Representations of principal $W$-algebra for the superalgebra $Q(n)$ and the super Yangian $YQ(1)$

TL;DR

This paper classifies irreducible representations of the finite W-algebra for the queer Lie superalgebra Q(n) and uses this to classify simple finite-dimensional representations of the super Yangian YQ(1).

Contribution

It provides the first complete classification of irreducible representations for these algebraic structures, linking W-algebras and super Yangians.

Findings

Classified irreducible representations of finite W-algebra for Q(n).

Derived classification of simple finite-dimensional representations of YQ(1).

Established connections between W-algebras and super Yangians.

Abstract

We classify irreducible representations of finite $W$-algebra of the queer Lie superalgebra $Q(n)$ associated with the principal nilpotent coadjoint orbits. We use this classification and our previous results to obtain a classification of simple finite-dimensional representations of the super Yangian $YQ(1)$.

Equations

$$\phi (f_{i,j})=0,\phi (e_{i,j})={\delta }_{i,j+1}.$$

$$W=\{\pi (y)\in U(\mathfrak{g})/I_{\phi }|\mathrm{ad}(x)y\in I_{\phi }\text{for all}x\in {\mathfrak{n}}^{-}\}.$$

$${\xi }_i:=(-1{)}^{i+1}{f}_{i,i},x_i:={\xi }_i^2=e_{i,i},$$

$$U(\mathfrak{h})\simeq \mathbb{C}[{\xi }_1,\dots ,{\xi }_n]/({\xi }_i{\xi }_j+{\xi }_j{\xi }_i{)}_{i<j\le n}.$$

$${\varphi }_0:=\sum _{i=1}^n{\xi }_i,{\varphi }_k:=T^k({\varphi }_0),$$

$$t_{ij}:=\left\{\begin{array}[]{cc}x_{j}&\text{if}\quad i<j,\\ -x_{j}&\text{if}\quad i>j.\\ \end{array}\right.$$

$$z_k:=[\sum _{i_1<i_2\dots <i_{k+1}}(x_{i_1}+(-1{)}^k{\xi }_{i_1})\cdots (x_{i_k}-{\xi }_{i_k})(x_{i_{k+1}}+{\xi }_{i_{k+1}}){]}_{even},$$

$$z_k:=\frac{1}{2}[{\varphi }_0,{\varphi }_k].$$

$$[{\varphi }_i,{\varphi }_j]=\{\begin{array}{l}(-1{)}^{i}2z_{i+j}\text{if}i+j\text{is even}\\ 0\text{if}i+j\text{is odd}\end{array}$$

$$z_0=x_1+\cdots +x_n,z_2=\frac{1}{3}\left((x_1^3+\cdots +x_n^3)-(x_1+\cdots +x_n{)}^3\right).$$

$${\bar{z}}_k=\sum _{i_1<i_2<\dots <i_{k+1}}{x}_{i_1}\cdots x_{i_{k+1}}.$$

$$\sum _{j=1}^m{b}_j{y}_j=0$$

$$u_k(0):=[\sum _{1\le i_1<i_2<\dots <i_k\le n}(x_{i_1}+(-1{)}^{k+1}{\xi }_{i_1})\cdots (x_{i_{k-1}}-{\xi }_{i_{k-1}})(x_{i_k}+{\xi }_{i_k}){]}_{even},$$

$$u_k(1):=[\sum _{1\le i_1<i_2<\dots <i_k\le n}(x_{i_1}+(-1{)}^{k+1}{\xi }_{i_1})\cdots (x_{i_{k-1}}-{\xi }_{i_{k-1}})(x_{i_k}+{\xi }_{i_k}){]}_{odd}.$$

$$U(\mathfrak{h})\simeq U({\mathfrak{h}}_i)\otimes U({\mathfrak{h}}_j).$$

$$u_k^{+}(0):=[\sum _{1\le i_1<i_2<\dots <i_k\le i}(x_{i_1}+(-1{)}^{k+1}{\xi }_{i_1})\cdots (x_{i_{k-1}}-{\xi }_{i_{k-1}})(x_{i_k}+{\xi }_{i_k}){]}_{even},$$

$$u_k^{+}(1):=[\sum _{1\le i_1<i_2<\dots <i_k\le i}(x_{i_1}+(-1{)}^{k+1}{\xi }_{i_1})\cdots (x_{i_{k-1}}-{\xi }_{i_{k-1}})(x_{i_k}+{\xi }_{i_k}){]}_{odd}.$$

$$u_k^{-}(0):=[\sum _{i+1\le i_1<i_2<\dots <i_k\le n}(x_{i_1}+(-1{)}^{k+1}{\xi }_{i_1})\cdots (x_{i_{k-1}}-{\xi }_{i_{k-1}})(x_{i_k}+{\xi }_{i_k}){]}_{even},$$

$$u_k^{-}(1):=[\sum _{i+1\le i_1<i_2<\dots <i_k\le n}(x_{i_1}+(-1{)}^{k+1}{\xi }_{i_1})\cdots (x_{i_{k-1}}-{\xi }_{i_{k-1}})(x_{i_k}+{\xi }_{i_k}){]}_{odd}.$$

$$u_k(d)=\sum _{e+f=d}\sum _{a+b=k}(-1{)}^{eb}{u}_a^{+}(e)u_b^{-}(f).$$

$$B_{\mathbf{s}}({\xi }_i,{\xi }_j)={\delta }_{ij}2s_i.$$

$$V(\mathbf{s})\simeq V(s_1,\dots ,s_p)⊠V(s_{p+1},\dots ,s_n).$$

$${\varphi }_k=\sum _{j\ne i}{y}_{kj}^{\prime }{\xi }_j\mathrm{mod}({\xi }_i).$$

$${\xi }_1\mapsto \left(\begin{array}{cc}0& \sqrt{s_1}\\ \sqrt{s_1}& 0\end{array}\right),{\xi }_2\mapsto \left(\begin{array}{cc}0& \sqrt{s_2}\mathbf{i}\\ -\sqrt{s_2}\mathbf{i}& 0\end{array}\right).$$

$${\varphi }_0={\xi }_1+{\xi }_2,{\varphi }_1=x_2{\xi }_1-x_1{\xi }_2,z_0=x_1+x_2,z_1^{\prime }=x_1{x}_2-{\xi }_1{\xi }_2$$

$${\varphi }_0\mapsto \left(\begin{array}{cc}0& \sqrt{s_1}+\sqrt{s_2}\mathbf{i}\\ \sqrt{s_1}-\sqrt{s_2}\mathbf{i}& 0\end{array}\right),{\varphi }_1\mapsto \sqrt{s_1{s}_2}\left(\begin{array}{cc}0& \sqrt{s_2}-\sqrt{s_1}\mathbf{i}\\ \sqrt{s_2}+\sqrt{s_1}\mathbf{i}& 0\end{array}\right),$$

$$z_0\mapsto (s_1+s_2)Id,z_1^{\prime }\mapsto \left(\begin{array}{cc}{s}_1{s}_2+\sqrt{s_1{s}_2}\mathbf{i}& 0\\ 0& s_1{s}_2-\sqrt{s_1{s}_2}\mathbf{i}\end{array}\right).$$

$$0\to \Pi {\Gamma }_{-s_1^2+s_1}\to V(\mathbf{s})\to {\Gamma }_{-s_1^2-s_1}\to 0,$$

$$D=\left(\begin{array}{cc}\sqrt{s_2}+\sqrt{s_1}\mathbf{i}& 0\\ 0& \sqrt{s_1}+\sqrt{s_2}\mathbf{i}\end{array}\right).$$

$$D{\varphi }_0{D}^{-1}=\left(\begin{array}{cc}0& \sqrt{s_2}+\sqrt{s_1}\mathbf{i}\\ \sqrt{s_2}-\sqrt{s_1}\mathbf{i}& 0\end{array}\right)$$

$$D{\varphi }_1{D}^{-1}=\sqrt{s_1{s}_2}\left(\begin{array}{cc}0& \sqrt{s_1}-\sqrt{s_2}\mathbf{i}\\ \sqrt{s_1}+\sqrt{s_2}\mathbf{i}& 0\end{array}\right).$$

$$0\to \Pi {\Gamma }_{-s_1^2-s_1}\to V({\mathbf{s}}^{\prime })\to {\Gamma }_{-s_1^2+s_1}\to 0.$$

Full text

1. Introduction

In the classical case a finite $W_{e}$-algebra is a quantization of the Slodowy slice to the adjoint orbit of a nilpotent element $e$ of a semisimple Lie algebra ${\mathfrak{g}}$. Finite-dimensional simple $W_{e}$-modules are used for classification of primitive ideals of $U({\mathfrak{g}})$. Losev’s work gives a new point of view on this classification, [8, 9, 10].

