# Spectra of Bethe subalgebras of $Y(\mathfrak{gl}_{n})$ in tame
representations

Aleksei Ilin, Inna Mashanova-Golikova and Leonid Rybnikov

###### Abstract.

We study the eigenproblem for Bethe subalgebras of the Yangian
$Y(\mathfrak{gl}_{n})$ in tame representations, i.e. in finite dimensional
representations which admit Gelfand-Tsetlin bases. Namely, we prove that for
any tensor product of skew modules
$V=\otimes_{i=1}^{k}V_{\lambda_{i}\setminus\mu_{i}}(z_{i})$ over the Yangian
$Y(\mathfrak{gl}_{n})$ with generic $z_{i}$’s, the family of Bethe subalgebras
$B(X)$ with $X$ being a regular element of the maximal torus of $GL_{n}$ (or,
more generally, with $X\in\overline{M_{0,n+2}}$) acts with a cyclic vector on
$V$. Moreover, for $X$ in the real form of $\overline{M_{0,n+2}}$ which is the
closure of regular unitary diagonal matrices we show, that the family of
subalgebras $B(X)$ acts with simple spectrum on
$\otimes_{i=1}^{k}V_{\lambda_{i}\setminus\mu_{i}}(z_{i})$ for generic
$z_{i}$’s where all $V_{\lambda_{i}\setminus\mu_{i}}(z_{i})$ are Kirillov-
Reshetikhin modules. In the subsequent paper we will use this to define a KR-
crystal structure on the spectrum of a Bethe subalgebra on $V$.

##  1\. Introduction

###  1.1. Yangian and Bethe subalgebras

The Yangian $Y(\mathfrak{gl}_{n})$ is a Hopf algebra, historically one of the
first examples of quantum groups. $Y(\mathfrak{gl}_{n})$ is in certain sense
unique Hopf algebra deforming the enveloping algebra $U(\mathfrak{gl}_{n}[t])$
(see [D]), where $\mathfrak{gl}_{n}[t]$ is the Lie algebra of
$\mathfrak{gl}_{n}$-valued polynomials. There is an action of the additive
group $\mathbb{C}$ by automorphisms on $Y(\mathfrak{gl}_{n})$ deforming the
action of $\mathbb{C}$ on $U(\mathfrak{gl}_{n}[t])$ which shifts the variable
$t$. For the details and links on $Y(\mathfrak{gl}_{n})$, we refer the reader
to the book [Mo] by A. Molev.

There is a flat family of maximal commutative subalgebras $B(C)\subset
Y(\mathfrak{gl}_{n})$, called _Bethe subalgebras_ , parameterized by
invertible diagonal matrices $C\in GL_{n}$ with pairwise different eigenvalues
see e.g. [NO] which are stable under the $\mathbb{C}$-action by shift
automorphisms of $Y(\mathfrak{gl}_{n})$. This family originates from the
integrable models in statistical mechanics and algebraic Bethe ansatz. More
precisely, the image of $B(C)$ in a tensor product of evaluation
representations of $Y(\mathfrak{gl}_{n})$ form a complete set of Hamiltonians
of the XXX Heisenberg magnet chain, cf. [B, KBI].

According to [IR18] the family of Bethe subalgebras extends to a bigger family
$B(X)$ of commutative subalgebras in $Y(\mathfrak{gl}_{n})$ with $X$ taking
values in the Deligne-Mumford space $\overline{M_{0,n+2}}$ of stable rational
curves with $n+2$ marked points: the subalgebra $B(C)$ corresponds to the non-
degenerate rational curve with the marked points being $0,\infty$ and the
eigenvalues of $C$, but there are also some new subalgebras corresponding to
degenerate curves $X\in\overline{M_{0,n+2}}$. In particular, the subalgebra
corresponding to the most degenerate _caterpillar_ curve corresponds to the
_Cartan_ subalgebra $H\subset Y(\mathfrak{gl}_{n})$, also known as the
_Gelfand-Tsetlin_ subalgebra, generated by all centers of the smaller Yangians
$Y(\mathfrak{gl}_{1})\subset Y(\mathfrak{gl}_{2})\subset\ldots\subset
Y(\mathfrak{gl}_{n})$ embedded in the standard way.

###  1.2. Representations of $Y(\mathfrak{gl}_{n})$

It is possible to obtain Yangian representations from representations of
$\mathfrak{gl}_{n}$ using the _evaluation homomorphism_

$\operatorname{ev}:Y(\mathfrak{gl}_{n})\to U(\mathfrak{gl}_{n})$

which gives a structure of $Y(\mathfrak{gl}_{n})$ module on every
$\mathfrak{gl}_{n}$-module called _evaluation_ $Y(\mathfrak{gl}_{n})$-module.
Also, from the $\mathbb{C}$-action on $Y(\mathfrak{gl}_{n})$, for any $z\
\in\mathbb{C}$ we have an automorphism $\tau_{z}:Y(\mathfrak{gl}_{n})\to
Y(\mathfrak{gl}_{n})$ (see Section 2 for precise definitions), so we can twist
any of the evaluation modules by such automorphism. Namely, for any
irreducible $\mathfrak{gl}_{n}$-module $V_{\lambda}$, we denote by
$V_{\lambda}(z)$ the $Y(\mathfrak{gl}_{n})$-module with the underlying space
$V_{\lambda}$ and the $Y(\mathfrak{gl}_{n})$ action given by
$\operatorname{ev}\circ\tau_{z}$.

It is possible to generalize this construction of
$Y(\mathfrak{gl}_{n})$-modules using the _centralizer construction_ of the
Yangian, due to Olshansky [O]. Namely, consider the embedding
$\mathfrak{gl}_{k}\subset\mathfrak{gl}_{n+k}$ as the subalgebra of lower-right
block $k\times k$-matrices, then for any $k\geq 0$ there is a homomorphism

$\eta_{k}:Y(\mathfrak{gl}_{n})\to U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}},$

which is surjective modulo the center of $U(\mathfrak{gl}_{n+k})$ (in
particular, we have $\eta_{0}=\operatorname{ev}$). Let $V_{\lambda}$ be an
irreducible representation of $\mathfrak{gl}_{n+k}$ with the highest weight
$\lambda$. Consider the restriction of $V_{\lambda}$ to $\mathfrak{gl}_{k}$:

$V_{\lambda}=\bigoplus_{\mu}M_{\lambda\mu}\otimes V_{\mu},$

where $M_{\lambda\mu}:=\operatorname{Hom}(V_{\mu},V_{\lambda})$ is the
multiplicity space with action of $U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$
and therefore is an irreducible representation of $Y(\mathfrak{gl}_{n})$.
Representations of this form are called skew representation of
$Y(\mathfrak{gl}_{n})$ because they depend on the _skew Young tableau_
$\lambda\setminus\mu$. If $M_{\lambda\mu}$ is any skew representation of
$Y(\mathfrak{gl}_{n})$ then we denote by $V_{\lambda\setminus\mu}(z)$ the
(irreducible) representation where the action of $Y(\mathfrak{gl}_{n})$ is
given by $\eta_{k}\circ\tau_{z}$. We also call these representations skew
representations of $Y(\mathfrak{gl}_{n})$.

We also note that Yangian is a Hopf algebra. This allows us to consider tensor
products of the representations above. In [NT], Nazarov and Tarasov introduce
the class of _tame_ representations i.e. representations of the form
$\bigotimes_{i=1}^{k}V_{\lambda_{i}\setminus\mu_{i}}(z_{i})$ such that
$z_{i}-z_{j}\not\in\mathbb{Z}$ for all $i\neq j$. This is the class of
irreducible representations of $Y(\mathfrak{gl}_{n})$ such that the Cartan
subalgebra $H\subset Y(\mathfrak{gl}_{n})$ acts without multiplicities. So it
is natural to expect similar properties for the action of Bethe subalgebras on
this class of representations of $Y(\mathfrak{gl}_{n})$. The eigenbasis for
the Cartan subalgebra $H\subset Y(\mathfrak{gl}_{n})$, known as the Gelfand-
Tsetlin basis, is naturally indexed by semistandard skew Young tableaux and is
described explicitly. The eigenbasis for a general Bethe subalgebra $B(X)$ is
then a deformation of the Gelfand-Tsetlin basis (being itself much less
explicit).

###  1.3. Bethe ansatz, cyclic vector and simplicity of spectrum

The main problem in the XXX integrable system is the diagonalization of the
subalgebras $B(C)$ in the corresponding representation of the Yangian. The
standard approach is the _algebraic Bethe ansatz_ which gives an explicit
formula the eigenvectors depending on auxiliary parameters satisfying some
system of algebraic equations called _Bethe ansatz equations_ , see for
example [KR].

The questions we address in the present paper are closely related to the
_completeness_ of the algebraic Bethe ansatz, i.e. to the problem whether the
eigenvectors obtained by Bethe ansatz form a basis in $V$. This problem is
extensively studied for many years, see e.g. [MV03, MTV07, MTV09, MTV14, T].
As the first step towards the solution of this problem, it is necessary that
the joint eigenvalues have no multiplicities. The latter is satisfied if and
only if the following two conditions hold: first, there is a _cyclic vector_
for the Bethe subalgebra in $V$ (i.e. $v\in V$ such that $B(C)v=V$) and,
second, the algebra $B(C)$ acts on $V$ semisimply.

Let $X\in\overline{M_{0,n+2}}$ and consider the Bethe subagebra $B(X)$. We
call $v\in V$ cyclic with respect to $B(X)$ if $B(X)\cdot v=V$.

######  Conjecture 1.1.

$B(X)$ has a cyclic vector in any tame representation of
$Y(\mathfrak{gl}_{n})$ for all $X\in\overline{M_{0,n+2}}$.

We will discuss some facts supporting this conjecture in Section 5.

In fact, it is easy to see that the Conjecture is true for _generic_
$X,z_{1},\ldots,z_{k}$. Indeed, consider the parameter space
$\overline{M_{0,n+2}}\times\mathbb{C}^{n}$. The condition that $B(X)$ acts
with a cyclic vector on $\bigotimes_{i=1}^{k}V_{i}(z_{i})$ determines a
Zariski open subset of $\overline{M_{0,n+2}}\times\mathbb{C}^{n}$, therefore
once we have a single point
$(X,z_{1},\ldots,z_{n})\in\overline{M_{0,n+2}}\times\mathbb{C}^{n}$ such that
$B(X)$ acts with a cyclic vector on $\bigotimes_{i=1}^{k}V_{i}(z_{i})$ we
automatically have the same property for generic $(X,z_{1},\ldots,z_{n})$. On
the other hand, according to [NT] the Gelfand-Tsetlin subalgebra of
$Y(\mathfrak{gl}_{n})$ (which is a particular case of $B(X)$) acts without
multiplicities on any tame representation, so has a cyclic vector – hence this
Zariski-open subset is non-empty. The problem with this argument is that it
does not give any representation such that $B(X)$ acts cyclicly for _all_
$X\in\overline{M_{0,n+2}}$. Now let us formulate the first main theorem of the
present paper.

Theorem A. _There is a Zariski open dense subset of $I\subset\mathbb{C}^{n}$
such that $B(X)$ has a cyclic vector in $V$ for all $X\in\overline{M_{0,n+2}}$
and $(z_{1},\ldots,z_{k})\in I$. _

Particularly, in a generic tame representation in the sense of [NT] every
Bethe subalgebra $B(X)$ with $X\in\overline{M_{0,n+2}}$ acts with a cyclic
vector. This allows to study the joint spectrum of $B(X)$ in a given tame
representation as a finite covering of $\overline{M_{0,n+2}}$ and reformulate
some properties of this spectrum in terms of geometry of Deligne-Mumford
compactifications. We will discuss other useful consequences of Theorem A in
Sections 5, 6, see also 1.6.

###  1.4. Quantum cohomology of quiver varieties.

In [MO] Maulik and Okounkov describe an action of $Y(\mathfrak{gl}_{n})$ on
the localized equivariant cohomology of type A quiver varieties such that some
elements from Bethe subalgebras with $C\in T^{reg}$ act as operators of
quantum multiplication by some cohomology classes (with $C$ being the quantum
parameter). According to a conjecture of [MO] the quantum cohomology ring
coincides with the image of the corresponding Bethe subalgebra.

The $Y(\mathfrak{gl}_{n})$-modules arising in this construction have the form
$\bigotimes_{i=1}^{k}W_{i}(z_{i})$ with certain parameters
$z_{i},i=1,\ldots,k$, where all $W_{i}$’s are fundamental representations of
$\mathfrak{gl}_{n}$. In particular, this conjecture implies that $B(C)$ acts
with a cyclic vector on any tensor product of fundamental
$Y(\mathfrak{gl}_{n})$-modules with generic evaluation parameters $z_{i}$,
since the unity class is always cyclic for the quantum cohomology ring.
Theorem A gives some evidence for this conjecture.

Also in [V] it is proved that the image of Gelfand-Tsetlin subalgebra
$H\subset Y(\mathfrak{gl}_{n})$, which has the form $B(X)$ for some
$X\in\overline{M_{0,n+2}}$ is generated bythe operators of (classical)
multiplication by equivariant cohomology classes of the quivier variety. Note
that $H$ corresponds to so-called caterpilar curves, i.e. one of the most
degenerate Bethe subalgebras corresponding to $0$-dimensional strata of
$\overline{M_{0,n+2}}$, so this particular degenerate curve corresponds to the
classical limit of the quantum cohomology ring. In this perspective it is
interesting to understand the geometric nature of other types of limit Bethe
subalgebras.

###  1.5. Plan of the proof.

The main idea of the proof is to reduce the eigenproblem for $XXX$ model to
that for the quantum _shift of argument subalgebras_ , cf. [R06]. The simplest
example is as follows. Suppose that $k=1$, i.e. we study the question about a
cyclic vector in the representation of the form $V(z)$, where $V$ is any
irreducible representation of $\mathfrak{gl}_{n}$. It is well-known, see [NO],
that $\operatorname{ev}_{z}(B(C))=\mathcal{A}_{C^{-1}}$ for any $C\in T^{reg}$
and $z\in\mathbb{C}$ with $\mathcal{A}_{C^{-1}}\subset U(\mathfrak{gl}_{n})$
being the shift of argument subalgebra defined in [R06]. On the other hand,
for the shift of argument subalgebras it is known (see [FFR]) that any
$\mathcal{A}_{C}$ with $C\in T^{reg}$ acts with a cyclic vector (moreover,
with simple spectrum) on any irreducible representation of
$\mathfrak{gl}_{n}$.

In the general situation, the set of values of the parameters $z_{i}$ such
that Theorem A is _not_ satisfied is Zariski-closed, since the variety
$\overline{M_{0,n+2}}$ is complete. So it is sufficient to prove that Theorem
A is satisfied at least for some values of the $z_{i}$’s. The most convenient
choice of such particular value of the evaluation parameters is
$z_{i}\in\mathbb{Z}$ with $|z_{i}-z_{j}|\gg 0$ since then
$\bigotimes_{i=1}^{k}V_{\lambda_{i}\setminus\mu_{i}}(z_{i})$ is in fact a skew
representation of the Yangian. This reduces Theorem A to the case of one
tensor factor which is a multiplicity space $M_{\lambda\mu}$. The image of
$B(X)$ under the centralizer construction map
$\eta_{k}:Y(\mathfrak{gl}_{n})\to U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$
turns to coincide with certain limit shift of argument subalgebra in
$U(\mathfrak{gl}_{n+k})$, hence we reduce Theorem A to the similar question on
shift of argument subalgebras. Finally, according to [HKRW], the corresponding
shift of argument subalgebra acts with a cyclic vector on any representation
of $U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$ in the multiplicity space
$M_{\lambda\mu}$ for any $\lambda$ and $\mu$.

