c}}))$, see (7.8).

By the induction assumption, there exists a polynomial vector field $\Gamma_{r,{\tilde{J}}}=\sum_{i=1}^{m-1}\gamma_{i}(\tilde{c})\frac{\partial}{\partial c_{i}}$ on ${\mathbb{C}}^{m-1}$ such that for all ${\tilde{c}}\in{\mathbb{C}}^{m-1}$ we have

[TABLE]

Proposition 8.3**.**

There exists a scalar polynomial $\gamma_{m}(\tilde{c},c_{m})$ on ${\mathbb{C}}^{m}$ such that the vector field $\Gamma_{r}=\Gamma_{r,{\tilde{J}}}+\gamma_{m}({\tilde{c}},c_{m})\frac{\partial}{\partial c_{m}}$ satisfies (8.1) for all $(\tilde{c},c_{m})\in{\mathbb{C}}^{m}$.

8.4. Proof of Proposition 8.3

Lemma 8.4**.**

We have

[TABLE]

for all $i\notin\{j_{m},2n+1-j_{m}\}$.

Proof.

The proof is the same as the proof of Lemma 5.5 in [VW]. Namely, by Theorem 4.7 we have $\mathfrak{m}_{i}\circ\mu^{J}(\tilde{c},c_{m})=\mathfrak{m}_{i}\circ\mu^{\tilde{J}}(\tilde{c})$ for all $i\notin\{j_{m},2n+1-j_{m}\}$. Hence,

[TABLE]

By Theorems 4.6 and 4.7, we have

[TABLE]

By the induction assumption, we have

[TABLE]

These three formulas prove the lemma. ∎

Lemma 8.5**.**

The difference \frac{\partial}{\partial t_{r}}\big{|}_{\mu^{J}(c)}-\mathfrak{L}_{\Gamma_{r,\tilde{J}}}\mu^{J}({\tilde{c}},c_{m}) has the form indicated in the right-hand side of formula ((i)) if $j_{m}\in\{1,\dots,n-1\}$, has the form indicated in the right-hand side of formula (7.14) if $j_{m}=0$, has the form indicated in the right-hand side of formula (7.15) if $j_{m}=n$.

Proof.

We have

[TABLE]

see formula (7.11). If $j_{m}\in\{1,\dots,n-1\}$, then \frac{\partial}{\partial t_{r}}\Big{|}_{\mu^{J}(c)}-\frac{\partial}{\partial t_{r}}\Big{|}_{\mu^{\tilde{J}}(\tilde{c})} has the form indicated in the right-hand side of formula ((i)) by Proposition 7.6 and $\mathfrak{L}_{\Gamma_{r,\tilde{J}}}g_{m}(x,{\tilde{c}},c_{m})\,h_{j_{m}}$ has that form since $h_{j_{m}}=-e_{j_{m},j_{m}}+e_{j_{m}+1,j_{m}+1}-e_{2n+1-j_{m},2n+1-j_{m}}+e_{2n+2-j_{m},2n+2-j_{m}}$. This proves the lemma for $j_{m}\in\{1,\dots,n-1\}$. The other two cases of the lemma are proved similarly. ∎

Let us finish the proof of Proposition 8.3. By Lemmas 8.4 and 8.5 the difference \frac{\partial}{\partial t_{r}}\big{|}_{\mu^{J}(c)}-\mathfrak{L}_{\Gamma_{r,\tilde{J}}}\mu^{J}({\tilde{c}},c_{m}) has the form indicated in the right-hand side of one of the formulas ((i))-(7.15) and lies in the kernels of the differentials of Miura maps $\mathfrak{m}_{i}$ for all $i\notin\{j_{m},2n+1-j_{m}\}$. By Propositions 7.9, 7.10, 7.11 we conclude that the difference has the form $\gamma_{m}(\tilde{c},c_{m})\frac{\partial\mu^{J}}{\partial c_{m}}$ for some scalar function $\gamma_{m}(\tilde{c},c_{m})$ on ${\mathbb{C}}^{m}$. Therefore, \frac{\partial}{\partial t_{r}}\big{|}_{\mu^{J}(\tilde{c},c_{m})}=\mathfrak{L}_{\Gamma_{r,\tilde{J}}}\mu^{J}({\tilde{c}},c_{m})+\\ \gamma_{m}(\tilde{c},c_{m})\frac{\partial\mu^{J}}{\partial c_{m}}(\tilde{c},c_{m}). If we set $\Gamma_{r}=\Gamma_{r,\tilde{J}}+\gamma_{m}(\tilde{c},c_{m})\frac{\partial}{\partial c_{m}}$, then the vector field $\Gamma_{r}$ will satisfy formula (8.1).

We need to prove that $\gamma_{m}(\tilde{c},c_{m})$ is a polynomial. The proof of that fact is the same as the proof of [VW, Proposition 5.9]. Proposition 8.3 is proved.

8.5. End of proof Theorem 8.1 for $m>1$

Proposition 8.3 implies the first statement of Theorem 8.1. The second statement says that if $r>4m$, then \frac{\partial}{\partial t_{r}}\big{|}_{\mu^{J}(c)}=0. But that follows from Corollary 7.5 and Lemma 2.3.