In the supercase the theory of the primitive ideals is even more complicated, [3]. It is interesting to generalize Losev’s result to the supercase. One step in this direction is to study representations of finite $W$-algebras for a Lie superalgebra ${\mathfrak{g}}$. In the case when ${\mathfrak{g}}=\mathfrak{gl}(m|n)$ and $e$ is the even principal nilpotent, Brown, Brundan and Goodwin classified irreducible representations of $W_{e}$ and explored the connection with the category $\mathcal{O}$ for ${\mathfrak{g}}$ using coinvariants functor, [1, 2].

In this paper, we study representations of finite $W$-algebras for the Lie superalgebra $Q(n)$ associated with the principal even nilpotent coadjoint orbit. Note that in this case the Cartan subalgebra ${\mathfrak{h}}$ of ${\mathfrak{g}}=Q(n)$ is not abelian and contains a non-trivial odd part. By our previous results ([17]), we realize $W$ as a subalgebra of the universal enveloping algebra $U({\mathfrak{h}})$. One of the main results of the paper is a classification of simple $W$-modules given in Theorem 4.7 (they are all finite-dimensional by [17]). The technique we use is completely different from one used in [1, 2] due to the lack of triangular decomposition of $W$ in our case. Instead, we can describe the restriction of simple $U({\mathfrak{h}})$-modules to $W$ and prove that any simple $W$-module occurs as a constituent of this restriction.

Next we proceed to classification of simple finite-dimensional modules over the super Yangian $YQ(1)$ associated with the Lie superalgebra $Q(1)$. The Yangians of type $Q$ were introduced by Nazarov in [13] and [14]. In [15] these super Yangians were realized as limits of certain centralizers in the universal enveloping algebras of type $Q$. We have shown previously in [17] that a principal $W$-algebra (for any $n$) is a quotient of $YQ(1)$. Hence a simple module over a $W$-algebra can be lifted to a simple $YQ(1)$-module. However, not every simple $YQ(1)$-module can be obtained in this way. We prove that any simple finite-dimensional $YQ(1)$-module is isomorphic to the tensor product of a module lifted from a $W$-algebra and some one-dimensional module (Theorem 5.14). We also obtain a formula for a generating function for the central character of a simple module. This generating function is rational and probably should be considered as an analogue of the Drinfeld polynomial, see [11] chapters 3, 4.

We also plan in a subsequent paper to study the coinvariants functor from the category $\mathcal{O}$ for $Q(n)$ to the category of $W$-modules.

As M. L. Nazarov pointed to us, it is also interesting to generalize the results of [7] to the case of $YQ(1)$ using the centralizer construction of $YQ(n)$ given in [15].

2. Notations and preliminary results

We work in the category of super vector spaces over $\mathbb{C}$. All tensor products are over $\mathbb{C}$ unless specified otherwise. By $\Pi$ we denote the functor of parity switch $\Pi(X)=X\otimes\mathbb{C}^{0|1}$.

Recall that if $X$ is a simple finite-dimensional $\mathcal{A}$-module for some associative superalgebra $\mathcal{A}$, then $\operatorname{End}_{\mathcal{A}}(X)=\mathbb{C}$ or $\operatorname{End}_{\mathcal{A}}(X)=\mathbb{C}[\epsilon]/(\epsilon^{2}-1)$, where the odd element $\epsilon$ provides an $\mathcal{A}$ isomorphism $X\rightarrow\Pi(X)$. We say that $X$ is of M-type in the former case and of Q-type in the latter (see [6, 4]).

If $X$ and $Y$ are two simple modules over associative superalgebras ${\mathcal{A}}$ and ${\mathcal{B}}$, we define the $\mathcal{A}\otimes\mathcal{B}$-module $X\boxtimes Y$ as the usual tensor product if at least one of $X$, $Y$ is of M-type and the tensor product over $\mathbb{C}[\epsilon]$ if both $X$ and $Y$ are of Q-type.

In this paper we consider the Lie superalgebra ${\mathfrak{g}}=Q(n)$ defined as follows (see [5]). Equip $\mathbb{C}^{n|n}$ with the odd operator $\zeta$ such that $\zeta^{2}=-\operatorname{Id}$. Then $Q(n)$ is the centralizer of $\zeta$ in the Lie superalgebra $\mathfrak{gl}(n|n)$. It is easy to see that $Q(n)$ consists of matrices of the form $\left(\begin{matrix}A&B\\ B&A\end{matrix}\right)$ where $A,B$ are $n\times n$-matrices. We fix the Cartan subalgebra ${\mathfrak{h}}\subset{\mathfrak{g}}$ to be the set of matrices with diagonal $A$ and $B$. By $\mathfrak{n}^{+}$ (respectively, $\mathfrak{n}^{-}$) we denote the nilpotent subalgebras consisting of matrices with strictly upper triangular (respectively, low triangular) $A$ and $B$. The Lie superalgebra ${\mathfrak{g}}$ has the triangular decomposition ${\mathfrak{g}}=\mathfrak{n}^{-}\oplus{\mathfrak{h}}\oplus\mathfrak{n}^{+}$ and we set ${\mathfrak{b}}=\mathfrak{n}^{+}\oplus{\mathfrak{h}}$.

Denote by $W$ the finite $W$-algebra associated with a principal111There is a unique open orbit in the nilpotent cone of the coadjoint representation, elements of this orbit are called principal. even nilpotent element $\varphi$ in the coadjoint representation of $Q(n)$. Let us recall the definition (see [19]). Let $\{e_{i,j},f_{i,j}\,|\,i,j=1,\dots,n\}$ denote the basis consisting of elementary even and odd matrices. Choose $\varphi\in{\mathfrak{g}}^{*}$ such that

[TABLE]

Let $I_{\varphi}$ be the left ideal in $U({\mathfrak{g}})$ generated by $x-\varphi(x)$ for all $x\in\mathfrak{n}^{-}$. Let $\pi:U({\mathfrak{g}})\rightarrow U({\mathfrak{g}})/I_{\varphi}$ be the natural projection. Then

[TABLE]

Using identification of $U({\mathfrak{g}})/I_{\varphi}$ with the Whittaker module $U({\mathfrak{g}})\otimes_{U({\mathfrak{n}})}\mathbb{C}_{\varphi}\simeq U({\mathfrak{b}})\otimes\mathbb{C}$ we can consider $W$ as a subalgebra of $U({\mathfrak{b}})$. The natural projection $\vartheta:U({\mathfrak{b}})\to U({\mathfrak{h}})$ with the kernel $\mathfrak{n}^{+}U({\mathfrak{b}})$ is called the Harish-Chandra homomorphism. It is proven in [17] that the restriction of $\vartheta$ to $W$ is injective.

The center of $U({\mathfrak{g}})$ is described in [21]. Set

[TABLE]

then

[TABLE]

The center of $U({\mathfrak{h}})$ coincides with $\mathbb{C}[x_{1},\dots,x_{n}]$ and the image of the center of $U({\mathfrak{g}})$ under the Harish-Chandra homomorphism is generated by the polynomials $p_{k}=x^{2k+1}_{1}+\dots+x_{n}^{2k+1}$ for all $k\in\mathbb{N}$. These polynomials are called $Q$-symmetric polynomials.

In [17] we proved that the center $Z$ of $W$ coincides with the image of the center of $U({\mathfrak{g}})$ and hence can be also identified with the ring of $Q$-symmetric polynomials.

3. The structure of $W$-algebra

Using Harish-Chandra homomorphism we realize $W$ as a subalgebra in $U({\mathfrak{h}})$. It is shown in [18] that $W$ has $n$ even generators $z_{0},\dots,z_{n-1}$ and $n$ odd generators $\phi_{0},\dots\phi_{n-1}$ defined as follows. For $k\geq 0$ we set

[TABLE]

where the matrix of $T$ in the standard basis $\xi_{1},\dots,\xi_{n}$ has [math] on the diagonal and

[TABLE]

For odd $k\leq n-1$ we define

[TABLE]

and for even $k\geq 0$ we set

[TABLE]

Let $W_{0}\subset W$ be the subalgebra generated by $z_{0},\dots,z_{n-1}$. By [17] Proposition 6.4, $W_{0}$ is isomorphic to the polynomial algebra $\mathbb{C}[z_{0},\dots,z_{n-1}]$. Furthermore there are the following relations

[TABLE]

Define the $\mathbb{Z}$-grading on $U({\mathfrak{h}})$ by setting the degree of $\xi_{i}$ to be $1$. It induces the filtration on $W$, for every $y\in W$ we denote by $\bar{y}$ the term of the highest degree.