###  1.6. Simple spectrum property and monodromy conjectures

Theorem A implies that once $B(C)$ acts semisimply, it has simple spectrum
(i.e. the joint eigenvalues have no multiplicities). The usual sufficient
condition for this is the existence of a Hermitian scalar product such that
$B(C)^{+}=B(C)$ i.e. all elements of $B(C)$ act by normal operators. We give
sufficient conditions on the representation of the Yangian guaranteeing that
such scalar product exists provided that $C$ belongs either to the closure of
the set of regular elements of the compact real torus $T_{comp}\subset T$ or
to that of the split real torus $T_{split}\subset T$.

The case of the compact torus goes back to Kirillov and Reshetikhin [KR].
Recall that Kirillov-Reshetikhin module is an irreducible
$\mathfrak{gl}_{n}$-module corresponding to a rectangular Young diagram
(equivalently, it is the highest irreducible component of a symmetric power of
a fundamental representation of $\mathfrak{gl}_{n}$). The second main result
of this paper is the following

Theorem B. _Suppose that all $V_{i}$’s are Kirillov-Reshetikhin modules. Let
$l_{i}\times r_{i}$ be the size of the corresponding Young diagram. Suppose
that $z_{i}=\frac{l_{i}}{2}-\frac{r_{i}}{2}+ix_{i}$, where
$x_{i}\in\mathbb{R}$. Then, for $(x_{1},\ldots,x_{k})\in\mathbb{R}^{k}$ from
Zariski open subset, $B(X)$ has simple spectrum on
$\bigotimes_{i=1}^{k}V_{i}(z_{i})$ for all $X\in\overline{M_{0,n+2}^{comp}}$._

Theorem B allows to regard the set of eigenlines for $B(X)$ in
$\bigotimes_{i=1}^{k}V_{i}(z_{i})$ as an _unramified_ covering of the space
$\overline{M_{0,n+2}^{comp}}$. In particular, we get the monodromy action of
the fundamental group $\pi_{1}(\overline{M_{0,n+2}^{comp}})$ (which is natural
to call (pure) _affine cactus group_) on the spectrum of Bethe subalgebras.
Moreover, it is possible to define the structure of a Kirillov-Reshetikhin
crystal on this spectrum, following the strategy of [HKRW] (see [KMR] and
Section 7 for the details). We expect that the action of the affine cactus
group on this set is given by partial Schutzenberger involutions on the KR-
crystal.

The closure of the set of regular points of the split real torus $T_{split}$
in $\overline{M_{0,n+2}}$ is the real form $\overline{M_{0,n+2}^{split}}$. Our
third main result is

Theorem C. _Let $V_{i},i=1,\ldots,k$ be a set of skew representations of
$Y(\mathfrak{gl}_{n})$. Then, for $(x_{1},\ldots,x_{k})\in\mathbb{R}^{k}$ from
Zariski open subset, $B(X)$ has simple spectrum on
$\bigotimes_{i=1}^{k}V_{i}(x_{i})$ for all
$X\in\overline{M_{0,n+2}^{split}}$._

Similarly to the case of the compact real form, this gives an action of the
usual cactus group $\pi_{1}(\overline{M_{0,n+2}^{split}})$ on the spectrum of
a Bethe algebra. Specializing the parameter of the Bethe algebra to the
_caterpillar_ point of $\overline{M_{0,n+2}^{split}}$ we get an action of the
cactus group on the Gelfand-Tsetlin basis in the tensor product of skew
representations. The latter is indexed by collections of semistandard skew
Young tableaux, and we conjecture that the action of the cactus group is given
by Bender-Knuth involutions, similarly to the construction of Chmutov, Glick
and Pylyavskyy [CGP].

###  1.7. The paper is organized as follows

In Section 2 we give some preliminaries on Yangians. In Section 3 we discuss
Deligne-Mumford compactification and its real forms. In Section 4 we define
the families of commutative subalgebras in $U(\mathfrak{gl}_{n})$ and in
$Y(\mathfrak{gl}_{n})$ and describe relations between them. In Section 5 we
prove Theorem A. In section 6 we prove Theorems B and C. In Section 7 we
discuss further directions of the research.

###  1.8. Acknowledgements

We thank Vasily Krylov for many valuable comments on the early draft of the
paper. We thank Joel Kamnitzer and Vitaly Tasrasov for extremely useful
discussions. The study has been funded within the framework of the HSE
University Basic Research Program. The work of A.I. and L.R. on Theorem C was
supported by the RSF grant 19-11-00056. I.M.-G. was supported by RFBR grant
number 19-31-90124. L.R. was partially supported by the Foundation for the
Advancement of Theoretical Physics and Mathematics “BASIS”. A.I. is a Young
Russian Mathematics award winner and would like to thank its sponsors and
jury.

##  2\. Yangian and universal enveloping algebra

###  2.1. Yangian of $\mathfrak{gl}_{n}$ and $\mathfrak{sl}_{n}$

######  Definition 2.1.

Yangian of $\mathfrak{gl}_{n}$ is a complex unital associative algebra with
countably many generators $t_{ij}^{(1)},t_{ij}^{(2)},\ldots$ where $1\leqslant
i,j\leqslant n$, and the defining relations

$[t_{ij}^{(r+1)},t_{kl}^{(s)}]-[t_{ij}^{(r)},t_{kl}^{(s+1)}]=t_{kj}^{(r)}t_{il}^{(s)}-t_{kj}^{(s)}t_{il}^{(r)},$

where $r,s\geqslant 0$ and $t_{ij}^{(0)}=\delta_{ij}$. This algebra is denoted
by $Y(\mathfrak{gl}_{n})$.

There is a filtration on the Yangian determined by $\deg t_{ij}^{(r)}=r$. The
$t_{ij}^{(r)}$’s are PBW generators with respect to this filtration.

It is convenient to consider the formal series

$t_{ij}(u)=\delta_{ij}+t_{ij}^{(1)}u^{-1}+t_{ij}^{(2)}u^{-2}+\ldots\in
Y(\mathfrak{gl}_{n})[[u^{-1}]].$

We denote by $T(u)$ the matrix whose $ij$-entry is $t_{ij}(u)$. We regard this
matrix as the following element of
$Y(\mathfrak{gl}_{n})[[u^{-1}]]\otimes\rm{End}\ \mathbb{C}^{n}$:

$T(u)=\sum_{i,j=1}^{n}t_{ij}(u)\otimes e_{ij},$

where $e_{ij}$ stands for the standard matrix units.

Consider the algebra

$Y(\mathfrak{gl}_{n})[[u^{-1}]]\otimes(\rm{End}\ \mathbb{C}^{n})^{\otimes n}.$

For any $a\in\\{1,\ldots,n\\}$ there is an embedding

$i_{a}:Y(\mathfrak{gl}_{n})[[u^{-1}]]\otimes\rm{End}\ \mathbb{C}^{n}\to
Y(\mathfrak{gl}_{n})[[u^{-1}]]\otimes(\rm{End}\ \mathbb{C}^{n})^{\otimes n}$

which is an identity on $Y(\mathfrak{gl}_{n})[[u^{-1}]]$ and embeds $\rm{End}\
\mathbb{C}^{n}$ as the $a$-th tensor factor in $(\rm{End}\
\mathbb{C}^{n})^{\otimes n}$. Denote by $T_{a}(u)$ the image of $T(u)$ under
this embedding.

Consider $R(u)=1-Pu^{-1}\in(\operatorname{End}\mathbb{C}^{n})^{\otimes 2}$,
where $P=\sum_{i,j=1}^{n}e_{ij}\otimes e_{ji}$. Then it is well-known that one
can rewrite defining relations for $Y(\mathfrak{gl}_{n})$ as follows:

$R(u-v)T_{1}(u)T_{2}(v)=T_{2}(u)T_{1}(v)R(u-v).$

$Y(\mathfrak{gl}_{n})$ is a Hopf algebra with the comultiplication:

$\Delta:Y(\mathfrak{gl}_{n})\to Y(\mathfrak{gl}_{n})\otimes
Y(\mathfrak{gl}_{n}),t_{ij}(u)\mapsto\sum_{k=1}^{n}t_{ik}(u)\otimes t_{kj}(u)$

and the antipode

$S:T(u)\to T^{-1}(u).$

For any $1\leq a_{1}\leq\ldots\leq a_{k}\leq n$ and $1\leq b_{1}\leq\ldots\leq
b_{k}\leq n$ by definition put

$t^{a_{1}\ldots a_{k}}_{b_{1}\ldots b_{k}}(u)=\sum_{\sigma\in
S_{k}}(-1)^{\sigma}\,\cdot\,t_{a_{\sigma(1)}b_{1}}(u)\ldots
t_{a_{\sigma(k)}b_{k}}(u-k+1)=\sum_{\sigma\in
S_{k}}(-1)^{\sigma}\,\cdot\,t_{a_{1}b_{\sigma(1)}}(u-k+1)\ldots
t_{a_{k}b_{\sigma(k)}}(u).$

We call series $t^{a_{1}\ldots a_{k}}_{b_{1}\ldots b_{k}}(u)$ quantum minors.

We also use Yangian $Y(\mathfrak{sl}_{n})$ for the Lie algebra
$\mathfrak{sl}_{n}$.

######  Definition 2.2.

Yangian $Y(\mathfrak{sl}_{n})$ is the subalgebra of $Y(\mathfrak{gl}_{n})$
which consists of all elements stable under the automorphisms $T(u)\to
f(u)T(u)$ for all $f(u)\in 1+u^{-1}\mathbb{C}[[u^{-1}]]$.

######  Proposition 2.1.

([Mo, Theorem 1.8.2, Proposition 1.84.])

  1. (1)

$Y(\mathfrak{gl}_{n})=Y(\mathfrak{sl}_{n})\otimes_{\mathbb{C}}Z(Y(\mathfrak{gl}_{n}))$,
where $Z(Y(\mathfrak{gl}_{n}))$ is the center of $Y(\mathfrak{gl}_{n})$;

  2. (2)

$Y(\mathfrak{sl}_{n})$ is a Hopf subalgebra of $Y(\mathfrak{gl}_{n})$.

######  Remark.

This means that one can also realize $Y(\mathfrak{sl}_{n})$ as the quotient of
$Y(\mathfrak{gl}_{n})$ by a maximal ideal of the center
$Z(Y(\mathfrak{gl}_{n}))$, see [Mo, Corollary 1.8.3].

###  2.2. Evaluation homomorphism and evaluation modules

Let $z\in\mathbb{C}$. It is well-known that the map

$\tau_{z}:Y(\mathfrak{gl}_{n})\to Y(\mathfrak{gl}_{n}),T(u)\mapsto T(u-z)$

is an automorphism of $Y(\mathfrak{gl}_{n})$.

Let $E=\sum\limits_{i,j=1}^{n}E_{ij}\otimes
e_{ij}=\left(e_{ij}\right)_{i,j=1,\ldots,n}\in U(\mathfrak{gl}_{n})\otimes
Mat_{n}$ be the $n\times n$-matrix with the $(i,j)$-th coefficient being
$E_{ij}\in U(\mathfrak{gl}_{n})$ (here $E_{ij}\in\mathfrak{gl}_{n}$ is the
matrix with 1 on $(i,j)$-entry and zeros on other entries). It is well known
that the map

$\operatorname{ev}:Y(\mathfrak{gl}_{n})\to U(\mathfrak{gl}_{n}),T(u)\mapsto
1+Eu^{-1}$

is a homomorphism of algebras. By definition put

$\operatorname{ev}_{z}:=\operatorname{ev}\circ\tau_{z}$

Any $\mathfrak{gl}_{n}$-module $V$ can be regarded as a
$Y(\mathfrak{gl}_{n})$-module by means of
$\operatorname{ev}=\operatorname{ev}_{0}$. Moreover, using
$\operatorname{ev}_{z}$ one can produce a $1$-parametric family $V(z)$ of
$Y(\mathfrak{gl}_{n})$-modules.

We also note that the elements $t_{ij}^{(1)},1\leq i,j\leq n$ generate a copy
of $U(\mathfrak{gl}_{n})$ in $Y(\mathfrak{gl}_{n})$. This allows to consider
any representation of $Y(\mathfrak{gl}_{n})$ as a representation of
$U(\mathfrak{gl}_{n})$, hence as a representation of the corresponding
connected simply connected group $\tilde{GL_{n}}$.

###  2.3. Some homomorphisms between Yangians

Let us define two different embedding of $Y(\mathfrak{gl}_{n})$ to
$Y(\mathfrak{gl}_{n+k})$:

$i_{k}:Y(\mathfrak{gl}_{n})\to Y(\mathfrak{gl}_{n+k});\ t^{(r)}_{ij}\mapsto
t^{(r)}_{ij}$ $\varphi_{k}:Y(\mathfrak{gl}_{n})\to Y(\mathfrak{gl}_{n+k});\
t^{(r)}_{ij}\mapsto t^{(r)}_{k+i,k+j}$

By definition, put

$\omega_{n}:Y(\mathfrak{gl}_{n})\to Y(\mathfrak{gl}_{n});\
T(u)\mapsto(T(-u-n))^{-1}.$

It is well-known that $\omega_{n}$ is an involutive automorphism of
$Y(\mathfrak{gl}_{n})$. We define a homomorphism

$\psi_{k}=\omega_{n+k}\circ\varphi_{k}\circ\omega_{n}:Y(\mathfrak{gl}_{n})\to
Y(\mathfrak{gl}_{n+k}).$

Note that $\psi_{k}$ is injective.

######  Proposition 2.2.

[IR18, Lemma 2.12] The homomorphisms $i_{k}$ and $\psi_{n}$ define an
embedding $i_{k}\otimes\psi_{n}:Y(\mathfrak{gl}_{n})\otimes
Y(\mathfrak{gl}_{k})\hookrightarrow Y(\mathfrak{gl}_{n+k})$.

