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abstract: 'We prove a Chevalley restriction theorem and its double analogue for the cyclic quiver.'
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The aim of this paper is to prove a Chevalley restriction theorem and its double analogue for the cyclic quiver. When the quiver is of type $\widehat{A}_0$, we recover the results for ${{\mathfrak{gl}}}_n$. The proof of our Chevalley restriction theorem is similar to the proof for ${{\mathfrak{gl}}}_n$; however, the proof of the double analogue uses a theorem of Crawley-Boevey on decomposition of quiver varieties. The double analogue is the limiting case of an isomorphism between a Calogero-Moser space and the center of a symplectic reflection algebra proved by Etingof and Ginzburg. It is also the associated graded version of a conjectural Harish-Chandra isomorphism for the cyclic quiver.

We now introduce our notations. Let $Q$ be the cyclic quiver with $m$ vertices. Let $\delta = (1, \ldots, 1)$ be the minimal positive imaginary root. Let ${{\mathcal{R}}}_{n} = \mathrm{Rep}(Q, n\delta)$ be the space of representations of $Q$ with dimension vector $n\delta$. Thus, $${{\mathcal{R}}}_{n} = \underbrace{{{\mathfrak{gl}}}_{n} \times \cdots \times {{\mathfrak{gl}}}_{n}}_{m}.$$ Next, let ${{\mathfrak{h}}}$ be the subspace of diagonal matrices in ${{\mathfrak{gl}}}_{n}$, and let $${{\mathcal{L}}}_n = \{ (z, \ldots, z)\in {{\mathcal{R}}}_{n}\ |\ z \in {{\mathfrak{h}}}\}.$$ Note that ${{\mathcal{L}}}_n$ is a $n$ dimensional subspace of ${{\mathcal{R}}}_{n}$. Let $$G_n = \underbrace{GL_{n} \times \cdots \times GL_{n}}_{m}.$$ An element $(g_{1}, \ldots, g_{m}) \in G_n$ acts on an element $(x_{1}, \ldots, x_{m}) \in {{\mathcal{R}}}_{n}$, giving $$(g_{2}^{-1}x_{1}g_{1},\ g_{3}^{-1}x_{2}g_{2},\ \ldots,
\ g_{1}^{-1}x_{m}g_{m}).$$ Let $\s_n$ be the symmetric group on $n$ letters, which we will also regard as the subgroup of permutation matrices in $GL_n$. Finally, let $$W_n = \s_{n} \ltimes (\mathbb{Z}/m\mathbb{Z})^{n}.$$ We have $W_n \hookrightarrow G_n$ via $$\begin{aligned}
(\sigma , \zeta_{1}, & \ldots, \zeta_{n}) \mapsto \\
& ( \sigma\cdot {{\mathrm{diag}}}(1, \ldots, 1), \sigma\cdot {{\mathrm{diag}}}(\zeta_{1}, 
\ldots, \zeta_{n}), \cdots, \sigma\cdot {{\mathrm{diag}}}(\zeta_{1}^{m-1},
\ldots, \zeta_{n}^{m-1})),\end{aligned}$$ where ${{\mathrm{diag}}}(\ldots)$ denotes the diagonal matrix with the indicated entries. Hence, $W_n$ acts on ${{\mathcal{R}}}_n$. Observe that the action of $W_n$ on ${{\mathcal{L}}}_n$ is stable. We remark that $W_n$ is the complex reflection group of type $G(m,1,n)$ and ${{\mathcal{L}}}_n$ is its reflection representation.

\[cr\] Restriction of functions from ${{\mathcal{R}}}_{n}$ to ${{\mathcal{L}}}_n$ gives an isomorphism $$\rho: \mathbb{C}[{{\mathcal{R}}}_{n}]^{G_n}
\stackrel{\sim}{\longrightarrow} \mathbb{C}[{{\mathcal{L}}}_n]^{W_n}.$$

[*Surjectivity of $\rho$*]{}: Write an element in ${{\mathcal{R}}}_{n}$ as $(x_{1}, \ldots, x_{m})$ and an element in ${{\mathcal{L}}}_n$ as $$({{\mathrm{diag}}}(z_{1}, \ldots, z_{n}), \ldots).$$ Note that $\mathbb{C}[{{\mathcal{L}}}_n]^{W_n}$ is a polynomial algebra generated by the elementary symmetric polynomials in $z_{1}^{m}, \ldots, z_{n}^{m}$. The homomorphism $\rho$ takes the coefficients of the characteristic polynomial of $x_{m}x_{m-1}\cdots x_{1}$ to the elementary symmetric polynomials in $z_{1}^{m}, \ldots, z_{n}^{m}$. This proved that $\rho$ is surjective.