Note that for even $k$, we have $z_{k}=\bar{z}_{k}$. Moreover, $z_{k}$ is in the image under the Harish-Chandra map of the center of the universal enveloping algebra $U(Q(n))$. Therefore by [21] $z_{2p}$ is a $Q$-symmetric polynomial in $\mathbb{C}[x_{1},\dots,x_{n}]$ of degree $2p+1$. For example,

[TABLE]

For odd $k$ the leading term is given by the elementary symmetric polynomial

[TABLE]

Lemma 3.1**.**

sl

1. (1)

$\operatorname{gr}W_{0}$ is isomorphic to the algebra of symmetric polynomials $\mathbb{C}[x_{1},\dots,x_{n}]^{S_{n}}=\mathbb{C}[\bar{z}_{0},\dots,\bar{z}_{n-1}]$ and the degree of $\bar{z}_{k}$ is $2k+2$; 2. (2)

$U({\mathfrak{h}})$ is a free right $W_{0}$-module of rank $2^{n}n!$.

Proof.

Since $\bar{z}_{0},\dots,\bar{z}_{n-1}$ are algebraically independent generators of $\mathbb{C}[x_{1},\dots,x_{n}]^{S_{n}}$ we obtain (1).

It is well-known fact that $\mathbb{C}[x_{1},\dots,x_{n}]$ is a free $\mathbb{C}[x_{1},\dots,x_{n}]^{S_{n}}$-module of rank $n!$, see, for example, [22] Chapter 4. Since $U({\mathfrak{h}})$ is a free $\mathbb{C}[x_{1},\dots,x_{n}]$-module of rank $2^{n}$ we get that $U({\mathfrak{h}})$ is a free $\mathbb{C}[x_{1},\dots,x_{n}]^{S_{n}}$-module of rank $m=2^{n}n!$. Let us choose a homogeneous basis $b_{1},\dots,b_{m}$ of $U({\mathfrak{h}})$ over $\mathbb{C}[x_{1},\dots,x_{n}]^{S_{n}}$. We claim that it is a basis of $U({\mathfrak{h}})$ as a right module over $W_{0}$. Indeed, let us prove first the linear independence. Suppose

[TABLE]

for some $y_{j}\in W_{0}$. Let $k=\max\{{\rm deg}y_{j}+{\rm deg}b_{j}\,|j=1,\dots,m\}$. If $J=\{j\,|\,{\rm deg}y_{j}+{\rm deg}b_{j}=k\}$ we have $\sum_{j\in J}\bar{b}_{j}y_{j}=0$. By above this implies $\bar{y}_{j}=0$ for all $j\in J$ and we obtain all $y_{j}=0$. On the other hand, it follows easily by induction on degree that $U({\mathfrak{h}})=\sum_{j=1}^{m}b_{j}W_{0}$. The proof of (2) is complete. ∎

Consider $U({\mathfrak{h}})$ as a free $U({\mathfrak{h}}_{\bar{0}})$-module and let $W_{1}$ denote the free $U({\mathfrak{h}}_{\bar{0}})$-submodule generated by $\xi_{1},\dots,\xi_{n}$. Then $W_{1}$ is equipped with $U({\mathfrak{h}}_{\bar{0}})$-valued symmetric bilinear form $B(x,y)=[x,y]$.

Lemma 3.2**.**

sl Let $p(x_{1},\dots,x_{n}):=\prod_{i<j}(x_{i}+x_{j})$ and $\Gamma$ denotes the Gram matrix $B(\phi_{i},\phi_{j})$. Then $\operatorname{det}\Gamma=cp^{2}x_{1}\cdots x_{n}$, where $c$ is a non-zero constant.

Proof.

Recall that $\phi_{k}=T^{k}\phi_{0}$. Since the matrix of the form $B$ in the basis $\xi_{1},\dots,\xi_{n}$ is the diagonal matrix $C=\operatorname{diag}(x_{1},\dots,x_{n})$, then $\Gamma=Y^{t}CY$, where $Y$ is the square matrix such that $\phi_{i}=\sum_{j=1}^{n}y_{ij}\xi_{j}$. Hence det $\Gamma=x_{1}\cdots x_{n}\operatorname{det}Y^{2}$. Since $B(\phi_{i},\phi_{j})$ is a symmetric polynomial in $x_{1},\dots,x_{n}$, the determinant of $\Gamma$ is also a symmetric polynomial. The degree of this polynomial is $n^{2}$. Therefore it suffices to prove that $(x_{1}+x_{2})^{2}$ divides $\operatorname{det}\Gamma$, or equivalently $x_{1}+x_{2}$ divides $\operatorname{det}Y$. In other words, we have to show that if $x_{1}=-x_{2}$, then $\phi_{0},\dots,\phi_{n-1}$ are linearly dependent. Indeed, one can easily see from the form of $T$ that the first and the second coordinates of $T^{k}\phi_{0}$ coincide, hence $\phi_{0},T\phi_{0},\dots,T^{n-1}\phi_{0}$ are linearly dependent. ∎

We also will use another generators in $W$ introduced in [18], Corollary 5.15:

[TABLE]

For convenience we assume $u_{k}(0)=u_{k}(1)=0$ for $k>n$.

Let $i+j=n$. We have the natural embedding of the Lie superalgebras $Q(i)\oplus Q(j)\hookrightarrow Q(n)$. If ${\mathfrak{h}}_{r}$ denotes the Cartan subalgebra of $Q(r)$, the above embedding induces the isomorphism

[TABLE]

The following lemma implies that we have also the embedding of $W$-algebras.

Lemma 3.3**.**

sl Let $i+j=n$. Then $W$ is a subalgebra in the tensor product $W^{i}\otimes W^{j}$, where $W^{r}\subset U({\mathfrak{h}}_{r})$ denotes the $W$-algebra for $Q(r)$.

Proof.

Introduce generators in $W^{i}$ and $W^{j}$:

[TABLE]

[TABLE]

Then for $d,e,f\in\mathbb{Z}/2\mathbb{Z}$ we have

[TABLE]

Here we assume $u_{0}^{\pm}(0)=1$ and $u_{0}^{\pm}(1)=0$. ∎

Corollary 3.4**.**

sl If $i_{1}+\dots+i_{p}=n$, then $W$ is a subalgebra in $W^{i_{1}}\otimes\dots\otimes W^{i_{p}}$.

4. Irreducible representations of $W$

4.1. Representations of $U({\mathfrak{h}})$

Let $\mathbf{s}=(s_{1},\dots,s_{n})\in\mathbb{C}^{n}$. We call $\mathbf{s}$ regular if $s_{i}\neq 0$ for all $i\leq n$ and typical if $s_{i}+s_{j}\neq 0$ for all $i\not=j\leq n$.

It follows from the representation theory of Clifford algebras that all irreducible representations of $U({\mathfrak{h}})$ up to change of parity can be parameterized by $\mathbf{s}\in\mathbb{C}^{n}$. Indeed, let $M$ be an irreducible representation of $U({\mathfrak{h}})$. By Schur’s lemma every $x_{i}$ acts on $M$ as a scalar operator $s_{i}\operatorname{Id}$. Let $I_{\mathbf{s}}$ denote the ideal in $U({\mathfrak{h}})$ generated by $x_{i}-s_{i}$, then the quotient algebra $U({\mathfrak{h}})/I_{\mathbf{s}}$ is isomorphic to the Clifford superalgebra $C_{\mathbf{s}}$ 222We consider Clifford algebras as superalgebras with the natural $\mathbb{Z}_{2}$-grading associated with the quadratic form:

[TABLE]

Then $M$ is a simple $C_{\mathbf{s}}$-module.

The radical $R_{\mathbf{s}}$ of $C_{\mathbf{s}}$ is generated by the kernel of the form $B_{\mathbf{s}}$. Let $m({\mathbf{s}})$ be the number of non-zero coordinates of ${\mathbf{s}}$, then $C_{\mathbf{s}}/R_{\mathbf{s}}$ is isomorphic to the matrix superalgebra $M(2^{\frac{m}{2}-1}|2^{\frac{m}{2}-1})$ for even $m$ and to the superalgebra $M(2^{\frac{m-1}{2}})\otimes\mathbb{C}[\epsilon]/(\epsilon^{2}-1)$ for odd $m$.

Therefore $C_{\mathbf{s}}$ has one (up to isomorphism) simple ${\mathbb{Z}}_{2}$-graded module $V(\mathbf{s})$ of type Q for odd $m({\mathbf{s}})$, and two simple modules $V(\mathbf{s})$ and $\Pi V(\mathbf{s})$ of type M for even $m({\mathbf{s}})$ (see [12]). In the case when $\mathbf{s}$ is regular, the form $B_{\mathbf{s}}$ is non-degenerate and the dimension of $V(\mathbf{s})$ equals $2^{k}$, where $k=\lceil n/2\rceil$. In general, $\dim V(\mathbf{s})=2^{\lceil m(\mathbf{s})/2\rceil}$.

Consider the embedding $Q(p)\oplus Q(q)\hookrightarrow Q(n)$ for $p+q=n$ and the isomorphism (3.7). It induces an isomorphism of $U({\mathfrak{h}})$-modules

[TABLE]

4.2. Restriction from $U({\mathfrak{h}})$ to $W$

We denote by the same symbol $V(\mathbf{s})$ the restriction to $W$ of the $U({\mathfrak{h}})$-module $V(\mathbf{s})$.