###  2.4. Centralizer construction

Consider the map

$\Phi_{k}:Y(\mathfrak{gl}_{n})\to U(\mathfrak{gl}_{n+k})\ \text{given by}\
\Phi_{k}=\operatorname{ev}\circ\omega_{n+k}\circ i_{k}.$

From [Mo, Proposition 8.4.2] it follows that ${\rm Im}\,\Phi_{k}\subset
U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$. Here we use an embedding

$\mathfrak{gl}_{k}\to\mathfrak{gl}_{n+k},\ E_{ij}\to E_{i+n,j+n}.$

Let $A_{0}=\mathbb{C}[\mathcal{E}_{1},\mathcal{E}_{2},\ldots]$ be the
polynomial algebra of infinite many variables. Define a grading on $A_{0}$ by
setting $\deg\mathcal{E}_{i}=i$. For any $k$ we have a surjective homomorphism

$z_{k}:A_{0}\to Z(U(\mathfrak{gl}_{n+k}));\
\mathcal{E}_{i}\to\mathcal{E}_{i}^{(n+k)}.$

where $\mathcal{E}_{i}^{(n+k)}$, $i=1,2,3,\ldots$ are the following generators
of $Z(U(\mathfrak{gl}_{n+k}))$ of degree $i$, see [Mo, Section 8.2]:

$1+\sum\limits_{i=1}^{n+k}\mathcal{E}_{i}u^{-i}=\operatorname{ev}(\operatorname{qdet}T(u))$

Consider the algebra $Y(\mathfrak{gl}_{n})\otimes A_{0}$. This algebra has a
well-defined ascending filtration given by

$\deg a\otimes b=\deg a+\deg b$

For any $k\geqslant 0$ we define homomorphisms of filtered algebras

$\eta_{k}:Y(\mathfrak{gl}_{n})\otimes A_{0}\to
U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}};\ a\otimes b\to\Phi_{k}(a)\cdot
z_{k}(b)$

These homomorphisms are known to be surjective. Denote by
$(Y(\mathfrak{gl}_{n})\otimes A_{0})_{N}$ the $N$-th filtered component, i.e.
the vector space of the elements of degree not greater than $N$. From [Mo,
Theorem 8.4.3] we have:

######  Theorem 2.3.

The sequence $\\{\eta_{k}\\}$ is an asymptotic isomorphism. This means that
for any $N$ there exists $K$ such that for any $k>K$ the restriction of
$\eta_{k}$ to the $N$-th filtered component $(Y(\mathfrak{gl}_{n})\otimes
A_{0})_{N}$ is an isomorphism of vector spaces $(Y(\mathfrak{gl}_{n})\otimes
A_{0})_{N}\simeq U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}_{N}$.

###  2.5. Skew representations of $Y(\mathfrak{gl}_{n})$

Let $V_{\lambda}$ be the finite-dimensional irreducible representation of
$U(\mathfrak{gl}_{n+k})$ of the highest weight
$(\lambda_{1},\ldots,\lambda_{n+k})$. Consider the restriction of
$V_{\lambda}$ to $\mathfrak{gl}_{k}$:
$V_{\lambda}=\sum_{\mu}M_{\lambda\mu}\otimes V_{\mu}$, where
$M_{\lambda\mu}:=\operatorname{Hom}(V_{\mu},V_{\lambda})$ is the multiplicity
space. It is well-known that multiplicity space is an irreducible
representation of $U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$. Restriction of
$\eta_{k}\circ\tau_{z}$ gives $M_{\lambda\mu}$ a structure of representation
of $Y(\mathfrak{gl}_{n})$. We denote this representation by
$V_{\lambda\setminus\mu}(z)$ and call skew representation of
$Y(\mathfrak{gl}_{n})$.

According to the Jacobson density theorem, any multiplicity space
$M_{\lambda\mu}$ is an irreducible
$U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$-module. Since $\eta_{k}$ is
surjective we have the following

######  Proposition 2.4.

[Mo, Section 6] Any skew representation of $Y(\mathfrak{gl}_{n})$ is
irreducible.

##  3\. $\overline{M}_{0,n+2}$ and its real forms.

###  3.1. Deligne-Mumford compactification.

Let $\overline{M_{0,n+2}}$ denote the Deligne-Mumford space of stable rational
curves with $n+2$ marked points. The points of $\overline{M_{0,n+2}}$ are
isomorphism classes of curves of genus $0$, with $n+2$ ordered marked points
and possibly with nodes, such that each component has at least $3$
distinguished points (either marked points or nodes). One can represent the
combinatorial type of such a curve as a tree with $n+2$ leaves with inner
vertices representing irreducible components of the corresponding curve, inner
edges corresponding to the nodes and the leaves corresponding to the marked
points. Informally, the topology of $\overline{M_{0,n+2}}$ is determined by
the following rule: when some $k$ of the distinguished points (marked or
nodes) of the same component collide, they form a new component with $k+1$
distinguished points (the new one is the intersection point with the old
component). In particular, the tree describing the combinatorial type of the
less degenerate curve is obtained from the tree corresponding to the more
degenerate one by contracting an edge.

The space $\overline{M_{0,n+2}}$ is a smooth algebraic variety. It can be
regarded as a compactification of the configuration space $M_{0,n+2}$ of
ordered $(n+2)$-tuples $(z_{0},z_{1},\ldots,z_{n+1})$ of pairwise distinct
points on $\mathbb{C}\mathbb{P}^{1}$ modulo the automorphism group
$PGL_{2}(\mathbb{C})$. Since the group $PGL_{2}(\mathbb{C})$ acts transitively
on triples of distinct points, we can fix the $0$-th and the $(n+1)$-th points
to be $0\in\mathbb{C}\mathbb{P}^{1}$ and $\infty\in\mathbb{C}\mathbb{P}^{1}$,
respectively. Then the space $M_{0,n+2}$ gets identified with the quotient
$\\{(z_{1},\ldots,z_{n})\in\mathbb{C}^{*n}\ |\ z_{i}\neq
z_{j}\\}/\mathbb{C}^{*}$, and the group $\mathbb{C}^{*}$ acts by dilations.
Under this identification of $M_{0,n+2}$, the space $\overline{M_{0,n+2}}$ is
the multiplicative version of the De Concini-Procesi wonderful closure of the
complement to a hyperplane arrangement introduced by De Concini and Giaffi in
[DCG], i.e. the wonderful closure of the complement to the arrangement of root
subtori in the maximal torus of the adjoint group on the toric variety $X$
attached to the fan formed by the root arrangement in the weight lattice of
$T=\mathbb{C}^{*n}/\mathbb{C}^{*}$. For $G=PGL_{n}$ we have
$X=\overline{L_{n+2}}$ is the _Losev-Manin moduli space_ , [LM]. The following
statement is well-known, see [K]:

######  Proposition 3.1.

$\overline{M_{0,n+2}}$ is the iterated blow-up of the subspaces of the form
$\\{z_{i_{1}}=z_{i_{2}}=\ldots=z_{i_{k}}\\}$ in $\overline{L_{n+2}}$.

###### Proof.

According to Kapranov [K], $\overline{M_{0,n+2}}$ is an iterated blow up of
the projective space $\mathbb{P}(\mathbb{C}^{n})$ at the subspaces of the form
$\\{0=z_{i_{1}}=z_{i_{2}}=\ldots=z_{i_{k}}\\}$ and
$\\{z_{i_{1}}=z_{i_{2}}=\ldots=z_{i_{k}}\\}$. The projective space
$\mathbb{P}(\mathbb{C}^{n})$ is a toric $T$-variety corresponding to the fan
formed by the cones $C_{i}=\\{x_{i}\geq x_{j}\ |\ \forall j\neq i\\}$ for all
$i=1,\ldots,n$. The fan formed by the root arrangement is a subdivision of the
fan formed by the cones $C_{i}$, and the additional faces correspond to the
subvarieties of the form $\\{0=z_{i_{1}}=z_{i_{2}}=\ldots=z_{i_{k}}\\}$ on
$\mathbb{P}(\mathbb{C}^{n})$. So the variety $\overline{L_{n+2}}$ is the blow-
up of $\mathbb{P}(\mathbb{C}^{n})$ at the subspaces of the form
$\\{0=z_{i_{1}}=z_{i_{2}}=\ldots=z_{i_{k}}\\}$. Hence $\overline{M_{0,n+2}}$
is the blow-up of the subvarieties of the form
$\\{z_{i_{1}}=z_{i_{2}}=\ldots=z_{i_{k}}\\}$ on $\overline{L_{n+2}}$. ∎

######  Remark.

It is well-known that $\overline{L_{n+2}}$ is also the closure of the maximal
torus $T$ in the De Concini-Procesi wonderful closure $\overline{G}$ of the
adjoint group $G=PGL_{n}$. See also 4.4.

###  3.2. Stratification of $\overline{M_{0,n+2}}$

The Deligne-Mumford variety $\overline{M_{0,n+2}}$ is stratified by the
combinatorial type of the curve with marked points. Namely, the strata are
indexed by trees whose leaves correspond to the marked points
$0,z_{1},\ldots,z_{n},\infty$ where the inner vertices represent the connected
components and the edges correspond to the nodal points. Clearly, the
dimension of the stratum is the sum of the degree minus $3$ over all inner
vertices. The natural partial order on the strata is given by contracting
edges of the trees. Clearly, the closure of each stratum is the direct product
$\prod\limits_{i\in I}\overline{M_{0,k_{i}}}$ where $I$ is the set of inner
vertices of the tree and the $i$-th inner vertex is $k_{i}$-valent. In
particular, starting from a point in the open stratum, one can approach any
boundary point of $\overline{M_{0,n+2}}$ just by successively approaching a
generic point in a codimension $1$ strata for subsequent
$\overline{M_{0,k_{i}}}$’s.

There are strata of the codimension $1$ parameterize $2$-component curves
(eqivalently, trees with exactly $2$ inner vertices). We distinguish the case
when the points $0$ and $\infty$ belong to different components and call them
_strata of the first type_. The corresponding curves have the form (these
strata are already visible on the Losev-Manin space):

All other codimension $1$ strata are called _strata of the second type_ and
have the form

Note that one can approach any boundary point by first passing successively to
strata of the first type (this corresponds to approaching boundary points in
$\overline{L_{n+2}}$) and next to strata of the second type in appropriate
$\overline{M_{0,k_{i}}}$ (this corresponds to approaching a boundary point in
a fiber of $\overline{M_{0,n+2}}$ over $\overline{L_{n+2}}$).

###  3.3. Real forms

In [C], Özgür Ceyhan studies real forms of the Deligne-Mumford space which
preserve the stratification. To any involution (including identity) $\sigma\in
S_{n+2}$ one can assign an involutive automorphism of $\overline{M_{0,n+2}}$
which permutes the marked points. Composing it with the complex conjugation
$z_{i}\mapsto\bar{z_{i}}$ we get a complex antilinear automorphism
$s:\overline{M_{0,n+2}}\to\overline{M_{0,n+2}}$. According to [C], these are
all complex antilinear automorphisms preserving the stratification of
$\overline{M_{0,n+2}}$. More precisely, to an involution $\sigma$ being the
product of $k$ independent transpositions corresponds the real form
$\overline{M_{2k,l}}$ (with $l+2k=n+2$) such that generic points of $M_{2k,l}$
have the form
$(z_{1},\ldots,z_{k},\bar{z}_{1},\ldots,\bar{z}_{k},u_{1},\ldots,u_{l})$ where
$u_{i}\in\mathbb{R}$.

Let $\overline{M_{0,n+2}}^{s}$ be the real locus of the Deligne-Mumford space
with respect to the real structure $s$. Note that if $s_{1}$ and $s_{2}$ are
two conjugated antiholomorphic involutions then $\overline{M_{0,n+2}}^{s_{1}}$
and $\overline{M_{0,n+2}}^{s_{2}}$ are diffeomorphic. The real forms of the
algebraic group $PGL_{n}$ have very similar description. Let $G_{r}$ be a real
form of $PGL_{n}$ and $T_{r}\subset G_{r}$ be a maximal torus which contains a
maximal split torus. We can assume that $T_{r}\subset T$. Recall that we have
a map from $T$ to $M_{0,n+2}$ so can regard the torus $T_{r}$ as a subset of
$M_{0,n+2}$. One can find an antiholomorphic involution $s$ of
$\overline{M_{0,n+2}}$ such that $T_{r}=(M_{0,n+2})^{s}$ hence an open subset
of $\overline{M_{0,n+2}}^{s}$. This means that the closure of $T_{r}$ in
$\overline{M_{0,n+2}}$ coincides with $\overline{M_{0,n+2}}^{s}$ which defines
a map from real forms of $PGL_{n}$ to real forms of $\overline{M_{0,n+2}}$. We
have the following

######  Proposition 3.2.

The above map from the equivalence classes of real forms of the complex
algebraic group $PGL_{n}$ to the real forms of $\overline{M_{0,n+2}}$ is
surjective. This map is one-to-one for odd $n$.

###### Proof.

Indeed, for $k>0$, $\overline{M_{2k,l}}$ corresponds to $U_{k+l-1,k-1}$ and
the “split” real form $k=0$ corresponds to $GL_{n}(\mathbb{R})$. ∎

We will be interested in two simplest cases, namely

  1. (1)

$\sigma_{0}=e\in S_{n+2}$. Then the set of $\mathbb{R}$-points of the
corresponding real form is the closure of the set regular points of the
_split_ form of the maximal torus $T_{split}\subset PGL_{n}(\mathbb{R})$, i.e.
the real locus is
$\overline{M_{0,n+2}^{split}}=\overline{\\{(z_{1},\ldots,z_{n})\ |\ z_{i}\neq
z_{j},z_{i}\in\mathbb{R}\\}/\mathbb{R}^{\times}}$. This is the usual real form
of $\overline{M_{0,n+2}}$ which plays the crucial role in _coboundary_
monoidal categories analogous to the role played by the space of
configurations of points on the complex line in braided monoidal categories,
see [HK, HKRW] for details.

  2. (2)

$\sigma_{1}=(1,n+2)\in S_{n+2}$. The corresponding real locus of
$\overline{M_{0,n+2}}$ is the closure of the set of regular points of the
_compact_ real form of the maximal torus $T_{comp}\subset PU_{u}$, i.e. the
real locus is $\overline{M_{0,n+2}^{comp}}=\overline{\\{(z_{1},\ldots,z_{n})\
|\ z_{i}\neq z_{j},|z_{i}|=1\\}/U_{1}}$. Namely, the points of
$\overline{M_{0,n+2}^{comp}}$ are curves where all marked points are on the
unit circle, and the points $0,\infty$ belongs to the same component. In
particular, this means that the projection of this real locus to the Losev-
Manin space is contained in the open $T$-orbit, i.e. there are no boundary
strata involving the degeneration of the first type.

For the future reference, we write down here the action of $\sigma_{1}$ on
codimension $1$ strata of the first and of the second type of
$\overline{M_{0,n+2}}$:

$\mapsto$

$\mapsto$

##  4\. Bethe and shift of argument subalgebras

###  4.1. Bethe subalgebras

The symmetric group $S_{n}$ acts on
$Y(\mathfrak{gl}_{n})[[u^{-1}]]\otimes(\rm{End}\ \mathbb{C}^{n})^{\otimes n}$
by permuting the tensor factors. This action factors through the embedding
$S_{n}\hookrightarrow(\rm{End}\ \mathbb{C}^{n})^{\otimes n}$ hence the group
algebra $\mathbb{C}[S_{n}]$ is a subalgebra of
$Y(\mathfrak{gl}_{n})[[u^{-1}]]\otimes(\operatorname{End}\
\mathbb{C}^{n})^{\otimes n}$. Let $S_{m}$ be the subgroup of $S_{n}$ permuting
the first $m$ tensor factors. Denote by $A_{m}$ the antisymmetrizer

$\sum_{\sigma\in S_{m}}(-1)^{\sigma}\sigma\in\mathbb{C}[S_{m}]\subset
Y(\mathfrak{gl}_{n})[[u^{-1}]]\otimes(\rm{End}\ \mathbb{C}^{n})^{\otimes n}.$

Suppose that $C\in GL_{n}$. For any $a\in\\{1,\ldots,n\\}$ denote by $C_{a}$
the element $i_{a}(1\otimes C)\in
Y(\mathfrak{gl}_{n})[[u^{-1}]]\otimes(\operatorname{End}\mathbb{C}^{n})^{\otimes
n}$. For any $1\leqslant p\leqslant n$ introduce the series with coefficients
in $Y(\mathfrak{gl}_{n})$ by

$\tau_{p}(u,C)=\operatorname{tr}A_{p}C_{1}\ldots C_{p}T_{1}(u)\ldots
T_{p}(u-p+1),$

where we take the trace over all $p$ copies of $\rm{End}\ \mathbb{C}^{n}$.