[*Injectivity in the $n=1$ case*]{}: Call an element $(x_{1}, \ldots, x_{m}) \in
{{\mathcal{R}}}_{1}$ generic if $x_{1}\cdots x_{m} \neq 0$. Observe that the set of generic elements are Zariski open dense in ${{\mathcal{R}}}_{1}$. Moreover, it is easy to see that in this case, two generic elements $(x_{1}, \ldots, x_{m})$ and $(x'_{1}, \ldots, x'_{m})$ are in the same $G_1$-orbit iff $x_{1}\cdots x_{m} = x'_{1}\cdots x'_{m}$. In particular, ${{\mathcal{L}}}_1$ intersects every generic orbit. Hence, $\rho$ is injective.

[*Injectivity in the general case*]{}: Call an element $(x_{1}, \ldots, x_{m})$ in ${{\mathcal{R}}}_n$ generic if $x_{m}x_{m-1}\cdots x_{1}$ has pairwise distinct nonzero eigenvalues. Denote the subset of generic elements in ${{\mathcal{R}}}_n$ by ${{\mathcal{R}}}'_n$, and let ${{\mathcal{L}}}'_n={{\mathcal{L}}}_n\cap{{\mathcal{R}}}'_n$. Observe that ${{\mathcal{R}}}'_n$ and ${{\mathcal{L}}}'_n$ are, respectively, Zariski open dense in ${{\mathcal{R}}}_n$ and ${{\mathcal{L}}}_n$. Moreover, ${{\mathcal{R}}}'_n$ is $G_n$-stable and ${{\mathcal{L}}}'_n$ is $W_n$-stable. The injectivity of $\rho$ follows from the $n=1$ case and the following claim.

*Claim*: If $(x_{1}, \ldots, x_{m})\in {{\mathcal{R}}}'_n$, then it can be diagonalized, i.e. $G_n$-conjugated to an element in $$\underbrace{R_{1} \times \cdots 
\times R_{1}}_{n} = \underbrace{{{\mathfrak{h}}}\times \cdots \times {{\mathfrak{h}}}}_{m}.$$

[*Proof of Claim*]{}: By our assumption, $x_1, \ldots, x_m$ are invertible matrices. Moreover, there exists an invertible matrix $g$ such that $g^{-1}x_{m}x_{m-1}\cdots x_{1}g$ is diagonal. Then, using $$(g, x_1g, x_2x_1g, \ldots, x_{m-1}\cdots x_1g)\in G_n,$$ we can conjugate $(x_1, \ldots, x_m)$ to $$(1, \ldots, 1, g^{-1}x_{m}x_{m-1}\cdots x_{1}g).$$ This proved the claim, and hence the theorem.

The Jacobian of the morphism ${{\mathcal{L}}}_n \rightarrow {{\mathcal{L}}}_n/W_n$ at a point $$(z, \ldots)=({{\mathrm{diag}}}(z_{1}, \ldots, z_{n}), \ldots) \in {{\mathcal{L}}}_n$$ is, up to a nonzero constant, equal to $$(z_1\cdots z_n)^{m-1}\prod_{i<j}(z_i^m-z_j^m).$$ Thus, ${{\mathcal{L}}}'_n$ is the set of points where the Jacobian is nonzero.

We now proceed to the double analogue of Theorem \[cr\]. Let ${{\mathcal{Z}}}_n$ be the zero set of the moment map of the $G_n$-action on $T^{*}{{\mathcal{R}}}_{n} = {\mathrm{Rep}}(\overline{Q}, n\delta)$, where $\overline{Q}$ is the double quiver of $Q$. Write an element in ${\mathrm{Rep}}(\overline{Q}, n\delta)$ as $$(x_{1}, \ldots, x_{m}, y_{1}, \ldots, y_{m})
\in \underbrace{{{\mathfrak{gl}}}_n\times \cdots \times {{\mathfrak{gl}}}_n}_{2m}.$$ Here, the arrow for $y_{i}$ is opposite to the arrow for $x_{i}$. In explicit terms, ${{\mathcal{Z}}}_n$ is defined by the moment map equations $$y_{1}x_{1} - x_{m}y_{m} = 0,\ y_{2}x_{2} - x_{1}y_{1} = 0,\ \ldots.$$ The action of an element $(g_{1}, \ldots, g_{m})\in G_n$ on ${\mathrm{Rep}}(\overline{Q}, n\delta)$ is given by the formula $$(g_{2}^{-1}x_{1}g_{1},\ \ldots, g_{1}^{-1}x_{m}g_{m}
,\ g_{1}^{-1}y_{1}g_{2},\ \ldots,\ g_{m}^{-1}y_{m}g_{1}).$$ Note that ${{\mathcal{Z}}}_n$ is stable under the $G_n$-action, and ${{\mathcal{L}}}_n \times {{\mathcal{L}}}_n \subset {{\mathcal{Z}}}_n$.

\[da\] Restriction of functions from ${{\mathcal{Z}}}_n$ to ${{\mathcal{L}}}_n \times {{\mathcal{L}}}_n$ gives an isomorphism $$\phi: \mathbb{C}[{{\mathcal{Z}}}_n]^{G_n} \stackrel{\sim}{\longrightarrow}
\mathbb{C}[{{\mathcal{L}}}_n \times {{\mathcal{L}}}_n]^{W_n}.$$