Proposition 4.1**.**

sl Let $S$ be a simple $W$-module. Then $S$ is a simple constituent of $V(\mathbf{s})$ for some $\mathbf{s}\in\mathbb{C}^{n}$.

Proof.

Since $W_{0}$ is commutative and $S$ is finite-dimensional (see [17]), there exists one dimensional $W_{0}$-submodule $\mathbb{C}_{\nu}\subset S$ with character $\nu$. Therefore $S$ is a quotient of $\operatorname{Ind}^{W}_{W_{0}}\mathbb{C}_{\nu}$. On the other hand, the embedding $W\hookrightarrow U({\mathfrak{h}})$ induces the embedding $\operatorname{Ind}^{W}_{W_{0}}\mathbb{C}_{\nu}\hookrightarrow\operatorname{Ind}^{U({\mathfrak{h}})}_{W_{0}}\mathbb{C}_{\nu}$. Thus, $S$ is a simple constituent of $\operatorname{Res}_{W}\operatorname{Ind}^{U({\mathfrak{h}})}_{W_{0}}\mathbb{C}_{\nu}$. By Lemma 3.1, $\operatorname{Ind}^{U({\mathfrak{h}})}_{W_{0}}\mathbb{C}_{\nu}$ is finite-dimensional, and hence has simple $U({\mathfrak{h}})$-constituents isomorphic to $V(\mathbf{s})$ for some $\mathbf{s}$. Hence $S$ must appear as a simple $W$-constituent of some $V(\mathbf{s})$. ∎

4.3. Typical representations

Theorem 4.2**.**

sl If $\mathbf{s}$ is typical, then $V(\mathbf{s})$ is a simple $W$-module.

Proof.

First, we assume that $\mathbf{s}$ is regular, i.e. $s_{i}\neq 0$ for all $i=1,\dots,n$. The specialization $x_{i}\mapsto s_{i}$ induces an injective homomorphism $\theta_{\mathbf{s}}:W/(I_{\mathbf{s}}\cap W)\hookrightarrow C_{\mathbf{s}}$ and a specialization of the quadratic form $B\mapsto B_{\mathbf{s}}$. By Lemma 3.2 $\operatorname{det}\Gamma(\mathbf{s})\neq 0$. Therefore $B_{\mathbf{s}}$ is non-degenerate and $\theta_{\mathbf{s}}$ is an isomorphism. Thus, $V(\mathbf{s})$ remains irreducible when restricted to $W$.

If $\mathbf{s}$ is typical non-regular, there is exactly one $i$ such that $s_{i}=0$. Let $\mathbf{s}^{\prime}=(s_{1},\dots,s_{i-1},s_{i+1},\dots s_{n})$. Note that $(\theta_{\mathbf{s}}(\xi_{i}))$ is a nilpotent ideal of $C_{\mathbf{s}}$ and hence $\xi_{i}$ acts by zero on $V(\mathbf{s})$. Then $V(\mathbf{s})$ is a simple module over the quotient $C_{\mathbf{s}^{\prime}}\simeq C_{\mathbf{s}}/(\theta_{\mathbf{s}}(\xi_{i}))$. Recall $Y$ from the proof of Lemma 3.2 and let $Y^{\prime}$ denote the minor of $Y$ obtained by removing the $i$-th column and the $i$-th row. Then

[TABLE]

Hence $\theta_{\mathbf{s}}(\phi_{0}),\dots,\theta_{\mathbf{s}}(\phi_{n-1})$ generate $C_{\mathbf{s}^{\prime}}\simeq C_{\mathbf{s}}/(\theta_{\mathbf{s}}(\xi_{i}))$ and the statement follows from the regular case for $n-1$. ∎

4.4. Simple $W$-modules for $n=2$

Let $n=2$, then by Theorem 4.2 $V(\mathbf{s})$ is simple as $W$-module if $s_{1}\neq-s_{2}$. The action of $U({\mathfrak{h}})$ in $V(s_{1},s_{2})$ is given by the following formulas in a suitable basis:

[TABLE]

Note that $W$ is generated by $\phi_{0},\phi_{1},z_{0}$ and $z^{\prime}_{1}$, where $z^{\prime}_{1}:=u_{2}(0)$. Using

[TABLE]

we obtain the following formulas for the generators of $W$:

[TABLE]

[TABLE]

Assume that $s_{1}=-s_{2}$. If $s_{1},s_{2}=0$ then $V(\mathbf{s})$ is isomorphic to $\mathbb{C}\oplus\Pi\mathbb{C}$, where $\mathbb{C}$ is the trivial module. If $s_{1}\neq 0$, we choose $\sqrt{s_{1}},\sqrt{s_{2}}$ so that $\sqrt{s_{2}}=\sqrt{s_{1}}\mathbf{i}$. Note that the choice of sign controls the choice of the parity of $V(\mathbf{s})$. The following exact sequence easily follows from (4.2) and (4.3):

[TABLE]

where $\Gamma_{t}$ is the simple module of dimension $(1|0)$ on which $\phi_{0},\phi_{1}$ and $z_{0}$ act by zero and $z^{\prime}_{1}$ acts by the scalar $t$. The sequence splits only in the case $s_{1}=0$, when $\Gamma_{0}\simeq\mathbb{C}$ is trivial. Thus, using Proposition 4.1, Theorem 4.2, and (4.4) we obtain

Lemma 4.3**.**

sl If $n=2$, then every simple $W$-module is isomorphic to one of the following

1. (1)

$V(s_{1},s_{2})$ or $\Pi V(s_{1},s_{2})$ for $s_{1}\neq-s_{2}$, $s_{1},s_{2}\neq 0$; 2. (2)

$V(s,0)$ if $s\neq 0$; 3. (3)

$\Gamma_{t}$ or $\Pi\Gamma_{t}$.

4.5. Invariance under permutations

Theorem 4.4**.**

sl Let $\mathbf{s}^{\prime}=\sigma(\mathbf{s})$ for some permutation of coordinates.

(1) If $\mathbf{s}$ is typical, then $V(\mathbf{s})$ is isomorphic to $V(\mathbf{s}^{\prime})$ as a $W$-module.

(2) If $\mathbf{s}$ is arbitrary, then $[V(\mathbf{s})]=[V(\mathbf{s}^{\prime})]$ or $[\Pi V(\mathbf{s}^{\prime})]$, where $[X]$ denotes the class of $X$ in the Grothendieck group.

Proof.

First, we will prove the statement for $n=2$. Assume first that $s_{2}\neq-s_{1}$. In this case $V(s_{1},s_{2})$ is a $(1|1)$-dimensional simple $W$-module.

Let

[TABLE]

Then by direct computation we have

[TABLE]

and

[TABLE]

Therefore $D$ defines an isomorphism between $V(s_{1},s_{2})$ and $V(s_{2},s_{1})$.

Now consider the case $s_{1}=-s_{2}$. Then the structure of $V(s_{1},-s_{1})$ is given by the sequence (4.4). Let $V(\mathbf{s}^{\prime})=V(-s_{1},s_{1})$, then analogously we have the exact sequence

[TABLE]

The statement (2) now follows directly from comparison of (4.4) and (4.5). Now we will prove the statement for all $n$. Note that it suffices to consider the case of the adjacent transposition $\sigma=(i,i+1)$.

The embedding of $Q(i-1)\oplus Q(2)\oplus Q(n-i-1)$ into $Q(n)$ provides the isomorphism

[TABLE]

where ${\mathfrak{h}}^{-}$, ${\mathfrak{h}}^{0}$ and ${\mathfrak{h}}^{+}$ are the Cartan subalgebras of $Q(i-1),Q(2)$ and $Q(n-i-1)$ respectively. Using twice the isomorphism (4.1) we obtain the following isomorphism of $U({\mathfrak{h}})$-modules

[TABLE]

Suppose that $s_{i}\not=-s_{i+1}$. Let $D_{i,i+1}=1\otimes D\otimes 1$. By Corollary 3.4 we have that $W$ is a subalgebra in $W^{i-1}\otimes W^{2}\otimes W^{n-i-1}$ and hence $D_{i,i+1}$ defines an isomorphism of $W$-modules $V(\mathbf{s})$ and $V(\mathbf{s}^{\prime})$.

If $s_{i}=-s_{i+1}$, then the statement follows from (4.4) and (4.5). This completes the proof of the theorem.