######  Definition 4.1.

_Bethe subalgebra_ $B(C)\subset Y(\mathfrak{gl}_{n})$ is generated by all
coefficients of the series $\tau_{p}(u,C)$ for $p=1,\ldots,n$.

It follows from the definition that $B(C)=B(a\cdot C)$ for any
$a\in\mathbb{C}$. We fix a maximal torus $T\subset GL_{n}$ (i.e. the subgroup
of diagonal matrices) and denote by $T^{reg}$ the set of regular elements of
$T$. Denote by $GL_{n}^{reg}$ the set of regular elements of the Lie group
$GL_{n}$. The following Proposition summarize known algebraic properties of
Bethe subalgebras, see e.g. [NO], [IR19], [I].

######  Theorem 4.1 (Properties of Bethe subalgebras).

  1. (1)

For any $C\in GL_{n}$, the subalgebra $B(C)$ is commutative.

  2. (2)

For $C\in T^{reg}$, the subalgebra $B(C)$ is a maximal commutative subalgebra.

  3. (3)

For any $C\in GL_{n}^{reg}$ the subalgebra $B(C)$ is freely generated by the
coefficients of $\tau_{p}(u,C)$ with $p=1,\ldots,n$.

  4. (4)

Let $C\in T^{reg}$ and let $\tilde{C}$ be an arbitrary representative of $C$
in the universal cover of $GL_{n}$. Then the subalgebra $B(C)$ generated by
all

$\operatorname{tr}_{V}\rho(\tilde{C})(\rho\otimes 1)(\hat{R}(u)),$

where $(\rho,V)$ ranges over all finite-dimensional representations of
$Y(\mathfrak{gl}_{n})$ and $\hat{R}(u)$ is the universal $R$-matrix for
$Y(\mathfrak{gl}_{n})$.

In fact there are no doubts that the assertion (2) is true for any $C\in
GL_{n}^{reg}$ and that (4) is true for any $C\in GL_{n}$ but still there is no
rigorous proof in the literature. Also, one can take (4) as the definition of
Bethe subalgebras in the case of any simple Lie algebra, see [I].

######  Remark.

_In the literature, Bethe subalgebras are often assigned to any matrix
$C\in\mathfrak{gl}_{n}$ (not necessarily invertible). One can regard these
subalgebras as the ones corresponding to some points in the wonderful
compactification of the group $GL_{n}$, see 4.4 for details._

######  Proposition 4.2.

Bethe subalgebra $B(C)$ of $Y(\mathfrak{gl}_{n})$ is the tensor product
$B^{\prime}(C)\otimes ZY(\mathfrak{gl}_{n})$ where $B^{\prime}(C)$ is a
commutative subalgebra in $Y(\mathfrak{sl}_{n})$ and $ZY(\mathfrak{gl}_{n})$
is the center of $Y(\mathfrak{gl}_{n})$.

###### Proof.

Let $d(u)\in ZY(\mathfrak{gl}_{n})[[u^{-1}]]$ be the (unique) series such that
$d(u)=1$ modulo $u^{-1}$ and

$\operatorname{qdet}T(u)=d(u)d(u-1)\ldots d(u-n+1).$

Then coefficients of series

$d(u)^{-1}\tau_{1}(u,C),(d(u)^{-1}d(u-1)^{-1})\tau_{2}(u,C),\ldots,(d(u)^{-1}d(u-1)^{-1}\ldots
d(u-n+2)^{-1})\tau_{n-1}(u,C)$

and $\tau_{n}(u,C)=\operatorname{qdet}T(u)$ are free generators of $B(C)$. All
coefficients of the first $n-1$ series then belong to $Y(\mathfrak{sl}_{n})$,
and the coefficients of $\operatorname{qdet}T(u)$ generate the center, [Mo,
Theorem 1.7.5]. The statement of the Proposition follows. ∎

We will call $B^{\prime}(C)$ _Bethe subalgebra of_ $Y(\mathfrak{sl}_{n})$ and
also denote it (slightly abusing notation) by $B(C)$. Theorem 4.1 holds for
Bethe subalgebras of $Y(\mathfrak{sl}_{n})$ as well. In the present paper we
restrict ourselves by $C\in T^{reg}$, i.e. we fix maximal torus and consider
the family of Bethe subalgebras parameterized by its regular points.

###  4.2. Quantum shift of argument subalgebras.

There is a family of commutative subalgebras in $U(\mathfrak{gl}_{n})$ closely
related to Bethe subalgebras, called quantum shift of argument subalgebras,
$\mathcal{A}_{\chi}\subset U(\mathfrak{gl}_{n})$ depending
$\chi\in\mathfrak{gl}_{n}$. One can describe these subalgebras as liftings of
shift of argument subalgebras in $S(\mathfrak{gl}_{n})$. Namely, by PBW
thorem, $\operatorname{gr}U(\mathfrak{gl}_{n})=S(\mathfrak{gl}_{n})$. This
determines a natural Poisson structure on $S(\mathfrak{gl}_{n})$ which is
defined on the generators as

$\\{x,y\\}=[x,y]\ \forall\ x,y\in\mathfrak{gl}_{n}$

.

Recall the matrix $E=\sum\limits_{i,j=1}^{n}E_{ij}\otimes e_{ij}\in
S(\mathfrak{gl}_{n})\otimes Mat_{n}$. Let
$E_{j_{1},\ldots,j_{i}}^{j_{1},\ldots,j_{i}}$ be the principal minor of $E$
corresponding to $j_{1},\ldots,j_{i}$ rows and columns of $E$. The Poisson
center of $S(\mathfrak{gl}_{n})$ is freely generated by $n$ homogeneous
generators $P_{1},\ldots,P_{n}$ of degrees $1,\ldots,n$. There are several
natural choices of the generators $P_{i}$; in particular, one can take either
the traces of powers $P_{i}=\operatorname{tr}E^{i},i=1,\ldots,n$ or the
coefficients of the characteristic polynomial $P_{i}=\sum_{1\leq
j_{1}\leq\ldots\leq j_{i}\leq n}E_{j_{1},\ldots,j_{i}}^{j_{1},\ldots,j_{i}}$.

######  Definition 4.2.

The subalgebra of $S(\mathfrak{gl}_{n})$ generated by elements

$\partial^{c_{i}}_{\chi}P_{i},1\leq c_{i}\leq i-1$

is called shift of argument subalgebra and denoted by
$\overline{\mathcal{A}_{\chi}}$.

######  Proposition 4.3.

[MF] For any $\chi\in\mathfrak{gl}_{n}$ subalgebra
$\overline{\mathcal{A}_{\chi}}$ is Poisson commutative.

######  Definition 4.3.

We say that the subalgebra $\mathcal{A}_{\chi}\subset U(\mathfrak{gl}_{n})$ is
a lifting of $\overline{\mathcal{A}_{\chi}}$ if $\mathcal{A}_{\chi}$ is
commutative and
$\operatorname{gr}\mathcal{A}_{\chi}=\overline{\mathcal{A}_{\chi}}$. We call
such subalgebras quantum shift of argument subalgebras.

######  Proposition 4.4.

([FM], [T03], [FFTL], [R06])

  * •

For any $\chi\in\mathfrak{gl}_{n}$ there exists a lifting $\mathcal{A}_{\chi}$
of $\overline{\mathcal{A}_{\chi}}$;

  * •

For $\chi\in\mathfrak{h}^{reg}$ this lifting is unique;

  * •

The subalgebras $\mathcal{A}_{\chi}$ with $\chi\in\mathfrak{gl}_{n}^{reg}$ are
free and maximal commutative;

In the present paper we restrict ourselves by $\chi\in\mathfrak{h}$, i.e. we
fix maximal torus $T$ and its tangent algebra $\mathfrak{h}$ and consider the
family of shift of argument subalgebras parameterized by $\mathfrak{h}$.

A natural question here is how to define explicitly quantum shift of argument
subalgebras? One possible way is as follows. Recall the symmetrization map

$S(\mathfrak{gl}_{n})\to U(\mathfrak{gl}_{n}),x_{1}\ldots
x_{s}\mapsto\frac{1}{s!}\sum_{\sigma\in S_{s}}x_{\sigma(1)}\ldots
x_{\sigma(s)}.$

Here $x_{1},\ldots,x_{s}\in\mathfrak{gl}_{n}$. Let $P_{i},i=1,\ldots,n$ be the
coefficients of characteristic polynomial. Then for any
$\chi\in\mathfrak{h}^{reg}$ elements

$\operatorname{sym}(\partial^{c_{i}}_{\chi}P_{i}),i=1,\ldots,n,1\leq c_{i}\leq
i-1$

are free generators of $\hat{A}_{\chi}$. Another possible way to lift shift of
argument subalgebras is use of Bethe subalgebras. Note that $GL_{n}$ is an
open subset in the matrix algebra $Mat_{n}$ which is $\mathfrak{gl}_{n}$. So
we will regard $GL_{n}$ as an open subset of its Lie algebra
$\mathfrak{gl}_{n}$, respectively, we regard $T$ as an open subset of
$\mathfrak{h}$. The following Proposition is a modified version of discussion
at the end of Section 2 of [NO].

######  Proposition 4.5.

Suppose that $C\in T$. Then

  * •

$\operatorname{ev}_{z}(B(C))=\mathcal{A}_{C^{-1}}$;

  * •

If $C\in T^{reg}$, then images of the coefficients at degrees
$u^{-1},\ldots,u^{-p}$ of $\tau_{p}(u,C)$ are free generators of
$\mathcal{A}_{C^{-1}}$.

###### Proof.

In [NO] the statement is proved for $C\in T^{reg}$. To prove the first
statement for any $C\in T$, the same argument gives the following inclusion

$\operatorname{gr}\operatorname{ev}_{z}(B(C))\supset\overline{\mathcal{A}_{C^{-1}}}.$

On the other hand, $\overline{\mathcal{A}_{C^{-1}}}$ is a maximal Poisson
commutative subalgebra in
$S(\mathfrak{gl}_{n})^{\mathfrak{z}_{\mathfrak{gl}_{n}}(C)}$ (where
$\mathfrak{z}_{\mathfrak{gl}_{n}}(C^{-1})$ is the centralizer of $C$ in
$\mathfrak{gl}_{n}$, see e.g. [IR18, Lemma 4.11]), so we have an equality
here. Hence $\operatorname{ev}_{z}(B(C))=\mathcal{A}_{C^{-1}}$.

∎

###  4.3. Limits of shift of argument subalgebras

There is a way to define more Poisson commutative subalgebras. Recall that for
any $\chi\in\mathfrak{h}^{reg}$ Poincaré series of subalgebra
$\overline{\mathcal{A}_{\chi}}\subset S(\mathfrak{gl}_{n})$ is the same. It
allows us to define the regular map $\theta_{i}:T^{reg}\to Gr(d(i),\dim
S^{i}(\mathfrak{gl}_{n}))$, where $S^{i}(\mathfrak{gl}_{n})$ is $i$-th graded
component of $S(\mathfrak{gl}_{n})$ and
$d(i)=\dim\overline{\mathcal{A}_{\chi}}\cap S^{i}(\mathfrak{gl}_{n})$.
Consider the closures $Z_{i}$ of $T^{reg}$ in Grasmannians and the inverse
limit $Z=\varprojlim Z_{i}$. Every point of $Z$ gives us some Poisson
commutative subalgebra with the same Poincaré series. We call this subalgebras
limit shift of argument subalgebras and new family of subalgebras as
compactification. For detailed definitions we refer the reader to [IR19,
Section 6].

The shift of argument subalgebras in $S(\mathfrak{gl}_{n})$ depend on a
parameter $C\in T^{reg}$ and do not change under the transformations $C\mapsto
aC+bE,a\in\mathbb{C}^{\times},b\in\mathbb{C}$. So the parameter space for the
subalgebras $\mathcal{A}_{C}\subset S(\mathfrak{gl}_{n})$ can be regarded as
$M_{0,n+1}$, i.e a point $C=\operatorname{diag}(z_{1},\ldots,z_{n})$
corresponds to $\mathbb{C}\mathbb{P}^{1}$ with an ordered set of marked points
$z_{1},\ldots,z_{n},\infty$. Let $\overline{M_{0,n+1}}$ be the Deligne-Mumford
compactification, see Section 3.

######  Theorem 4.6.

([Sh], [AFV], [T02], [IR18])

  1. (1)

For the family of shift of argument subalgebras $Z\simeq\overline{M_{0,n+1}}$.

  2. (2)

Limit subalgebras are free polynomial algebras.

  3. (3)

Limit subalgebras are maximal commutative.

In [IR18] there is a practical way to assign a commutative subalgebra to any
$X\in\overline{M_{0,n+1}}$. We will extensively use this description so we
reproduce it here.

Let $X_{\infty}$ be the irreducible component of $X$ containing the marked
point $\infty$. To any distinguished point $\alpha\in X_{\infty}$ we assign
the number $k_{\alpha}$ of marked points on the (reducible) curve $X_{\alpha}$
attached to $X_{\infty}$ at $\alpha$. Let $C$ be the diagonal matrix with the
eigenvalues $\alpha$ of multiplicity $k_{\alpha}$ for all distinguished points
$\alpha\in X_{\infty}$. Then the corresponding shift of argument subalgebra
$F(C)\subset S(\mathfrak{gl}_{n})$ is centralized by the Lie subalgebra
$\bigoplus\limits_{\alpha}\mathfrak{gl}_{k_{\alpha}}$ in $\mathfrak{gl}_{n}$
and contains the Poisson center
$S(\bigoplus\limits_{\alpha}\mathfrak{gl}_{k_{\alpha}})^{\bigoplus\limits_{\alpha}\mathfrak{gl}_{k_{\alpha}}}$.
The subalgebra corresponding to the curve $X$ is just the product of
$F(C)\subset S(\mathfrak{gl}_{n})$ and the subalgebras corresponding to
$X_{\alpha}$ in $S(\mathfrak{gl}_{k_{\alpha}})\subset S(\mathfrak{gl}_{n})$
for all distinguished points $\alpha\in X_{\infty}$ (for this we need to
define the point $\infty$ on each $X_{\alpha}$ – it is just the intersection
with $X_{\infty}$).

######  Proposition 4.7.

([IR18, Proposition 4.7]) The subalgebra $\overline{\mathcal{A}_{X}}$
corresponding to a degenerate curve $X$ is the tensor product
$\overline{\mathcal{A}_{C}}\otimes_{S(\bigoplus\limits_{\alpha}\mathfrak{gl}_{k_{\alpha}})^{\bigoplus\limits_{\alpha}\mathfrak{gl}_{k_{\alpha}}}}\bigotimes\limits_{\alpha}\overline{\mathcal{A}_{X_{\alpha}}}$.