[*Surjectivity of $\phi$*]{}: Write an element of ${{\mathcal{L}}}_n \times {{\mathcal{L}}}_n$ as $$({{\mathrm{diag}}}(z_{1}, \ldots, z_{n}), \ldots, {{\mathrm{diag}}}(z'_{1}, \ldots, z'_{n}),
\ldots).$$ By a result of Weyl [@We], the algebra $\mathbb{C}[{{\mathcal{L}}}_n \times {{\mathcal{L}}}_n]^{W_n}$ is generated by $$z_{1}^{r}{z'_{1}}^{s} + \cdots + z_{n}^{r}{z'_{n}}^{s},$$ where $r, s \geq 0$ and $r-s$ is divisible by $m$. The homomorphism $\phi$ takes $${\mathrm{Tr}}
( \underbrace{y_{1}\cdots y_{m}y_{1}\cdots y_{m}\cdots y_{1}
\cdots y_{j}}_{s}
\underbrace{x_{j}\cdots x_{1}\cdots x_{m}\cdots x_{1}
x_{m}\cdots x_{1}}_{r})$$ to $z_{1}^{r}{z'_{1}}^{s} + \cdots + z_{n}^{r}{z'_{n}}^{s}$. Hence, $\phi$ is surjective.

[*Injectivity in the $n=1$ case*]{}: Suppose $x_{1}\cdots x_{m} = d^{m} \neq 0$ for some $d$. Let $g_{i} = x_{1}\cdots x_{i-1}/d^{i-1}$. Then $g_{i+1}^{-1}x_{i}g_{i} = d$ and $g_{i}^{-1}y_{i}g_{i+1} = x_{i}y_{i}/d$. But $x_{1}y_{1} = x_{2}y_{2} = \cdots x_{m}y_{m}$ by the moment map equations. Hence, the $G_1$-saturation of ${{\mathcal{L}}}_1\times {{\mathcal{L}}}_1$ is Zariski dense in ${{\mathcal{Z}}}_1$. It follows that $\phi$ is injective.

[*Injectivity in the general case*]{}: Observe that $$\underbrace{{{\mathcal{Z}}}_1\times\cdots\times{{\mathcal{Z}}}_1}_{n} 
= \big(\underbrace{{{\mathfrak{h}}}\times\cdots\times{{\mathfrak{h}}}}_{m}\big)\cap{{\mathcal{Z}}}_n
\hookrightarrow {{\mathcal{Z}}}_n.$$ This inclusion induces the map $f$ in the following commutative diagram. $$\begin{CD}
{{\mathbb{C}}}[{{\mathcal{Z}}}_n]^{G_n} @>f>> \big(({{\mathbb{C}}}[{{\mathcal{Z}}}_1]^{G_1})^{\otimes n}\big)^{\s_n} \\
@V{\phi}VV @VV{\psi}V \\
{{\mathbb{C}}}[{{\mathcal{L}}}_n\times {{\mathcal{L}}}_n]^{W_n} @>>> \big(({{\mathbb{C}}}[{{\mathcal{L}}}_1\times 
{{\mathcal{L}}}_1]^{W_1})^{\otimes n}\big)^{\s_n}
\end{CD}$$ In this diagram, the map $f$ is injective by [@CB Theorem 3.4], and the map $\psi$ is injective by the $n=1$ case which we have proved. Hence, $\phi$ must be injective.

When $Q$ is of type $\widehat{A}_0$, i.e. the ${{\mathfrak{gl}}}_n$-case, the injectivity of $\phi$ was due to Gerstenhaber [@Ge], cf. [@Ri].

Let $\Gamma$ be any finite subgroup of $SL_2({{\mathbb{C}}})$ and let ${\mathbf{\Gamma}}_n = \s_n \ltimes \Gamma^n$. In [@EG], Etingof and Ginzburg defined the Calogero-Moser space $\mathcal{M}_{\Gamma,n,c}$ associated to $\Gamma$ and a parameter $c$. They proved that, for generic $c$, there is an isomorphism ${{\mathbb{C}}}[\mathcal{M}_{\Gamma,n,c}]
\stackrel{\sim}{\rightarrow} \mathsf{Z}_{0,c}({\mathbf{\Gamma}}_n)$, where $\mathsf{Z}_{0,c}({\mathbf{\Gamma}}_n)$ is the center of the symplectic reflection algebra associated to the group ${\mathbf{\Gamma}}_n$, see [@EG Theorem 11.16]. The isomorphism $\phi$ in Theorem \[da\] is the $c=0$ case of their isomorphism when $\Gamma=
\mathbb{Z}/m\mathbb{Z}$; it is also the associated graded version of the isomorphism in [@EG Conjecture 11.22] for this $\Gamma$.

[*Acknowledgment*]{}. I thank Victor Ginzburg for suggesting the problem and for helpful discussions.

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.;\
[*E-mail address*]{}: [wlgan@math.mit.edu]{}