∎

4.6. Construction of simple $W$-modules

Now we give a general construction of a simple $W$-module. Let $r,p,q\in\mathbb{N}$ and $r+2p+q=n$, $\mathbf{t}=(t_{1},\dots,t_{p})\in\mathbb{C}^{p}$, $t_{1},\dots,t_{p}\neq 0$, and $\lambda=(\lambda_{1},\dots,\lambda_{q})\in\mathbb{C}^{q}$, $\lambda_{1},\dots,\lambda_{q}\neq 0$, such that $\lambda_{i}+\lambda_{j}\neq 0$ for any $1\leq i\not=j\leq q$. Recall that by Corollary 3.4 we have an embedding $W\hookrightarrow W^{r}\otimes(W^{2})^{\otimes p}\otimes W^{q}$. Set

[TABLE]

where the first term $\mathbb{C}$ in the tensor product denotes the trivial $W^{r}$-module. For $q=0$ we use the notation $S(\mathbf{t},0)$ and set $V(\lambda)=\mathbb{C}$.

Remark 4.5*.*

The dimension of $S(\mathbf{t},\lambda)$ equals $2^{\frac{q}{2}}$ for even $q$ and $2^{\frac{q+1}{2}}$ for odd $q$. Furthermore, $S(\mathbf{t},\lambda)$ is isomorphic to $\Pi S(\mathbf{t},\lambda)$ if and only if $q$ is odd.

Lemma 4.6**.**

sl All $u_{k}(1)$ act by zero on $S(\mathbf{t},0)$. The action of $u_{k}(0)$ is given by the formula

[TABLE]

where $\sigma_{a}$ denote the elementary symmetric polynomials, $0\leq a\leq p$.

Proof.

The first assertion is trivial. We prove the second assertion by induction on $p$. For $p=1$ it is a consequence of the definition of $\Gamma_{t}$ for $Q(2)$. For $p>1$ we consider the embedding $Q(n-2)\oplus Q(2)\hookrightarrow Q(n)$. The formula (3.10) degenerates to

[TABLE]

As $z_{0}$ acts by zero on $\Gamma_{t_{p}}$ the statement now follows from the obvious identity

[TABLE]

∎

Theorem 4.7**.**

sl

1. (1)

$S(\mathbf{t},\lambda)$ is a simple $W$-module; 2. (2)

Every simple $W$-module is isomorphic to $S(\mathbf{t},\lambda)$ up to change of parity.

Proof.

Let $u^{-}_{k}(d)$, $d\in\mathbb{Z}/2\mathbb{Z},\,1\leq k\leq n$ be as in (3.9) where indices are taken in the interval $[n-q+1,n]$. If $q=0$ we set $u^{-}_{k}(0)=1$ and $u^{-}_{k}(1)=0$. Using Lemma 4.6 and formula (3.10) we can easily write the action of $u_{k}(d)$ in $S(\mathbf{t},\lambda)$ in terms of $u^{-}_{k}(d)$ after identifying $S(\mathbf{t},\lambda)$ with $V(\lambda)$:

[TABLE]

From these formulas we see that $u^{-}_{k}(d)$ and $u_{k}(d)$ generate the same subalgebra in $\operatorname{End}_{\mathbb{C}}(V(\lambda))$. By Theorem 4.2 this proves irreducibility of $S(\mathbf{t},\lambda)$.

To show (2) we use Proposition 4.1. Every simple $W$-module is a subquotient of $V(\mathbf{s})$. By Theorem 4.4 (2) we may assume that $s_{1}=\dots=s_{r}=0$, $s_{i}\neq 0$ for $i>r$, $s_{r+1}=-s_{r+2},\dots,s_{r+2p-1}=-s_{r+2p}$. We can compute $W^{r}\otimes(W^{2})^{\otimes p}\otimes W^{q}$-simple constituents of $V(\mathbf{s})$. They are $S(\mathbf{t},\lambda)$ (up to change of parity) with $t_{j}=-s_{r+2j}^{2}\pm s_{r+2j}$ and $\lambda_{i}=s_{r+2p+i}$ (we can assume that all $s_{i}\not=\pm 1$). By (1) $S(\mathbf{t},\lambda)$ remains simple when restricted to $W$. Hence the statement. ∎

Remark 4.8*.*

$\Gamma_{0}\simeq\mathbb{C}\boxtimes\mathbb{C}$ as $W^{2}$-modules ($r=2$, $p=q=0$).

4.7. Central characters

Recall that the center of $U(Q(n))$ coincides with the center $Z$ of $W$, see Section 2. Every $\mathbf{s}$ defines the central character $\chi_{\mathbf{s}}:Z\to\mathbb{C}$. Furthermore, Theorem 4.7 (2) implies that every simple $W$-module admits central character $\chi_{\mathbf{s}}$ for some $\mathbf{s}$. For every $\mathbf{s}=(s_{1},\dots,s_{n})$ we define the core $c(\mathbf{s})=(s_{i_{1}},\dots,s_{i_{m}})$ as a subsequence obtained from $\mathbf{s}$ by removing all $s_{j}=0$ and all pairs $(s_{i},s_{j})$ such that $s_{i}+s_{j}=0$. Up to a permutation this result does not depend on the order of removing. Thus, the core is well defined up to permutation. We call $m$ the length of the core. The notion of core is very useful for describing the blocks in the category of finite-dimensional $Q(n)$-modules, see [16] and [20].

Example 4.9**.**

Let $\mathbf{s}=(1,0,3,-1,-1)$, then $c(\mathbf{s})=(3,-1)$.

The following is a reformulation of the central character description in [21].

Lemma 4.10**.**

sl Let $\mathbf{s},\mathbf{s}^{\prime}\in\mathbb{C}^{n}$. Then $\chi_{\mathbf{s}}=\chi_{\mathbf{s}^{\prime}}$ if and only if $\mathbf{s}$ and $\mathbf{s}^{\prime}$ have the same core (up to permutation).

It follows from Lemma 4.10 that the core depends only on the central character $\chi_{\mathbf{s}}$, we denote it $c(\chi)$. By Theorem 4.4 we obtain the following.

Corollary 4.11**.**

sl Let $\chi:Z\to\mathbb{C}$ be a central character with core $c(\chi)$ of length $m$. Then $W^{m}$-module $V(c(\chi))$ is well-defined. From now on we denote it by $V(\chi)$ and call it the core representation.

The category $W-\mathrm{mod}$ of finite dimensional $W$-modules decomposes into direct sum $\bigoplus W^{\chi}-\mathrm{mod}$, where $W^{\chi}-\mathrm{mod}$ is the full subcategory of modules admitting generalized central character $\chi$.

Lemma 4.12**.**

sl A simple $W$-module $S$ belongs to $W^{\chi}-\mathrm{mod}$ if and only if it is isomorphic (up to change of parity) to $S(\mathbf{t},\lambda)$ with $\lambda=c(\chi)$.

Proof.

We have to compute the central character of $S(\mathbf{t},\lambda)$. For a $Q$-symmetric polynomial $p_{k}=x^{2k+1}_{1}+\dots+x^{2k+1}_{n}$ we have $p_{k}(\mathbf{t},\lambda)=\lambda_{1}^{2k+1}+\dots+\lambda_{q}^{2k+1}$. Since $p_{k}$ generate the center of $W$ the statement follows. ∎

Proposition 4.13**.**

sl Two simple modules $S(\mathbf{t},\lambda)$ and $S(\mathbf{t}^{\prime},\lambda^{\prime})$ are isomorphic (up to change of parity) if and only if $p^{\prime}=p$, $q^{\prime}=q$, $\mathbf{t}^{\prime}=\sigma(\mathbf{t})$ and $\lambda^{\prime}=\tau(\lambda)$ for some $\sigma\in S_{p}$ and $\tau\in S_{q}$.

Proof.

First, (4.6) and Theorem 4.4 imply the “if” statement. To prove the “only if” statement, assume that $S(\mathbf{t},\mathbf{\lambda})$ and $S(\mathbf{t}^{\prime},\lambda^{\prime})$ are isomorphic. Then these modules admit the same central character. Therefore by Lemma 4.12 $\lambda^{\prime}=\tau(\lambda)$ for some $\tau\in S_{q}$. Hence without loss of generality we may assume that $q^{\prime}=q$ and $\lambda^{\prime}=\lambda$.

Denote by $\operatorname{tr}x$ and $\operatorname{tr}^{\prime}x$ the trace of $x\in W$ in $S(\mathbf{t},\lambda)$ and $S(\mathbf{t}^{\prime},\lambda)$ respectively. Then we must have

[TABLE]

Using the formula (4.6) we get

[TABLE]

[TABLE]

Let $b_{j}:=\operatorname{tr}_{V(\lambda)}u_{j}^{-}(0)$. Without loss of generality we may assume that $p\geq p^{\prime}$. Then we can rewrite our formula with $p=p^{\prime}$ assuming $t_{i}^{\prime}=0$ for $p\geq i>p^{\prime}$. Then the above implies

[TABLE]

[TABLE]

where we assume $b_{i}=0$ for $i>q$. Since $b_{0}=\dim V(\lambda)\neq 0$ the above equations imply $\sigma_{a}(t_{1},\dots,t_{p})=\sigma_{a}(t^{\prime}_{1},\dots,t_{p}^{\prime})$ for all $a=1,\dots,p$. Therefore $\mathbf{t}^{\prime}=\sigma(\mathbf{t})$ for some $\sigma\in S_{p}$ and in particular, $p^{\prime}=p$. ∎

We denote by $\mathcal{P}^{l}$ the subcategory of $W^{l}$-modules which admit trivial generalized central character.