Let us describe the simplest limit subalgebras corresponding to the case when
all the curves $X_{\alpha}$ are irreducible.

######  Proposition 4.8.

[IR18, Lemma 4.9] Suppose that $n=k_{1}+\ldots+k_{l}$, where
$k_{1},\ldots,k_{l}\in\mathbb{Z}_{\geq 1}$ Let
$C_{0}=\operatorname{diag}(\underbrace{a_{1},\ldots,a_{1}}_{k_{1}},\ldots,\underbrace{a_{l},\ldots,a_{l}}_{k_{l}})$
and
$C_{i}=\operatorname{diag}(\underbrace{b_{i,1},\ldots,b_{i,k_{i}}}_{k_{i}})$
for $i=1,\ldots,l$ such that $a_{r}\neq a_{s}$ and $b_{i,r}\neq b_{i,s}$ for
$r\neq s$. Then the element

$\displaystyle C(t):=C_{0}+t\cdot\operatorname{diag}(C_{1},\ldots,C_{l})$

belongs to $T^{reg}$ for small $t\not=0$. The subalgebra $\lim_{t\to
0}\overline{\mathcal{A}_{C(t)}}$ is the tensor product

$\overline{\mathcal{A}_{C_{0}}}\otimes_{Z(S(\mathfrak{gl}_{k_{1}}\oplus\ldots\oplus\mathfrak{gl}_{k_{l}}))}(\overline{\mathcal{A}_{C_{1}}}\otimes\ldots\otimes\overline{\mathcal{A}_{C_{l}}}).$

Here $\overline{\mathcal{A}_{C_{i}}}$ is a subalgebra in
$S(\mathfrak{gl}_{k_{i}})\subset
S(\mathfrak{gl}_{k_{1}}\oplus\ldots\oplus\mathfrak{gl}_{k_{l}})$.

Note that any limit can be obtained by iterations of limit from Proposition
4.8, which we call simple limits.

######  Theorem 4.9.

([T03]) Let $X\in\overline{M_{0,n+1}}$. There exists the unique lifting
$\mathcal{A}_{X}$ of the algebra $\overline{\mathcal{A}_{X}}$. The structure
of subalgebras $\mathcal{A}_{X}$ is the same as that of
$\overline{\mathcal{A}_{X}}$ described in Propositions 4.7 and 4.8.

Note that from Theorem 4.9 we cannot conclude that the parameter space for the
closure of the corresponding family of subalgebras in $U(\mathfrak{gl}_{n})$
is the same. The following Theorem is [HKRW, Theorem 10.8]. We give here
another proof specific to $\mathfrak{gl}_{n}$ case.

######  Theorem 4.10.

The closure of the family $\mathcal{A}_{\chi},\chi\in\mathfrak{h}^{reg}$ is
isomorphic to $\overline{M_{0,n+1}}$.

###### Proof.

Let $\tilde{Z}$ be the closure of the family $U(\mathfrak{gl}_{n})$. It is
enough to prove that the size of a limit filtered subalgebra is not bigger
than that of a generic subalgebra from this family: this implies that the
associated graded of any limit subalgebra is isomorphic to some limit of
associated graded subalgebras, so we obtain the proper birational map
$\tilde{Z}_{i}\to Z_{i}$ which is bijective from Theorem 4.9, thus
$\tilde{Z}\simeq Z$. The following Lemma implies that the size of the limit
subalgebras does not actually jump.

######  Lemma 4.11.

([T03, Theorem 4]) For any $X\in\overline{M_{0,n+1}}$ subspace
$\bigoplus_{m=0}^{k}\overline{\mathcal{A}_{X}}^{(m)}$ is a maximal commutative
subspace of $S(\mathfrak{gl}_{n})^{(m)}$.

∎

Using the description of limit subalgebras we can refine the statement about
uniqueness of $\mathcal{A}_{\chi}$ for $\chi\in\mathfrak{h}$. Note that
$\overline{\mathcal{A}_{\chi}}$ belongs to
$S(\mathfrak{gl}_{n})^{\mathfrak{z}_{\mathfrak{gl}_{n}}(\chi)}$.

######  Proposition 4.12.

The lifting of $\overline{\mathcal{A}_{\chi}}$ to
$U(\mathfrak{gl}_{n})^{\mathfrak{z}_{\mathfrak{gl}_{n}}(\chi)}$ is unique for
any $\chi\in\mathfrak{h}$.

###### Proof.

Let us fix some regular semisimple
$\chi_{1}\in\mathfrak{z}_{\mathfrak{gl}_{n}}(\chi)$ such that
$\chi+t\cdot\chi_{1}$ is regular for small $t\not=0$. Consider the curve
$X\in\overline{M_{0,n+1}}$ corresponding to limit $t\to 0$. The subalgebra
corresponding to this curve is

$\mathcal{A}_{\chi}\otimes_{ZU(\mathfrak{z}_{\mathfrak{gl}_{n}}(\chi))}\mathcal{A}_{\chi_{1}}.$

If there are more than one lifting of $\overline{\mathcal{A}_{\chi}}$ to
$U(\mathfrak{gl}_{n})^{\mathfrak{z}_{\mathfrak{gl}_{n}}(\chi)}$ then we have
at least two liftings of subalgebra $\mathcal{A}_{X}$ which is in
contradiction with Theorem 4.9. ∎

###  4.4. Limits of Bethe subalgebras

One can describe the closure of the family of Bethe subalgebras parameterized
by $C\in T^{reg}$ similarly to that for shift of argument subalgebras. Bethe
subalgebras do not change under transformations of the form $C\to
aC,a\in\mathbb{C}^{\times}$. So the parameter space for $B(C),C\in T^{reg}$
can be regraded as $M_{0,n+2}$, i.e a point
$C=\operatorname{diag}(z_{1},\ldots,z_{n})$ corresponds to
$\mathbb{C}\mathbb{P}^{1}$ with an ordered set of marked points
$0,z_{1},\ldots,z_{n},\infty$. Let $\overline{M_{0,n+2}}$ be the Deligne-
Mumford compactification, see Section 3.

######  Theorem 4.13.

([IR18, Theorem 5.1])

  1. (1)

The closure of the family of Bethe subalgebras corresponding to $C\in T^{reg}$
is parameterized by $\overline{M_{0,n+2}}$.

  2. (2)

Limit subalgebras are free, maximal commutative and of the same size as $B(C)$
with $C\in T^{reg}$.

######  Remark.

_In Section 3 we discuss Losev-Manin $\overline{L_{n+2}}$ space such that
$\overline{M_{0,n+2}}$ is its blow-up. In [IR19] the family of Bethe
subalgebras $B(C)$ is extended to the De Concini-Procesi wonderful closure
$\overline{PGL_{n}}$ of the adjoint group $PGL_{n}$. The Losev-Manin space
$\overline{L_{0,n+2}}$ is the closure of the torus $T$ inside
$\overline{PGL_{n}}$ and its boundary points correspond to the first type of
limits below. This means that one can construct limit subalgebras in two
steps, firstly extend to parameter space to the closure of group and then blow
up subspaces where corresponding subalgebras have smaller size to obtain a
flat family of subalgebras parameterized be a complete variety._

In [IR18] we describe explicitly the limit subalgebra corresponding to a point
$X\in\overline{M_{0,n+2}}$. We will extensively use this description so we
reproduce it here. Let $X_{\infty}$ be the irreducible component of
$X\in\overline{M_{0,n+2}}$ containing the marked point $\infty$. We identify
$X_{\infty}$ with the standard $\mathbb{CP}^{1}$ in such a way that the marked
point $\infty$ is $\infty$ and the point where the curve containing the marked
point $0$ touches $X_{\infty}$ is $0$. To any distinguished point $\alpha\in
X_{\infty}$ we assign the number $k_{\alpha}$ of nonzero marked points on the
maximal (possibly reducible) curve $X_{\alpha}$ attached to $X_{\infty}$ at
$\alpha$ (we set $k_{\alpha}=1$ if $X_{\alpha}$ is a (automatically, marked)
point). Let $C$ be the diagonal $(n-k_{0})\times(n-k_{0})$-matrix with the
eigenvalues $\alpha$ of multiplicity $k_{\alpha}$ for all distinguished points
$0\neq\alpha\in X_{\infty}$. Then the subalgebra $i_{k_{0}}(B(C))$ centralized
by the Lie subalgebra $\bigoplus\limits_{\alpha\neq
0}\mathfrak{gl}_{k_{\alpha}}$ in $\mathfrak{gl}_{n-k_{0}}\subset
i_{k_{0}}(Y(\mathfrak{gl}_{n-k_{0}}))\subset Y(\mathfrak{gl}_{n})$ and by the
complement sub-Yangian $\psi_{n-k_{0}}(Y(\mathfrak{gl}_{k_{0}}))$.

######  Theorem 4.14.

([IR18, Theorem 5.2])

  1. (1)

The limit Bethe subalgebra corresponding to the curve
$X\in\overline{M_{0,n+2}}$ is the product of the following $3$ commuting
subalgebras: first, $i_{k_{0}}(B(C))\subset
i_{k_{0}}(Y(\mathfrak{gl}_{n-k_{0}}))\subset Y(\mathfrak{gl}_{n})$, second,
the subalgebra corresponding to $X_{0}$ in the complement sub-Yangian
$\psi_{n-k_{0}}(Y(\mathfrak{gl}_{k_{0}}))\subset Y(\mathfrak{gl}_{n})$ and
third, the limit shift of argument subalgebras $\mathcal{A}_{X_{\alpha}}$ in
$U(\mathfrak{gl}_{k_{\alpha}})\subset
i_{k_{0}}(Y(\mathfrak{gl}_{n-k_{0}}))\subset Y(\mathfrak{gl}_{n})$ for all
distinguished points $\alpha\neq 0$ (again we define the point $\infty$ on
each $X_{\alpha}$ just as the intersection with $X_{\infty}$).

  2. (2)

$i_{k_{0}}(B(C))$ contains the center of every
$U(\mathfrak{gl}_{k_{\alpha}})\subset i_{k_{0}}(Y(\mathfrak{gl}_{n-k_{0}}))$.
The above product is in fact the tensor product

$\psi_{n-k_{0}}(B(X_{0}))\otimes_{\mathbb{C}}i_{k_{0}}(B(C))\otimes_{ZU(\bigoplus_{\alpha\neq
0}\mathfrak{gl}_{k_{\alpha}})}\bigotimes_{\alpha\neq
0}\mathcal{A}_{X_{\alpha}}.$

Again as in the case of shift of argument subalgebras there are simple limits
such that any limit can be obtained by its iteration.

######  Theorem 4.15.

([IR18, Theorem 5.3]) 1) Let $C_{0}=\operatorname{diag}(a_{1},\ldots,a_{n-k})$
and $C_{1}=\operatorname{diag}(b_{1},\ldots,b_{k})$. Suppose that
$C(t)=\operatorname{diag}(C_{0},\underbrace{0,\ldots,0}_{k})+t\cdot\operatorname{diag}(\underbrace{0,\ldots,0}_{n-k},C_{1})\in
T^{reg}$ (particulary both $C_{0}$ and $C_{1}$ are regular and non-
degenerate). Then

$\lim_{t\to 0}B(C(t))=i_{k}(B(C_{0}))\otimes\psi_{n-k}(B(C_{1})).$

2) Let
$C_{0}=\operatorname{diag}(\underbrace{a_{1},\ldots,a_{1}}_{k_{1}},\ldots,\underbrace{a_{l},\ldots,a_{l}}_{k_{l}})$
be a non-degenerate matrix and
$C_{i}=\operatorname{diag}(\underbrace{b_{i,1},\ldots,b_{i,k_{i}}}_{k_{i}})$
for $i=1,\ldots,l$ such that $a_{r}\neq a_{s}$ and $b_{i,r}\neq b_{i,s}$ for
$r\neq s$. Let

$\displaystyle C(t):=C_{0}+t\cdot\operatorname{diag}(C_{1},\ldots,C_{l}).$

Then

$\lim_{t\to
0}B(C(t))=B(C_{0})\otimes_{\bigotimes\limits_{i=1}^{l}Z(U(\mathfrak{gl}_{k_{i}}))}\bigotimes\limits_{i=1}^{l}\mathcal{A}_{C_{i}},$

where $\mathcal{A}_{C_{i}}$ is the shift of argument subalgebra in
$U(\mathfrak{gl}_{k_{i}})\subset U(\mathfrak{gl}_{n})\subset
Y(\mathfrak{gl}_{n})$ (the copy of $U(\mathfrak{gl}_{n})$ in the Yangian
$Y(\mathfrak{gl}_{n})$ is generated by $t_{ij}^{(1)}$).

The last Theorem allows us to describe explicitly the image of
$B(X),X\in\overline{M_{0,n+2}}$ under the evaluation homomorphism.

######  Proposition 4.16.

For any $X\in\overline{M_{0,n+2}}$ we have

$\operatorname{ev}(B(X))=\mathcal{A}_{\sigma_{1}(X)},$

where $\sigma_{1}$ is the involution (3.3) on $\overline{M_{0,n+2}}$ .

###### Proof.

It is enough to prove the Proposition for the two types of simple limits. Let
$n=k_{1}+\ldots+k_{l}$, where $k_{1},\ldots,k_{l}\in\mathbb{Z}_{\geq 1}$ and
let
$C_{0}=\operatorname{diag}(\underbrace{a_{1},\ldots,a_{1}}_{k_{1}},\ldots,\underbrace{a_{l},\ldots,a_{l}}_{k_{l}})$
and
$C_{i}=\operatorname{diag}(\underbrace{b_{i,1},\ldots,b_{i,k_{i}}}_{k_{i}})$
for $i=1,\ldots,l$ such that $a_{r}\neq a_{s}$ and $b_{i,r}\neq b_{i,s}$ for
$r\neq s$. For the second type of limits, the statement follows from
Proposition 4.5:

$B(C_{0})\otimes_{\bigotimes\limits_{i=1}^{l}Z(U(\mathfrak{gl}_{k_{i}}))}\bigotimes\limits_{i=1}^{l}\mathcal{A}_{C_{i}}$

maps to
$\mathcal{A}_{C_{0}^{-1}}\otimes_{\bigotimes\limits_{i=1}^{l}Z(U(\mathfrak{gl}_{k_{i}}))}\bigotimes\limits_{i=1}^{l}\mathcal{A}_{C_{i}}$.