Lemma 4.14**.**

slLet $\chi:Z\to\mathbb{C}$ be a central character with core $c(\chi)$ of length $m$. Then the functor $W^{n-m}-\mathrm{mod}\to W-\mathrm{mod}$ defined by $F(M)=\operatorname{Res}_{W}(M\otimes V(\chi))$ 333We consider here the usual exterior tensor product in contrast with $\boxtimes$ restricts to the functor $\Phi:\mathcal{P}^{n-m}\to W^{\chi}-\mathrm{mod}$. Furthermore, $\Phi$ is an exact functor which sends a simple object to a simple object.

Proof.

The first assertion is immediate consequence of Lemma 4.12 and the second follows from the construction of $S(\mathbf{t},\lambda)$. ∎

Conjecture 4.15**.**

The functor $\Phi:\mathcal{P}^{n-m}\to W^{\chi}-\mathrm{mod}$ defines an equivalence of categories.

5. Representations of the super Yangian of type $Q(1)$

Recall that in [13] the Yangians $YQ(n)$ associated with Lie superalgebras $Q(n)$ were defined. In [17] and [18] (Corollary 5.16) we have shown the existence of the surjective homomorphism $\varphi_{n}:YQ(1)\to W^{n}$, where $W^{n}$ is the finite $W$-algebra associated with principal nilpotent orbit in $Q(n)^{*}$. In this section we classify irreducible finite-dimensional representations of $YQ(1)$ and explore its connections with irreducible representations of $W^{n}$.

Recall that $YQ(1)$ is the associative unital superalgebra over ${\mathbb{C}}$ with the countable set of generators

[TABLE]

The ${\mathbb{Z}}_{2}$-grading of the algebra $YQ(1)$ is defined as follows:

[TABLE]

To write down defining relations for these generators we employ the formal series

in $YQ(1)[[u^{-1}]]$:

[TABLE]

Then for all possible indices $i,j,k,l$ we have the relations

[TABLE]

where $v$ is a formal parameter independent of $u$, so that (5.2) is an equality in the algebra of formal Laurent series in $u^{-1},v^{-1}$ with coefficients in $YQ(1)$.

For all indices $i,j$ we also have the relations

[TABLE]

Note that the relations (5.2) and (5.3) are equivalent to the following defining relations:

[TABLE]

[TABLE]

where $m,r=1,\ldots$ and $T_{i,j}^{(0)}=\delta_{ij}$.

Recall that $YQ(1)$ is a Hopf superalgebra, see [14], with comultiplication given by the formula

[TABLE]

There exists a surjective homomorphism $\varphi_{n}:YQ(1)\to W^{n}$ defined as follows:

[TABLE]

[TABLE]

Note that $\varphi_{n}(T_{1,1}^{(k)})=\varphi_{n}(T_{-1,1}^{(k)})=0$ if $k>n$.

Lemma 5.1**.**

sl Let

[TABLE]

1. (1)

The relation (3.5) holds. 2. (2)

The elements $\{Z_{2i}\mid i\in\mathbb{N}\}$ are algebraically independent generators of the center of $YQ(1)$. 3. (3)

The elements $\eta_{0}$ and $\{T_{1,1}^{(2i)}\mid i\in\mathbb{N}\}$ generate $YQ(1)$.

Remark 5.2*.*

We have the following correspondence between generators of $YQ(1)$ and $W^{n}$

[TABLE]

Proof.

It follows from the similar statements for $W^{n}$ for all $n$ and the fact that $\bigcap_{n\in\mathbb{N}}\operatorname{Ker}\varphi_{n}=0$. ∎

It is easy to see the following commutative diagram:

[TABLE]

where the bottom horizontal arrow is the composition of the flip $W^{n}\otimes W^{m}\to W^{m}\otimes W^{n}$ with the map $W^{m+n}\to W^{n}\otimes W^{m}$ defined in Lemma 3.3. The appearance of the flip is due to the fact that the flip is used in the identification of $U(\mathfrak{h})\subset U(Q(l))$ with $U(Q(1))^{\otimes l}$, see the formula before Theorem 5.8 and Theorem 5.14 in [18].

Let $M$ be a simple $YQ(1)$-module. Then $M$ admits the central character $\chi$. We set $\chi_{2k}=\chi(Z_{2k})$ and consider the generating function

[TABLE]

Lemma 5.3**.**

sl Let $M$ be a finite-dimensional simple $YQ(1)$-module admitting central character $\chi$. Then $\chi(u)$ is a rational function of the form $\frac{a_{0}u^{-1}+\dots+a_{p}u^{-2p-1}}{1+c_{1}u^{-2}+\dots+c_{q}u^{-2q}}$.

Proof.

Let $\mathbf{C}\subset YQ(1)$ denote the unital subalgebra generated by $\{\eta_{i}\mid i\in\mathbb{N}\}$. Let $\mathbf{C}_{\chi}$ denote the quotient of $\mathbf{C}$ by the ideal $(\{Z_{2i}-\chi_{2i}\mid i\in\mathbb{N}\})$. Then the relations (3.5) imply that $\mathbf{C}_{\chi}$ is isomorphic to the infinite-dimensional Clifford algebra $\mathbf{Cliff}(V,B_{\chi})$ on the space $V$ with basis $\{\eta_{i}\mid i\in\mathbb{N}\}$ and the symmetric form $B_{\chi}$ defined by the formula

[TABLE]

Note that $M$ by definition restricts to a certain $\mathbf{C}_{\chi}$-module. On the other hand, $\mathbf{Cliff}(V,B_{\chi})$ admits a finite-dimensional representation if and only if $B_{\chi}$ has a finite rank. Look at the infinite symmetric matrix of $B_{\chi}$ in the basis $\{\eta_{i}\}$. Then every column of this matrix is a linear combination of the first $k$ columns for some $k$. The formula (5.8) implies that for some positive integers $q$ and $s$ and the coefficients $c_{1},\dots,c_{q}$ we have a recurrence relation

[TABLE]

This condition is equivalent to the rationality of $\chi(u)$. ∎

Recall the $W^{n}$-module $V(\mathbf{s})$ constructed in Section 4. Using the homomorphism $\varphi_{n}$ we equip $V(\mathbf{s})$ with a $YQ(1)$-module structure. Our next goal is to compute the central character of $V(\mathbf{s})$. For this we need to compute the $\{z_{2i}\}$ in terms of symmetric polynomials. Recall the notations of Section 3. Note that for any $n$ the elements $\{z_{2i}\}$ of the center can be expressed in terms of symmetric polynomials of $x_{1},\dots,x_{n}$ and this expression stabilizes as $n\to\infty$. Thus, $z_{2i}$ is a particular element in the ring of symmetric functions of degree $2i+1$.

Lemma 5.4**.**

sl We have the following expression

[TABLE]

where $\sigma_{p}=\sum_{i_{1}<\dots<i_{p}}x_{i_{1}}\dots x_{i_{p}}$ is the elementary symmetric function.

Proof.

We proved in [17], Lemma 5.5 that for $W^{n}$ the characteristic polynomial $\operatorname{det}(\lambda\operatorname{Id}-T)$ of $T$ equals $\lambda^{n}+\sum_{i=1}^{\lfloor n/2\rfloor}\sigma_{2i}\lambda^{n-2i}$. As

[TABLE]

the Hamilton–Cayley identity implies that for $2k\geq n$ we have

[TABLE]

Since the degree of $z_{2k}$ is $2k+1$ it is a polynomial of $\sigma_{1},\dots,\sigma_{2k+1}$. Therefore it suffices to prove (5.10) for $n=2k+1$. We do it by induction on $k$ using the fact that $z_{2k}(x_{1},\dots,x_{2k+1})$ is $Q$-symmetric. Indeed, we already know that

[TABLE]

therefore from substituting $x_{2k+1}=0$ we get

[TABLE]

It remains to find the coefficient $A$. By $Q$-symmetry

[TABLE]

This leads to the identity

[TABLE]

[TABLE]

Furthermore, by induction assumption we have

[TABLE]

[TABLE]

Hence $A=1$. ∎

Corollary 5.5**.**

sl $YQ(1)$-module $V(\mathbf{s})$ admits central character $\chi$ where

[TABLE]

Corollary 5.6**.**

sl The elements $\{z_{2k},\mid k=0,\dots,\lfloor\frac{n-1}{2}\rfloor\}$ and $\{\sigma_{2k},\mid k=1,\dots,\lfloor\frac{n}{2}\rfloor\}$ form an algebraically independent set of generators in the ring of symmetric polynomials in $n$ variables.