Let $C_{0}=\operatorname{diag}(a_{1},\ldots,a_{n-k})$ and
$C_{1}=\operatorname{diag}(a_{n-k+1},\ldots,a_{n})$. Suppose that
$\operatorname{diag}(C_{0},\underbrace{0,\ldots,0}_{k})+t\cdot\operatorname{diag}(\underbrace{0,\ldots,0}_{n-k},C_{1})\in
T^{reg}$. For the first type of limits we need to consider a limit subalgebra
of the form

$i_{k}(B(C_{0}))\otimes_{\mathbb{C}}\psi_{n-k}(B(C_{1}))$

and prove that it maps to

$\mathcal{A}_{C_{0}^{-1}}\otimes_{ZU(\mathfrak{gl}_{k})}\mathcal{A}_{(\underbrace{0,\ldots,0}_{k},C_{1}^{-1})}.$

By Proposition 4.5, the subalgebra $i_{k}(B(C_{0}))$ maps to
$\mathcal{A}_{C_{0}^{-1}}\subset U(\mathfrak{gl}_{n-k})$. We need to prove
that $\psi_{n-k}(B(C_{1}))$ maps to $\mathcal{A}_{(0,\ldots,0,C_{1}^{-1})}$.
According to the proof of Theorem 6.5 in [IR19], the coefficients of the
following series are the generators of $\psi_{n-k}(B(C_{1}))$:

$t_{1\ldots n-k}^{1\ldots
n-k}(u+n-k)^{-1}\cdot\left(\prod_{i=1}^{n-k}a_{i}\right)\sum_{1\leqslant
a_{1}\leqslant
k}a_{n-k+a_{1}}t_{1,\ldots,n-k,n-k+a_{1}}^{1,\ldots,n-k,n-k+a_{1}}(u)$
$t_{1\ldots n-k}^{1\ldots
n-k}(u+n-k)^{-1}\cdot\left(\prod_{i=1}^{n-k}a_{i}\right)\sum_{1\leqslant
a_{1}<a_{2}\leqslant
k}a_{n-k+a_{1}}a_{n-k+a_{2}}t_{1,\ldots,n-k,n-k+a_{1},n-k+a_{2}}^{1,\ldots,n-k,n-k+a_{1},n-k+a_{2}}(u)$

…

$\tau_{n}(u,X)=t_{1\ldots n-k}^{1\ldots n-k}(u+n-k)^{-1}\cdot a_{1}\ldots
a_{n}t_{1\ldots n}^{1\ldots n}(u).$

The coefficients of $t_{1\ldots n-k}^{1\ldots n-k}(u+n-k)^{-1}$ map to some
elements in the center of $U(\mathfrak{gl}_{k})$ hence to the elements of
$\mathcal{A}_{C_{0}^{-1}}$.

Therefore the images of coefficients of the following series are contained in
the image of the corresponding Bethe subalgebra:

$a_{n-k+1}^{-1}\ldots a_{n}^{-1}\sum_{1\leqslant a_{1}\leqslant
k}a_{n-k+a_{1}}t_{1,\ldots,n-k,n-k+a_{1}}^{1,\ldots,n-k,n-k+a_{1}}(u)$
$a_{n-k+1}^{-1}\ldots a_{n}^{-1}\sum_{1\leqslant a_{1}<a_{2}\leqslant
k}a_{n-k+a_{1}}a_{n-k+a_{2}}t_{1,\ldots,n-k,n-k+a_{1},n-k+a_{2}}^{1,\ldots,n-k,n-k+a_{1},n-k+a_{2}}(u)$

…

$t_{1\ldots n}^{1\ldots n}(u)=\operatorname{qdet}T(u).$

The leading term of these elements are the derivatives of coefficients of
characteristic polynomial along $(0,\ldots,0,C_{0}^{-1})\in\mathfrak{h}$.
Therefore the images of coefficients of these series generate
$\mathcal{A}_{(0,\ldots,0,C_{1}^{-1})}$.

∎

###  4.5. The family of shift of argument subalgebras in centralizer

Let $k\geq 0$. Consider the family $\mathfrak{U}$ of shift of argument
subalgebras in $S(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$ of the form
$\mathcal{A}_{(\chi,\underbrace{0,\ldots,0}_{k})},\chi\in\mathfrak{h}^{reg}$.

######  Proposition 4.17.

([IR18, Lemma 4.11])

  1. (1)

The closure of the parameter space of the family $\mathfrak{U}$ is isomorphic
to $\overline{M_{0,n+2}}$.

  2. (2)

For any $X\in\overline{M_{0,n+2}}$ subalgebra $\mathcal{A}_{X}$ is free and
maximal commutative subalgebra of
$S(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$.

Similarly to the proof of Proposition 4.12 one can prove that for any
$X\in\overline{M_{0,n+2}}$ a lifting of $\mathcal{A}_{X}\subset
S(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$ to
$U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$ is unique. Moreover, the closure
of the family of subalgebras of the form
$\mathcal{A}_{(\chi,\underbrace{0,\ldots,0})}\subset
U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}},\chi\in\mathfrak{h}^{reg}$ is
isomorphic to $\overline{M_{0,n+2}}$ as well because the size of a limit
subalgebra from this family is the same as for generic one, i.e. the proof is
analagous to the proof of Theorem 4.10. Let us describe explicitly the
subalgebra $B(X)\subset U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$ for any
$X\in\overline{M_{0,n+2}}$.

Let $X_{\infty}$ be the irreducible component of $X\in\overline{M_{0,n+2}}$
containing the marked point $\infty$. We identify $X_{\infty}$ with the
standard $\mathbb{CP}^{1}$ in such a way that the marked point $\infty$ is
$\infty$ and the point where the curve containing the marked point $0$ touches
$X_{\infty}$ is $0$. To any distinguished point $\alpha\in X_{\infty}$ we
assign the number $k_{\alpha}$ of nonzero marked points on the maximal
(possibly reducible) curve $X_{\alpha}$ attached to $X_{\infty}$ at $\alpha$
(we set $k_{\alpha}=1$ if $X_{\alpha}$ is a (automatically, marked) point).
Let $\chi$ be the diagonal $(n-k_{0})\times(n-k_{0})$-matrix with the
eigenvalues $\alpha$ of multiplicity $k_{\alpha}$ for all distinguished points
$0\neq\alpha\in X_{\infty}$. Then the subalgebra
$\mathcal{A}_{(\chi,\underbrace{0,\ldots,0}_{k_{0}+k})}$ is centralized by the
Lie subalgebra $\bigoplus\limits_{\alpha\neq 0}\mathfrak{gl}_{k_{\alpha}}$ in
$\mathfrak{gl}_{n-k_{0}}\subset U(\mathfrak{gl}_{n-k_{0}})\subset
U(\mathfrak{gl}_{n})$ and by the complement $U(\mathfrak{gl}_{k_{0}})$.

######  Proposition 4.18.

The subalgebra in $U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$ corresponding
to $X\in\overline{M_{0,n+2}}$ has the form

$\mathcal{A}_{X_{0}}\otimes_{ZU(\mathfrak{gl}_{k})}(\mathcal{A}_{(\chi,\underbrace{0,\ldots,0}_{k_{0}+k})}\otimes_{ZU(\bigoplus_{\alpha\not=0}\mathfrak{gl}_{k_{\alpha}})}\bigotimes_{i=1}^{l}\mathcal{A}_{X_{\alpha_{i}}})$

###### Proof.

Follows from the explicit description of limit subalgebras in
$U(\mathfrak{gl}_{n+k})$. ∎

This means that one can get any limit subalgebra by iterating the following
two simple limits. The first simple limit has the form

$\mathcal{A}_{(C_{0},\underbrace{0,\ldots,0}_{m+k})}\otimes_{ZU(\mathfrak{gl}_{k})}\mathcal{A}_{(C_{1},\underbrace{0,\ldots,0}_{k})}$

The second simple limit has the form

$\mathcal{A}_{(C_{0},\underbrace{0,\ldots,0}_{k})}\otimes_{\bigotimes\limits_{i=1}^{l}Z(U(\mathfrak{gl}_{k_{i}}))}\bigotimes_{i=1}^{l}\mathcal{A}_{C_{i}}$

###  4.6. Centralizer construction and Bethe subalgebras

Let $X\in\overline{M_{0,n+2}}$. We have the following

######  Proposition 4.19.

[IR18] The maps $\eta_{k}$ are an asymptotic isomorphism between $B(X)\otimes
A_{0}$ and $\mathcal{A}_{X}\subset
U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$.

In fact the statement of Proposition 4.19 can be refined.

######  Proposition 4.20.

For any $k\geq 0$ we have $\eta_{k}(B(X)\otimes A_{0})=\mathcal{A}_{X}\subset
U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}.$

###### Proof.

Let $C\in T$.

######  Lemma 4.21.

  1. (1)

$\eta_{k}(\mathcal{A}_{C}\otimes 1)=\mathcal{A}_{C}\subset
U(\mathfrak{gl}_{n})\subset U(\mathfrak{gl}_{n+k})$

  2. (2)

$\eta_{k}(B(C)\otimes A_{0})=\mathcal{A}_{(C,\underbrace{0,\ldots,0)}_{k}}$

###### Proof.

1) $\eta_{k}(\mathcal{A}_{C}\otimes 1)=\operatorname{ev}\circ\omega_{n+k}\circ
i_{k}(\mathcal{A}_{C})=\mathcal{A}_{C}\subset U(\mathfrak{gl}_{n})\subset
U(\mathfrak{gl}_{n+k})$, because the restriction of $\omega_{n+k}$ to
$U(\mathfrak{gl}_{n+k})$ is the identity map.
2) $\eta_{k}(B(C)\otimes A_{0})=\operatorname{ev}\circ\omega_{n+k}\circ
i_{k}(B(C))\cdot
ZU(\mathfrak{gl}_{n+k})=\operatorname{ev}\circ\omega_{n+k}\circ
i_{k}\circ\omega_{n}(B(C^{-1}))\cdot
ZU(\mathfrak{gl}_{n+k})=\operatorname{ev}\circ\psi_{k}(B(C^{-1}))\cdot
ZU(\mathfrak{gl}_{n+k})=\mathcal{A}_{(C,0,\ldots,0)}$ according to the proof
of Proposition 4.16. ∎

As in the proof of Proposition 4.16 it is enough to show that the map is
compatible with the elementary limits. Consider a limit of the second type.
Suppose that $X\in\overline{M_{0,n+2}}$ such that

$B(X)=B(C_{0})\otimes_{\bigotimes\limits_{i=1}^{l}Z(U(\mathfrak{gl}_{k_{i}}))}\bigotimes\limits_{i=1}^{l}\mathcal{A}_{C_{i}}$

From Lemma it follows that

$\eta_{k}(B(X)\otimes
A_{0})=\mathcal{A}_{(C_{0},\underbrace{0,\ldots,0)}_{k}}\otimes_{\bigotimes\limits_{i=1}^{l}Z(U(\mathfrak{gl}_{k_{i}}))}\mathcal{A}_{C_{i}}=\mathcal{A}_{X}\subset
U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$

Consider the first type of a limit subalgebra. Let $X\in\overline{M_{0,n+2}}$
such that

$B(X)=i_{m}(B(C_{0}))\otimes\psi_{n-m}(B(C_{1})).$

From [IR18, Lemma 2.15] it follows, that the restriction of $\eta_{k}$ to
$i_{m}(Y(\mathfrak{gl}_{n-m}))$ is $\eta_{k+m}$ and that the restriction of
$\eta_{k}$ to $\psi_{n-m}(Y(\mathfrak{gl}_{m}))$ is $\eta_{k}$. Applying Lemma
4.21 we get

$\eta_{k}(B(X)\otimes
A_{0})=\mathcal{A}_{(C_{0},\underbrace{0,\ldots,0)}_{k+m}}\otimes\mathcal{A}_{(C_{1},\underbrace{0,\ldots,0}_{k})}=\mathcal{A}_{X}\subset
U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$

∎

###  4.7. Spectra of quantum shift of argument subalgebras

Let $V_{\lambda}$ be an irreducible representation of $\mathfrak{gl}_{n}$ with
the arbitrary highest weight $\lambda$.

######  Theorem 4.22.

For any $X\in\overline{M_{0,n+1}}$ the subalgebra $\mathcal{A}_{X}$ acts with
a cyclic vector in $V_{\lambda}$.

Let $k\geq 0$ and let $V_{\lambda}$ be an irreducible representation of
$\mathfrak{gl}_{n+k}$ with the highest weight
$(\lambda_{1},\ldots,\lambda_{n+k})$.

Let

$V=\bigoplus_{\mu}M_{\lambda\mu}\otimes V_{\mu}$

be the decomposition of the restriction of $V$ to $\mathfrak{gl}_{k}$ into
irreducible representations of $\mathfrak{gl}_{k}$. The following Proposition
is well-known.

######  Proposition 4.23.

1) $M_{\lambda\mu}\not=0$ if and only if $\mu$ is a subdiagram of $\lambda$
and $\mu_{i}-\lambda_{i+n}\geq 0\ \forall\,i=1,\ldots,n$;
2) $M_{\lambda\mu}$ is an irreducible representation of
$U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$.

The following proposition is a refined version of Theorem 11.2, [HKRW].

######  Proposition 4.24.

For any $X\in\overline{M_{0,n+2}}$ subalgebra $\mathcal{A}_{X}\subset
U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$ has cyclic vector on any
irreducible representation of $U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$ of
the form $M_{\lambda\mu}$ for some integer $\mathfrak{gl}_{n+k}$-weight
$\lambda$ and integer $\mathfrak{gl}_{k}$-weight $\mu$.

###### Proof.

For $X\in T^{reg}$ the statement is a consequence of Theorem 11.2 [HKRW]. For
other $X$ the statement of the Proposition follows from Proposition 4.18 using
the induction on the number of irreducible components of the curve. ∎

######  Proposition 4.25.

For any $X\in\overline{M_{0,n+2}^{split}}$ subalgebra $\mathcal{A}_{X}\subset
U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$ acts with simple spectrum on any
irreducible representation of $U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$ of
the form $M_{\lambda\mu}$ for some integer $\mathfrak{gl}_{n+k}$-weight
$\lambda$ and some integer $\mathfrak{gl}_{k}$-weight $\mu$.

###### Proof.

For $X\in T^{reg}_{split}$ we know that there is a cyclic vector on such
representation. Moreover $\mathcal{A}_{X}=\mathcal{A}_{X_{0}}\cap
U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}\subset U(\mathfrak{gl}_{n+k})$ for
some $X_{0}\in\overline{M_{0,n+k+2}}$. We know [FFR] that
$\mathcal{A}_{X_{0}}$ acts by Hermitian operators with respect to _some_
positive definite Hermitian form. This implies that $\mathcal{A}_{X}$ acts by
Hermitian operators as well with respect to restriction of this form to the
multiplicity space. So the statement follows.

For other $X\in\overline{M_{0,n+2}^{split}}$, the statement of the Proposition
follows from Proposition 4.18 using the induction on the number of irreducible
components of the curve. ∎

##  5\. Cyclic vector

###  5.1. General conjecture.

Let $X\in\overline{M_{0,n+2}}$ and consider $B(X)\subset
Y(\mathfrak{gl}_{n})$. Let
$V_{\lambda_{i}\setminus\mu_{i}}(z_{i}),i=1,\ldots,k$ be a set of skew
representations of $Y(\mathfrak{gl}_{n})$. Consider the representation
$W=\bigotimes_{i=1}^{k}V_{\lambda_{i}\setminus\mu_{i}}(z_{i})$ such that
$z_{i}-z_{j}\not\in\mathbb{Z}$.

######  Conjecture 5.1.

$B(X)$ has a cyclic vector in $W$ for all $X\in\overline{M_{0,n+2}}$.

The following observations support the conjecture. Note that in the conjecture
we do not suppose that representation $W$ is irreducible.

  1. (1)

Gelfand-Tsetlin subalgebra $H$ is $B(X)$ for $X\in\overline{M_{0,n+2}}$ being
the so-called caterpillar curve.

It is known that if $z_{i}-z_{j}\not\in\mathbb{Z}$ then $H$ acts with simple
spectrum on $W$.

  2. (2)

From the geometric point of view there is an action of $Y(\mathfrak{gl}_{n})$
on the equivariant cohomology of quivier varieties of type A. The direct sum
of cohomology spaces is the tensor product of fundamental Yangian
representations with the equivariant parmeters being the evaluation
parameters. There is a conjecture of [MO] that $B(C),C\in T^{reg}$ acts by the
operators of quantum multiplication by cohomology classes thus has the unity
class as a cyclic vector;

  3. (3)

The $Y(\mathfrak{gl}_{2})$-case of this conjecture is proved in [Ma];

  4. (4)

In [MTV14] the authors consider representation of the form
$\bigotimes_{i=1}^{k}\mathbb{C}^{n}(z_{i})$ and prove that $B(C),C\in T^{reg}$
has a cyclic vector on it if $z_{i}\not=z_{j}+1$ for $i>j$. Note that it
includes some cases where representation is not irreducible.

In the present paper we prove the conjecture for generic values of the
parameters $z_{i}$.

###  5.2. Tensor product of skew representations

We are going to prove that Bethe subalgebra of $Y(\mathfrak{gl}_{n})$ acts
with a cyclic vector on any tensor product skew representation with generic
values of the evaluation parameters. From Schur lemma and Proposition 2.1 it
follows that the restriction of a skew representation to
$Y(\mathfrak{sl}_{n})$ is an irreducible representation of
$Y(\mathfrak{sl}_{n})$ and that existence of a cyclic vector for a Bethe
subalgebra of $Y(\mathfrak{sl}_{n})$ implies the existence of a cyclic vector
for the corresponding Bethe subalgebra of $Y(\mathfrak{gl}_{n})$ in the
original representation.

Irreducible representations of $Y(\mathfrak{sl}_{n})$ are classified by means
of Drinfeld polynomials, see e.g. [Mo, Paragraph 3]. The following Proposition
describes Drinfeld polynomials of a skew representation. Let
$\lambda\setminus\mu$ be a skew diagram such that $M_{\lambda\mu}\not=0$. For
any cell $\alpha\in\lambda\setminus\mu$ let $c(\alpha)$ be its content, i.e.
$c(\alpha)=j-i$ if $\alpha$ belongs to $i$-th row and $j$-th column.

######  Proposition 5.2.

([Mo, Corollary 8.5.5]) Drinfeld polynomials $P_{1}(u),\ldots,P_{n-1}(u)$ of
$V_{\lambda\setminus\mu}$ as $Y(\mathfrak{sl}_{n})$ representation are

$P_{k}(u)=\prod_{\alpha}(u+c(\alpha)),$

where the product is over all $\alpha$ be the upper cells in columns of height
$k$ in $\lambda/\mu$.

######  Proposition 5.3.

A representation of the form
$\bigotimes_{i=1}^{k}V_{\lambda_{i}\setminus\mu_{i}}(z_{i})$ with
$|z_{i}-z_{j}|\gg 0$ and non-positive integers $z_{i}$’s is a skew
representation of $Y(\mathfrak{sl}_{n})$.

###### Proof.

It is enough to choose skew diagram $\lambda\setminus\mu$ such that
$M_{\lambda\mu}\not=0$ and Drinfeld polynomials of $V_{\lambda\setminus\mu}$
coincides with Drinfeld polynomials of
$\bigotimes_{i=1}^{k}V_{\lambda_{i}\setminus\mu_{i}}(z_{i})$. The following
facts are well-known, see [Mo].
1) If the tensor product of irreducible representations is irreducible then
the Drinfeld polynomial of the tensor product is the product of the Drinfeld
polynomials of tensor factors;
2) $V_{\lambda_{i}\setminus\mu_{i}}(z_{i})$ is isomorphic to
$V_{\tilde{\lambda}_{i}\setminus\tilde{\mu}_{i}}(0)$ where
$\tilde{\lambda}\setminus\tilde{\mu}$, where
$\tilde{\lambda}_{i}=\lambda_{i}-z_{i},\tilde{\mu}_{i}=\mu_{i}-z_{i}$.
3) If $|z_{i}-z_{j}|\gg 0$ for all pairs $i,j$ then the tensor product
$\bigotimes_{i=1}^{k}V_{\lambda_{i}\setminus\mu_{i}}(z_{i})$ is an irreducible
representation of $Y(\mathfrak{sl}_{n})$.

This means that it is enough to choose such Young diagrams $\lambda$ and $\mu$
that $\lambda\setminus\mu$ is a disjoint union of the skew diagrams
$\tilde{\lambda}_{i}\setminus\tilde{\mu}_{i}$ such that the rows’ and columns’
numbers of the components are sufficiently distant from each other. Note that
in this case
$M_{\lambda\mu}=\bigotimes_{i=1}^{k}V_{\tilde{\lambda}_{i}\setminus\tilde{\mu}_{i}}(0)=\bigotimes_{i=1}^{k}V_{\lambda_{i}\setminus\mu_{i}}(z_{i})$
as a $Y(\mathfrak{sl}_{n})$-module. ∎

###  5.3. Cyclic vector for Bethe subalgebras of $Y(\mathfrak{gl}_{n})$

From Proposition 4.20 it follows that the image of $B(X)$ under $\eta_{k}$ is
$\mathcal{A}_{X}\subset U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$.

######  Theorem 5.4.

For all $X\in\overline{M_{0,n+2}}$, the subalgebra $B(X)$ has a cyclic vector
on any skew representation $V$ of $Y(\mathfrak{gl}_{n})$.

###### Proof.

Follows from Proposition 4.24. ∎

######  Theorem 5.5.

Let $I$ be the subset of $\mathbb{C}^{n}$ such that all
$B(X),X\in\overline{M_{0,n+2}}$ has a cyclic vector on $V$. The the set $I$ is
a Zariski open subset of $\mathbb{C}^{n}$.

###### Proof.

Consider the space $\overline{M_{0,n+2}}\times\mathbb{C}^{n}$. Let
$\pi_{\mathbb{C}^{n}}:\overline{M_{0,n+2}}\times\mathbb{C}^{n}\to\mathbb{C}^{n}$
be the projection to second factor. Let
$J\subset\overline{M_{0,n+2}}\times\mathbb{C}^{n}$ be the set such that for
any $(X,(z_{1},\ldots,z_{k}))$ subalgebra $B(X)$ does not have a cyclic vector
on $W=\bigotimes_{i=1}^{k}V_{\lambda_{i}\setminus\mu_{i}}(z_{i})$.

######  Lemma 5.6.

The set $\pi_{\mathbb{C}^{n}}(J)$ is a proper closed subset (in Zariski
topology) of $\mathbb{C}^{n}$.

###### Proof.

Note that $\overline{M_{0,n+2}}$ is a projective algebraic variety hence is a
complete algebraic variety. It follows that for any algebraic variety $Y$ the
projection $\pi_{Y}:\overline{M_{0,n+2}}\times Y\to Y$ is a closed map (with
respect to Zariski topology). Consider $Y=\mathbb{C}^{k}$ and the projection
$\pi_{\mathbb{C}^{n}}:\overline{M_{0,n+2}}\times\mathbb{C}^{n}\to\mathbb{C}^{n}$.
Note that the condition to have a cyclic vector is open hence the set
$J\subset\overline{M_{0,n+2}}\times\mathbb{C}^{n}$ is a closed subset thus its
image $\pi_{\mathbb{C}^{n}}(J)$ is also closed. From Proposition 5.3 it
follows that there exist parameters $z_{i},i=1,\ldots,k$ such that $W$ is a
skew representation of $Y(\mathfrak{gl}_{n})$. Then from Theorem 5.4 it
follows that $\pi_{\mathbb{C}^{n}}(J)$ is a proper closed subset of
$\mathbb{C}^{n}$. ∎

From the Lemma it follows that $I$ is Zariski open and non-empty.

∎

######  Corollary 5.7.

For all $X\in\overline{M_{0,n+2}}$ the subalgebra $B(X)$ has a cyclic vector
on any representation of the form
$\bigotimes_{i=1}^{k}V_{\lambda_{i}\setminus\mu_{i}}(z_{i})$ for all
$|z_{i}-z_{j}|\gg 0$ on any irreducible curve in $\mathbb{C}^{n}$ which is not
contained in $J$.

###### Proof.

If the curve in $\mathbb{C}^{n}$ does not belong to $J$ then
$J\cap\mathbb{C}^{n}$ is a proper closed subset of the curve thus finite.

∎

##  6\. Real forms of $\overline{M_{0,n+2}}$ and simplicity of spectra

In this section we give a proof of modified Proposition 4.2 from [KR]
following the argument of Section 1 of [Re].

###  6.1. Hermitian product on a tensor product of evaluation representations

Let $W$ be a representation of $Y(\mathfrak{gl}_{n})$ and $\rho_{W}$ be the
corresponding morphism to $\operatorname{End}(W)$. Let $T^{(p)}(u)$ be a
$\binom{n}{p}\times\binom{n}{p}$ matrix whose entries are the principal
quantum $p\times p$ minors. One can also define $T^{(p)}(u)$ by the following
two equalities:

$T^{(p)}(u)=A_{p}T_{1}(u)\ldots T_{p}(u-p+1)=T_{p}(u-p+1)\ldots T_{1}(u)A_{p}$

We consider $T^{(p)}(u)$ as an element of $Mat_{\binom{n}{p}}\otimes
Y(\mathfrak{gl}_{n})[[u^{-1}]]$. By definition put

$\mathcal{T}^{(p)}(u)=\operatorname{id}\otimes\rho_{W}(T^{(p)}(u)).$

If $W=V(z)$, where $V$ is an irreducible $\mathfrak{gl}_{n}$ module and
$z\in\mathbb{C}$ then

$\mathcal{T}^{(1)}(u)=1+\left(\sum_{1\leqslant i,j\leqslant
n}e_{ij}\otimes\rho_{V}(E_{ij})\right)(u+z)^{-1}$

where $e_{ij}\in$ $Mat_{n}$ are matrix identities and $E_{ij}$ form a basis of
$\mathfrak{gl}_{n}$.

Let

$W=V_{1}(z_{1})\otimes\ldots\otimes V_{k}(z_{k})$

be a product of evaluation modules. Then we have

$\mathcal{T}_{W}^{(1)}(u)=\prod_{i=1}^{k}\mathcal{T}_{V_{i}(z_{i})}^{(1)}(u)$

Each $V_{i}$ has a standard Hermitian form which makes $V_{i}$ a unitary
$\mathfrak{gl}_{n}$ representation. By taking the product of these forms, we
obtain a Hermitian form on $W$. We denote by + the conjugate operator with
respect to the Hermitian form. By t we denote the transpose of the operators
with respect to matrix argument, i.e. the $(i,j)$-th matrix element of
$T(u)^{t}$ is $t_{ji}(u)$ and the same for $\mathcal{T}^{(1)}(u)^{t}$.

######  Proposition 6.1.

Suppose that for each $i$, $1\leqslant i\leqslant k$,

$\left(\mathcal{T}^{(1)}_{V_{i}(z_{i})}(u)\right)^{-1}=f_{i}(u)\left((\mathcal{T}^{(1)}_{V_{i}(z_{i})}(-u+c))^{t}\right)^{+}$

with some $f_{i}(u)\in 1+\mathbb{C}[[u^{-1}]]$ and $c\in\mathbb{C}$ and that
$C\in T^{comp}$. Then the subalgebra $B(C)$ acts by normal operators on $W$.

###### Proof.

From the definition of $T^{(p)}(u)$ it follows that for any representation
$V_{i}(z_{i})$ of $Y(\mathfrak{gl}_{n})$ we have

$\mathcal{T}^{(p)}(u)=A_{k}\mathcal{T}^{(1)}_{1}(u)\ldots\mathcal{T}^{(1)}_{k}(u-p+1)$

Using the fact that $T(u)\to T^{t}(u),T(u)\to T(-u)$ and the conjugation are
anti-automorphisms, we get

$\displaystyle(\mathcal{T}^{(p)}(-u+c)^{t})^{+}=A_{p}(\mathcal{T}^{(1)}_{p}(-u+p-1+c)^{t})^{+}\ldots(\mathcal{T}^{(1)}_{1}(-u+c)^{t})^{+}=$
$\displaystyle f_{i}(u)^{-1}\ldots
f_{i}(u-p+1)^{-1}A_{p}(\mathcal{T}^{(1)}_{p})^{-1}(-u+p-1)\ldots(\mathcal{T}^{(1)}_{1})^{-1}(u)=g_{pi}(u)(\mathcal{T}^{(p)}(u))^{-1}$

Since

$\mathcal{T}^{(p)}_{W}(u)=\prod_{i=1}^{k}\mathcal{T}^{(p)}_{V_{i}(z_{i})}(u)$

we obtain

$\left(\mathcal{T}^{(p)}_{W}(u)\right)^{-1}=\prod_{i=1}^{k}g_{pi}(u)\left((\mathcal{T}^{(p)}_{W}(-u+c))^{t}\right)^{+}$

for some $g_{pi}(u)\in 1+\mathbb{C}[[u^{-1}]]$ and $1\leq p\leq n$.

Abusing notation we denote by $\tau_{p}(u,C)$ the image of $\tau_{p}(u,C)$ in
$W$. Multiplying the last equality by
$\rho_{\Lambda^{p}(\mathbb{C}^{n})}(C^{-1})=\rho_{\Lambda^{p}(\mathbb{C}^{n})}(\overline{C})$
and taking the trace we get that

$S(\tau_{p}({u,C^{-1}}))=\left(\prod_{i=1}^{k}g_{pi}(u)\right)\tau_{p}(-u+c,C)^{+}$

But $S(B(C^{-1}))=B(C)$, see e.g [IR18]. From this we see that coefficients of
$\tau_{k}(u,C)^{+}$ belong to $B(C)$ hence $B(C)$ acts by normal operators.

∎

######  Corollary 6.2.

Under the conditions of Proposition 6.1, the subalgebra $B(X)$ acts by normal
operators on $W$ for any $X\in\overline{M_{0,n+2}^{comp}}$.

###### Proof.

Indeed, the condition $[B(X),B(X)^{+}]=0$ determines a Zariski closed subset
in $\overline{M_{0,n+2}^{comp}}$. On the other hand, it is satisfied on a
Zariski open dense subset $T_{comp}^{reg}\subset\overline{M_{0,n+2}^{comp}}$.
Hence it is satisfied for any $X\in\overline{M_{0,n+2}^{comp}}$. ∎

### 6.2.

Now we are going to find representations of $Y(\mathfrak{gl}_{n})$ which
satisfy the conditions of Proposition 6.1. Let $\omega_{r},1\leqslant
r\leqslant n-1$ be the fundamental weights of $\mathfrak{sl}_{n}$.