Proposition 5.7**.**

sl For any rational $\chi(u)$ there exist $n$ and $\mathbf{s}$ such that $V({\mathbf{s}})$ admits central character $\chi$.

Proof.

It follows immediately from Corollary 5.5. Indeed, by Lemma 5.3

[TABLE]

Let $n=\max(2p+1,2q)$ and assume that $a_{i}=0$ for $i>p$, $c_{j}=0$ for $j>q$. One can choose $\mathbf{s}=(s_{1},\dots,s_{n})$ so that $\sigma_{2k}(s_{1},\dots,s_{n})=c_{k}$ and $\sigma_{2k+1}(s_{1},\dots,s_{n})=a_{k}$. ∎

Corollary 5.8**.**

sl Any simple finite-dimensional $\mathbf{C}$-module is either trivial or isomorphic to $V(\mathbf{s})$ or $\Pi V(\mathbf{s})$ for some typical regular $\mathbf{s}$.

Proof.

Recall the notations of Section 4. Consider a homomorphism $\mathbf{C}\to C_{\mathbf{s}}$ defined as the composition

[TABLE]

This homomorphism is surjective if $\mathbf{s}$ is typical regular, see Theorem 4.2. For any central character $\chi$ there exists one up to isomorphism and parity change simple $\mathbf{C}_{\chi}$-module. By Proposition 5.7 it must be isomorphic to $V(\mathbf{s})$. ∎

Remark 5.9*.*

If $\mathbf{s}=(s_{1},\dots,s_{n})$ and $\mathbf{s}^{\prime}=(s_{1},\dots,s_{n},s,-s)$ then $V(\mathbf{s})$ and $V(\mathbf{s}^{\prime})$ admit the same central character. We can see it now from the formula

[TABLE]

Lemma 5.10**.**

sl We have the following expression

[TABLE]

Proof.

Note that according to (6.4) and (6.5) from [17]

[TABLE]

Hence $[T_{1,1}^{(2k)},T_{1,-1}^{(1)}]=2T_{1,-1}^{(2k)}$. Note also that

[TABLE]

[TABLE]

Using (6.9) from [17] we have that $[T_{1,1}^{(2)},T_{1,1}^{(2k)}]=0$. Hence

[TABLE]

Next,

[TABLE]

Furthermore,

[TABLE]

[TABLE]

[TABLE]

Thus

[TABLE]

which gives (5.11). ∎

Corollary 5.11**.**

sl Let $\mathbf{A}$ be the commutative subalgebra in $YQ(1)$ generated by $T_{1,1}^{(2k)}$ for $k\geq 0$. Then $YQ(1)=\mathbf{C}\mathbf{A}=\mathbf{A}\mathbf{C}$.

Proof.

We will show that $\mathbf{C}\mathbf{A}\subset\mathbf{A}\mathbf{C}$. (The proof of the opposite inclusion is similar.) Let $D_{i}$ denote the span of $\eta_{j}$ for $j<i$. By Lemma 5.10 for $i\geq 2$ we have that $\eta_{i}T^{(2k)}_{1,1}=T^{(2k)}_{1,1}\eta_{i}$ modulo $D_{i}\mathbf{A}+\mathbf{A}D_{i}$. Therefore, it suffices to show that $\eta_{i}\mathbf{A}\in\mathbf{A}\mathbf{C}$ for $i=0,1$. Furthermore, the relations in the second line of (5.11) imply that it suffices to show that $T_{1,-1}^{(m)}\in\mathbf{C}\mathbf{A}\cap\mathbf{A}\mathbf{C}$. This can be done by induction on $m$. The case $m=1$ is trivial as $\eta_{0}=T_{1,-1}^{(m)}$. For the step of induction if $m$ is even we employ (5.12) and if $m$ is odd (5.13) and the relation

[TABLE]

Finally, since $\mathbf{A}$ and $\mathbf{C}$ generate $YQ(1)$ we get $YQ(1)=\mathbf{C}\mathbf{A}=\mathbf{A}\mathbf{C}$.

∎

Recall that for any Hopf superalgebra $R$ the ideal $(R_{1})$ generated by all odd elements is a Hopf ideal and the quotient $R/(R_{1})$ is a Hopf algebra.

Lemma 5.12**.**

sl The quotient $YQ(1)/(YQ(1)_{1})$ is isomorphic to $\mathbf{A}\simeq\mathbb{C}[T_{1,1}^{(2k)}]_{k>0}$ with comultiplication

[TABLE]

where $T_{1,1}(u^{-2})=\sum T_{1,1}^{(2k)}u^{-2k}$.

Proof.

Since all $\eta_{i}$ generate $(YQ(1)_{1})$, Lemma 5.1 implies $YQ(1)=\mathbf{A}+(YQ(1)_{1})$. Therefore there exists a surjective homomorphism

[TABLE]

To prove that it is injective we need to show that $\mathbf{A}\cap(YQ(1)_{1})=\{0\}$. It suffices to check that for any $y\in\mathbf{A}$ there exists a one-dimensional $YQ(1)$-module $\Gamma$ such that $y\Gamma\neq 0$. Let $y=P(T_{1,1}^{(2)}\dots T_{1,1}^{(2k)})$ for some polynomial $P$ and consider the module $\Gamma=S(\mathbf{t},0)$ as in Lemma 4.6. Then $y$ acts on $\Gamma$ by $P(\sigma_{1}(\mathbf{t}),\dots,\sigma_{k}(\mathbf{t}))$. By a suitable choice of $\mathbf{t}$ we can get $P(\sigma_{1}(\mathbf{t}),\dots,\sigma_{k}(\mathbf{t}))\neq 0$. The comultiplication formula is straightforward as all $T^{(2k+1)}_{1,1}\in(YQ(1)_{1})$. ∎

Let $f(u)=1+\sum_{k>0}f_{2k}u^{-2k}$. We denote by $\Gamma_{f}$ the one-dimensional $\mathbf{A}$-module, where the action of $T_{1,1}(u^{-2})$ is given by the generating function $f(u)$.

Lemma 5.13**.**

sl The isomorphism classes of one-dimensional $YQ(1)$-modules are in bijection with the set $\{\Gamma_{f}\}$. Furthermore, we have the identity $\Gamma_{f}\otimes\Gamma_{g}\simeq\Gamma_{fg}$.

Proof.

Lemma 5.12 reduces the statement to classification of one-dimensional $\mathbf{A}$-modules which is straightforward. ∎

Theorem 5.14**.**

sl Any simple finite-dimensional $YQ(1)$-module is isomorphic to $V(\mathbf{s})\otimes\Gamma_{f}$ or $\Pi V(\mathbf{s})\otimes\Gamma_{f}$ for some regular typical $\mathbf{s}$ and $f(u)=1+\sum_{k>0}f_{2k}u^{-2k}$. Furthermore, $V(\mathbf{s})\otimes\Gamma_{f}$ and $V(\mathbf{s}^{\prime})\otimes\Gamma_{g}$ are isomorphic up to change of parity if and only if $\mathbf{s}^{\prime}$ is obtained from $\mathbf{s}$ by permutation of coordinates and $f(u)=g(u)$.

Proof.

We start with regular typical $\mathbf{s}$ and identify $V(\mathbf{s})$ with $V(\mathbf{s})\otimes\Gamma_{\mathbf{1}}$. Let $\chi$ be the central character of $V(\mathbf{s})$ and consider only simple modules with central character $\chi$. We denote by $YQ(1)^{\chi}$ the quotient of $YQ(1)$ by the ideal generated by $\operatorname{Ker}\chi$. Note that $YQ(1)^{\chi}=\mathbf{C}_{\chi}\mathbf{A}$.

Note that the central characters of $V(\mathbf{s})$ and $V(\mathbf{s})\otimes\Gamma_{f}$ are the same and they are isomorphic as $\mathbf{C}_{\chi}$-modules. For any finite-dimensional $YQ(1)$-module $M$ and $\theta(u)=1+\sum\theta_{i}u^{-2i}$ set $M^{\theta}=\bigcap_{k>0,m>0}\operatorname{Ker}(T_{1,1}^{(2k)}-\theta_{k})^{m}$. Clearly, we have an isomorphism of $\mathbf{A}$-modules

[TABLE]

for some finite set $P(M)$. Furthermore, we have the following obvious relations

[TABLE]

This implies that $P(V(\mathbf{s})\otimes\Gamma_{f})=P(V(\mathbf{s})\otimes\Gamma_{g})$ if and only if $f=g$. Therefore we obtain the second assertion of the theorem.