######  Proposition 6.3.

Let $V$ be an irreducible representation of $\mathfrak{gl}_{n}$ and $V(z)$ be
a evaluation representation of $Y(\mathfrak{gl}_{n})$. Then the condition of
Proposition 6.1 is equivalent to the following two conditions:

  1. (1)

The highest weight of $V$ is equal to $l\omega_{r}$ for some $l\in\mathbb{Z}$,
$1\leqslant r\leqslant n-1$;

  2. (2)

the evaluation parameter $z=\frac{l+n-r-\bar{c}+1}{2}+iv$, with
$v\in\mathbb{R}$.

Here $c$ is from Proposition 6.1.

###### Proof.

We can rewrite the condition from Proposition 6.1 as

$\mathcal{T}^{(1)}_{V(z)}(u)^{t}\mathcal{T}^{(1)}_{V(z)}(-u+c)^{+}=f(u)\mathrm{Id}\otimes\mathrm{Id}.$

for some $f(u)\in 1+\mathbb{C}[[u^{-1}]]$.

As we know from (6.1),

$\mathcal{T}_{V(z)}^{(1)}(u)=1+\left(\sum_{1\leqslant i,j\leqslant
n}e_{ij}\otimes\rho_{V}(E_{ij})\right)(u+z)^{-1}.$

Putting this together and multiplying by $(u+z)(-u+\bar{z}+\bar{c})$, on the
LHS we get

$(u+z)(-u+\bar{z}+\bar{c})+(z+\bar{z}+\bar{c})\left(\sum_{1\leqslant
i,j\leqslant n}e_{ij}\otimes\rho_{V}(E_{ij})\right)+\sum_{1\leqslant
i,j,k\leqslant n}e_{ij}\otimes\rho_{V}(E_{ik})\rho_{V}(E_{kj}).$

Denote $P=\sum_{1\leqslant i,j\leqslant
n}e_{ij}\otimes\rho_{V}(E_{ij})\in\operatorname{End}(\mathbb{C}^{n})\otimes\operatorname{End}(V)$.
Therefore for the condition (6.2) to be satisfied we need

$P^{2}=-(z+\bar{z}+\bar{c})P.$

So the statement of the Proposition reduces to the following lemma

######  Lemma 6.4.

Suppose that $V$ is an irreducible $\mathfrak{gl}_{n}$ representation. Then we
have $P^{2}=aP$ for some $a\in\mathbb{C}$ if and only if the following two
conditions are satisfied:

  1. (1)

The highest weight of $V$ is $l\omega_{r}$ for some $l\in\mathbb{Z}_{>0},1\leq
r\leq n-1$;

  2. (2)

$a=l+n-r$.

###### Proof.

Suppose that $V$ has the highest weight of the form $l\omega_{r}$ for some
$r,l$. Then $P$ acts proportionally to $P^{2}$ on $\mathbb{C}^{n}\otimes V$.
We know that $P$ acts as an endomorphism of $\mathfrak{gl}_{n}$ representation
$\mathbb{C}^{n}\otimes V$. Hence both $P$ and $P^{2}$ are endomorphisms of
this representation. If the highest weight of $V$ equals $l\omega_{r}$ for
some $r,l$, then by Littlewood-Richardson rule the product
$\mathbb{C}^{n}\otimes V$ is isomorphic to the sum of two non-isomorphic
irreducible representations of $\mathfrak{gl}_{n}$. Hence
$\operatorname{End}(\mathbb{C}^{n}\otimes V)$ is two-dimensional and thus
$P^{2}=\alpha P+\beta\mathrm{Id}$. If we compare the action of
$\operatorname{tr}P^{2}$, $\operatorname{tr}P$ and
$\operatorname{tr}\mathrm{Id}$ and then compare the action of coefficient at
$e_{11}$ for $P^{2}$, $P$ and $\mathrm{Id}$, we get that $\beta=0$ and thus
the claim follows.

Suppose that $P$ acts proportionally to $P^{2}$ on $\mathbb{C}^{n}\otimes V$.
Note that the $i$-th diagonal element of $P$ is $E_{ii}$ and that of $P^{2}$
is $\sum_{j=1}^{n}E_{ij}E_{ji}$. Comparing the action of these matrix elements
on the highest vector of $V$ we obtain the condition on the representation and
the proportionality coefficient $l+n-r$.

∎

Comparing the coefficients of $P$ we get

$l+n-r=-(a+\bar{a}+\bar{c}).$

This implies that $a=\frac{-l-n+r-\bar{c}}{2}+iv$, $v\in\mathbb{R}$.

∎

The $Y(\mathfrak{gl}_{n})$-module $V_{l\omega_{r}}(z)$ is called _Kirillov-
Reshetikhin modules_. Its highest weight with respect to $\mathfrak{gl}_{n}$
is the rectangular Young diagram of the size $l\times r$.

###  6.3. Compact real form.

######  Theorem 6.5.

Suppose that all $V_{i}$’s are Kirillov-Reshetikhin modules. Let $l_{i}\times
r_{i}$ be the size of the corresponding Young diagram. Then, for
$(x_{1},\ldots,x_{k})\in\mathbb{R}^{k}$ from a Zariski open subset, subalgebra
$B(X)$ for all $X\in\overline{M_{0,n+2}^{comp}}$ has simple spectrum on
$\bigotimes_{i=1}^{k}V_{i}\left(\frac{l_{i}}{2}-\frac{r_{i}}{2}+ix_{i}\right)$.

###### Proof.

From Lemma 5.6 we know that for all $X\in\overline{M_{0,n+2}}$ there is a
Zariski open subset $I$ of $\mathbb{C}^{k}$ such that $B(X)$ for all
$X\in\overline{M_{0,n+2}}$ has a cyclic vector on
$\bigotimes_{i=1}^{k}V_{i}(z_{i})$. Then $I\cap\mathbb{R}^{k}$ is a Zariski
open subset of $\mathbb{R}^{k}$ such that for all $X\in\overline{M_{0,n+2}}$
the subalgebra $B(X)$ has a cyclic vector on
$\bigotimes_{i=1}^{k}V_{i}\left(\frac{l_{i}}{2}-\frac{r_{i}}{2}+ix_{i}\right)$.
If we restrict ourselves to $X\in\overline{M_{0,n+2}^{comp}}$ then we also
have the semismplicity of action of $B(X)$ from Proposition 6.3 which
concludes the proof. ∎

###  6.4. Split real form.

######  Theorem 6.6.

Let $V_{i},i=1,\ldots,k$ be a set of skew representations of
$Y(\mathfrak{gl}_{n})$. Then, for $(x_{1},\ldots,x_{k})\in\mathbb{R}^{k}$ from
a Zariski open subset, the subalgebra $B(X)$ has simple spectrum on
$\bigotimes_{i=1}^{k}V_{i}(x_{i})$ for all $X\in\overline{M_{0,n+2}^{split}}$.

###### Proof.

The proof is analogous to the proof of Theorem 5.5. There are parameters
$(x_{1},\ldots,x_{n})$ such that $\bigotimes_{i=1}^{k}V_{i}(x_{i})$ is
isomorphic to a skew representation. Thus for such parameters for any
$X\in\overline{M_{0,n+2}^{split}}$ subalgebra $B(X)$ acts as
$\mathcal{A}_{X}\subset U(\mathfrak{gl}_{n+k})^{\mathfrak{gl}_{k}}$, see
Proposition 4.20. But the latter subalgebra acts with simple spectrum, see
Proposition 6.3. It means that we find at least one set of parameters
$(x_{1},\ldots,x_{k})$ such that for all $X\in\overline{M_{0,n+2}^{split}}$
subalgebra $B(X)$ acts with simple spectrum. Using the fact that
$\overline{M_{0,n+2}^{split}}$ is projective variety and hence complete we
obtain the result analogous to Theorem 5.5.

∎

##  7\. Further development and conjectures

The cyclicity property implies that there are only finitely many joint
eigenlines for the Bethe subalgebra $B(X)$ in the given
$Y(\mathfrak{gl}_{n})$-module. Following [HKRW, R18] we can regard these joint
eigenlines at once as a _covering_ over the Deligne-Mumford space
$\overline{M_{0,n+2}}$ (possibly, ramified, as the operators from $B(C)$ can
sometimes have non-trivial Jordan blocks). The Hermitian property implies that
this covering is unramified over the corresponding real form of
$\overline{M_{0,n+2}}$, so is determined by the action of the fundamental
group on some fiber. We expect that this action of the fundamental group has a
combinatorial description in terms of partial Schutzenberger involutions on
Kirillov-Reshetikhin crystals in the compact case an in terms of Bender-Knuth
involutions on Gelfand-Tsetlin polytopes in the split case. Let us give a more
precise description of the combinatorial structures arising in these two
cases.

One defines the fundamental groupoid of
$\overline{M_{0,n+2}^{split}}=\overline{M_{0,n+2}}(\mathbb{R})$ as follows.
The objects are the components of the open stratum which are given by fixing
the increasing order of the $z_{i}$’s. The mophisms are the homotopy classes
of paths which connect some inner points of the components of the open stratum
and cross the strata of codimension $1$ transversely. This groupoid is in fact
the orbifold fundamental group of $\overline{M_{0,n+2}}(\mathbb{R})/S_{n+1}$
usually denoted by $J_{n+1}$. Clearly, the group $J_{n+1}$ is generated by the
homotopy classes of paths connecting _neighboring_ open cells (i.e. the open
cells having common face of codimension $1$). Thus there are the following
generators of $J_{n+1}$.

For positive integers $p\leq q$, denote by $[p,q]$ the set
$\\{p,p+1,\ldots,q-1,q\\}$. Let $\overline{s_{p,q}}\in S_{n+1}$ be the
involution reversing the segment $[p,q]\subset[0,n]$. Denote by $s_{p,q}$ the
element of $J_{n}$ corresponding to the shortest path connecting the cells
which differ by $\overline{s_{p,q}}$ (they have a common face). Then the
elements $s_{p,q}$ with $0\leq p<q\leq n$ generate $J_{n+1}$ and the defining
relations are

$\begin{array}[]{l}s_{p,q}^{2}=e;\\\
s_{p_{1},q_{1}}s_{p_{2},q_{2}}=s_{p_{2},q_{2}}s_{p_{1},q_{1}}\ \text{if}\
q_{1}<p_{2};\\\
s_{p_{1},q_{1}}s_{p_{2},q_{2}}s_{p_{1},q_{1}}=s_{p_{1}+q_{1}-q_{2},p_{1}+q_{1}-p_{2}}\
\text{if}\ p_{1}\leq p_{2}<q_{2}\leq q_{1}.\end{array}$

We refer the reader to [HK] for more details.

The fundamental group $PJ_{n+1}:=\pi_{1}(\overline{M_{0,n+1}(\mathbb{R})})$ is
the kernel of the natural homomorphism $J_{n+1}\to S_{n+1}$ which maps
$s_{p,q}$ to $\overline{s_{p,q}}$. By the analogy with braid groups,
$PJ_{n+1}$ is called the _pure cactus group_.

Similarly, one can define the fundamental groupoid for
$\overline{M_{0,n+2}^{comp}}$ whose generators correspond to flipping a
segment in the cyclic ordering of the marked points $z_{1},\ldots,z_{n}$ and
the relations are similar. It is natural to call this _affine_ cactus group.

###  7.1. Compact case.

According to Theorem 6.5 for any collection of Kirillov-Reshetikhin modules
$V_{i}$ there is a Zariski open dense subset $U\subset\mathbb{C}^{N}$ such
that for any $(z_{1},\ldots,z_{N})\in U$ the algebra $B(X)$ acts with simple
spectrum on the tensor product $V_{1}(z_{1})\otimes\ldots\otimes V_{N}(z_{N})$
for all $X\in\overline{M_{0,n+2}^{comp}}$. In [KMR] we define the structure of
a Kirillov-Reshetikhin crystal on this spectrum following the strategy of
[HKRW]. One can write the generators of the fundamental groupoid of
$\overline{M_{0,n+2}^{comp}}$ as the paths connecting two neighboring
components of the open stratum which cross transversely the codimension $1$
stratum at a generic point. Such generators correspond to the longest elements
of Weyl groups for all connected Dynkin subdiagrams in the affine Dynkin
diagram.

We expect that the following is true:

######  Conjecture 7.1.

The above generators of the fundamental groupoid of
$\overline{M_{0,n+2}^{comp}}$ act on this KR crystal via partial
Schutzenberger involutions assigned to any connected subdiagram of the affine
Dynkin diagram $\tilde{A}_{n+1}$.

###  7.2. Split case.

Let $V$ be a tensor product of skew representations of the Yangian
$Y(\mathfrak{gl}_{n})$ with sufficiently distant real parameters, as in
Theorem 6.6. Then according to Theorem 6.6 the algebra $B(X)$ acts on $V$ with
simple spectrum for all $X\in\overline{M_{0,n+2}(\mathbb{R})}$. This gives an
unramified covering $\mathcal{E}_{V}$ over $\overline{M_{0,n+2}(\mathbb{R})}$
whose fiber over $X\in\overline{M_{0,n+2}(\mathbb{R})}$ is the set of
eigenlines for $B(X)$ on $V$.

######  Remark.

_This agrees with the results of Nazarov and Tarasov[NT] stating that the
Gelfand-Tsetlin subalgebra (or the Cartan subalgebra of the Yangian) generated
by the elements $A_{i}(u)$ in Drinfeld’s new realization act with simple
spectrum on the same class of representations. Namely, the Gelfand-Tsetlin
subalgebra $H$ is $B(X_{0})$ for $X_{0}$ being the caterpillar point of
$\overline{M_{0,n+2}(\mathbb{R})}$._

The eigenbasis for $H=B(X_{0})$ in a skew representation of
$Y(\mathfrak{gl}_{n})$ is indexed by keystone Gelfand-Tsetlin patterns (or,
equivalently, by semistandard skew Young tableaux). So it is convenient to
choose $X_{0}$ as the base point and regard the monodromy of the eigenlines as
combinatorial transformations of the Gelfand-Tsetlin patterns. Moreover, we
can assign to any connected component of the open stratum of
$\overline{M_{0,n+2}^{split}}$ a caterpillar point in its closure, the only
one with the given ordering of marked points. We expect that the following is
true:

######  Conjecture 7.2.

The paths along $1$-dimensional strata connecting the caterpillar points
generate the fundamental group $J_{n+1}$ and the action of these generators on
the set of semistandard skew tableau is given by Bender-Knuth involutions (see
[BK] for definitions).

In [CGP] Chmutov, Glick and Pylyavskyy show that Bender-Knuth involutions
generate some quotient of the cactus group. We hope that the above conjecture
gives the geometric explanation of this fact, which is just rewriting the
cactus groupoid in dual terms (of vertices and edges instead of the open cells
and codimension one faces).

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Aleksei Ilin
National Research University Higher School of Economics,
Russian Federation,
Department of Mathematics, 6 Usacheva st, Moscow 119048;
<EMAIL_ADDRESS>

Inna Mashanova-Golikova
National Research University Higher School of Economics,
Russian Federation,
Department of Mathematics, 6 Usacheva st, Moscow 119048
<EMAIL_ADDRESS>

Leonid Rybnikov
National Research University Higher School of Economics,
Russian Federation,
Department of Mathematics, 6 Usacheva st, Moscow 119048;
Institute for Information Transmission Problems of RAS;
<EMAIL_ADDRESS>