Consider the natural homomorphism

[TABLE]

Lemma 5.15**.**

sl Let $J_{\chi}={\rm Ker}F_{\chi}$. Then

1. (1)

$J_{\chi}=\mathbf{A}R_{\chi}=R_{\chi}\mathbf{A}$, where $R_{\chi}=\operatorname{Ann}_{\mathbf{C}_{\chi}}V(\mathbf{s})$ is the Jacobson radical of $\mathbf{C}_{\chi}$; 2. (2)

$J_{\chi}$ acts by zero on any simple finite-dimensional $YQ(1)^{\chi}$-module.

Proof.

Let us prove (1). Note that $T_{1,1}^{(2k)}$ acts on $V(\mathbf{s})\otimes\Gamma_{f}$ as $\sum_{i=0}^{k}T_{1,1}^{2i}\otimes T_{1,1}^{2k-2i}$ and $\eta_{0}$ acts as $T_{1,-1}^{(1)}\otimes 1$. Therefore by (5.6) $\eta_{i}$ acts as $\eta_{i}\otimes 1$ for all $i\geq 0$ and hence every $\zeta\in\mathbf{C}$ acts as $\zeta\otimes 1$. This implies $R_{\chi}\subset J_{\chi}$. Assume

[TABLE]

Set $f=1+u^{-2k}$. Then since $X$ annihilates both $V(\mathbf{s})$ and $V(\mathbf{s})\otimes\Gamma_{f}$ and $X$ acts on the latter module as $X\otimes 1+\zeta_{k}\otimes 1$ we obtain that $\zeta_{k}\in R_{\chi}$. Repeating this argument we obtain that all $\zeta_{i}\in R_{\chi}$. Thus, $J_{\chi}=R_{\chi}\mathbf{A}$. The equality $\mathbf{A}R_{\chi}=R_{\chi}\mathbf{A}$ follows from $\mathbf{A}\mathbf{C}_{\chi}=\mathbf{C}_{\chi}\mathbf{A}$ by symmetry.

To prove (2) note that $J_{\chi}=\mathbf{A}R_{\chi}$ annihilates the induced module $YQ(1)^{\chi}\otimes_{\mathbf{C}_{\chi}}V(\mathbf{s})$ and hence any its quotient. On the other hand, up to switch of parity, any simple finite-dimensional $YQ(1)^{\chi}$-module is a quotient of this induced module. Hence the statement. ∎

Corollary 5.16**.**

slLet $M$ be a finite-dimensional simple $YQ(1)^{\chi}$-module. Then $M$ is isomorphic to $V(\mathbf{s})$ or $\Pi V(\mathbf{s})$ for the regular typical $\mathbf{s}$ as a module over $\mathbf{C}_{\chi}$.

Proof.

The algebra $YQ(1)^{\chi}/J_{\chi}$ is a subalgebra in the product of matrix algebras of the size $\dim V(\mathbf{s})$. Hence $\dim M\leq\dim V(\mathbf{s})$.444 By the Amitsur-Levitzki identity and the Jacobson density theorem Since $R_{\chi}$ annihilates $M$, the module $M$ is isomorphic to a direct sum of several copies of $V(\mathbf{s})$ and $\Pi V(\mathbf{s})$ as a module over $\mathbf{C}_{\chi}$. This implies the statement. ∎

Remark 5.17*.*

By Corollary 5.8, $\mathbf{C}_{\chi}/R_{\chi}\simeq C_{\mathbf{s}}$. Furthermore, $J_{\chi}\cap\mathbf{C}_{\chi}=R_{\chi}$.

Denote by $\mathbf{1}$ the function $\theta(u)=1$ and assume that $M$ is a simple finite-dimensional $YQ(1)^{\chi}$-module such that $M_{0}^{\mathbf{1}}\neq 0$. Then $M$ is a quotient of the induced module

[TABLE]

Note that

[TABLE]

but we will see later that the equality takes place.

Lemma 5.18**.**

sl Let M be a simple $YQ(1)^{\chi}$-module such that $M_{0}^{\mathbf{1}}\neq 0$ and $M$ remains simple after restriction to $\mathbf{C}_{\chi}$. Then there exists a quotient $U$ of $I$ with all simple subquotients isomorphic to $M$ and length equal to $\dim M_{0}^{\mathbf{1}}$.

Proof.

Let $U=M\otimes(M_{0}^{\mathbf{1}})^{*}$. It obviously has a filtration with all quotients isomorphic to $M$ and hence it satisfies the desired property. It remains to construct a surjective map $I\to U$. By Frobenius reciprocity we have a canonical isomorphism

[TABLE]

Consider the identity map in ${\rm Hom}_{\mathbf{A}}(\Gamma_{\mathbf{1}},M^{\mathbf{1}}\otimes(M_{0}^{\mathbf{1}})^{*})$ and denote by $\gamma$ the corresponding map in ${\rm Hom}_{YQ(1)}(I,U)$. Let us prove that $\gamma$ is surjective. First, observe that any $y\in\mathbf{C}$ acts on $M\otimes(M_{0}^{\mathbf{1}})^{*}$ as $y\otimes 1$ by the same argument as in the proof of Lemma 5.15. Choose a basis $\{v_{1},\dots,v_{r}\}$ in $M_{0}^{\mathbf{1}}$ and let $\{w_{1},\dots,w_{r}\}$ be the corresponding dual basis in $(M_{0}^{\mathbf{1}})^{*}$. By construction $\sum v_{i}\otimes w_{i}\in{\rm Im}\gamma$. Since $M$ is a simple $\mathbf{C}_{\chi}$-module, by the Jacobson density theorem for every $i=1,\dots r$ there exists $y_{i}\in\mathbf{C}_{\chi}$ such that $y_{i}v_{j}=\delta_{i,j}v_{1}$. This implies $v_{1}\otimes w_{i}\in{\rm Im}\gamma$ for all $i$ and hence $M\otimes w_{i}\in{\rm Im}\gamma$ for all $i$. The surjectivity of $\gamma$ follows immediately. ∎

Now let us prove the first assertion of the theorem. Consider first the case $\mathbf{s}=(s_{1},\dots,s_{n})$ when $n$ is even. Then $\dim V(\mathbf{s})=2^{n/2}$, $V(\mathbf{s})$ is not isomorphic to $\Pi V(\mathbf{s})$ and $\dim(\mathbf{C}_{\chi}/R_{\chi})=2^{n}$. By Lemma 5.18 and (5.14) for every $\theta\in P(V(\mathbf{s}))$ we have

[TABLE]

On the other hand, $\dim I\leq\dim(\mathbf{C}_{\chi}/R_{\chi})$. Hence any simple subquotient of $I$ is isomorphic to $V(\mathbf{s})\otimes\Gamma_{\theta^{-1}}$ or $\Pi V(\mathbf{s})\otimes\Gamma_{\theta^{-1}}$ and $\dim I=\dim(\mathbf{C}_{\chi}/R_{\chi})$. Therefore every simple $YQ(1)^{\chi}$-module $M$ with $\mathbf{1}\in P(M)$ is isomorphic to $V(\mathbf{s})\otimes\Gamma_{\theta^{-1}}$ or $\Pi V(\mathbf{s})\otimes\Gamma_{\theta^{-1}}$. If $f\in P(M)$ then $M$ is isomorphic to $V(\mathbf{s})\otimes\Gamma_{f\theta^{-1}}$ or $\Pi V(\mathbf{s})\otimes\Gamma_{f\theta^{-1}}$. This implies the statement.

Let us consider the case of odd $n$. Then $\dim V(\mathbf{s})=2^{(n+1)/2}$, $V(\mathbf{s})$ is isomorphic to $\Pi V(\mathbf{s})$ and $\dim(\mathbf{C}_{\chi}/R_{\chi})=2^{n}$. By Lemma 5.18 and (5.14) for every $\theta\in P(V(\mathbf{s}))$ we have

[TABLE]

By counting dimensions we again obtain that every simple subquotient of $I$ is isomorphic to $V(\mathbf{s})\otimes\Gamma_{\theta^{-1}}$. The end of the proof is the same as in the previous case. ∎

Let us conclude by stating the relation between $W$-modules and $YQ(1)$-modules.

Proposition 5.19**.**

slThe simple $YQ(1)$-module $V(\mathbf{s})\otimes\Gamma_{f}$ is lifted from some $W^{m+n}$-module if and only if $f\in\mathbb{C}[u^{-2}]$. Moreover, the smallest $m$ is equal to the degree of the polynomial $f$.

Remark 5.20*.*

Note that $m=2p$ is even. Then Theorem 4.7 and the diagram (5.7) imply $S(t_{1},\dots,t_{p},\lambda)\simeq V(\lambda)\otimes\Gamma_{f}$ where

[TABLE]

Proof.

Immediately follows from Theorem 4.7. ∎

Acknowledgments

This work was supported by a grant from the Simons Foundation (#354874, Elena Poletaeva) and the NSF grant (DMS-1701532, Vera Serganova). We would like to thank V. G. Kac and M. L. Nazarov for useful comments.