Lusztig sheaves and integrable highest weight modules in symmetrizable case

Given a symmetrizable generalized Cartan matrix $C=(c_{ij})_{i,j\in I^{\prime}}=DB$, where $D=\mathbf{diag}(\frac{1}{s_{i}})_{i\in I^{\prime}}$ is a diagonal matrix and $B=(b_{ij})_{i,j\in I^{\prime}}$ is a symmetric matrix. It determines a bilinear form $\langle,\rangle:{\mathbb{Z}}[I^{\prime}]\times{\mathbb{Z}}[I^{\prime}]\rightarrow{\mathbb{Z}}$ by $\langle i,j\rangle=c_{ij}$. Then the quantum group $\mathbf{U}$ is the $\mathbb{Q}(v)$-algebra generated by $E_{i},F_{i},K_{\nu}$ for $i\in I^{\prime},\nu\in{\mathbb{Z}}[I^{\prime}]$, subject to the following relations:

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  (a) $K_{0}=1$, and $K_{\nu}K_{\nu^{\prime}}=K_{\nu+\nu^{\prime}}$ for any $\nu,\nu^{\prime}\in{\mathbb{Z}}[I^{\prime}]$;

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  (b) $K_{\nu}E_{i}=v^{\langle\nu,i\rangle}E_{i}K_{\nu}$ for $i\in I^{\prime},\nu\in{\mathbb{Z}}[I^{\prime}]$;

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  (c) $K_{\nu}F_{i}=v^{-\langle\nu,i\rangle}F_{i}K_{\nu}$ for $i\in I^{\prime},\nu\in{\mathbb{Z}}[I^{\prime}]$;

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  (d) $E_{i}F_{j}-F_{j}E_{i}=\delta_{ij}\frac{\tilde{K}_{i}-\tilde{K}_{-i}}{v_{i}-v_{i}^{-1}},$

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  (e) $\sum\limits_{p+q=1-c_{ij}}(-1)^{p}E_{i}^{(p)}E_{j}E_{i}^{(q)}=0$;

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  (f) $\sum\limits_{p+q=1-c_{ij}}(-1)^{p}F_{i}^{(p)}F_{j}F_{i}^{(q)}=0$.

Here we use the notation $v_{i}:=v^{s_{i}}$. For $\nu=\sum\limits_{i\in I^{\prime}}\nu_{i}i\in{\mathbb{Z}}[I^{\prime}]$, we denote $\tilde{K_{\nu}}:=\prod\limits_{i\in I^{\prime}}K_{s_{i}\nu_{i}i}$, and $E_{i}^{(n)}:=E_{i}^{n}/[n]^{!}_{i}$, $F_{i}^{(n)}:=F_{i}^{n}/[n]^{!}_{i}$ are the divided power of $E_{i},F_{i}$, where $[n]_{i}:=\frac{v_{i}^{n}-v_{i}^{-n}}{v_{i}-v_{i}^{-1}}$, $[n]^{!}_{i}:=\Pi_{s=1}^{n}[s]_{i}$.

Let $M$ be a $\mathbf{U}$-module with a decomposition $M=\oplus_{\lambda\in{\mathbb{Z}}[I^{\prime}]}M^{\lambda}$, such that for any $m\in M^{\lambda}$, $K_{\nu}m=v^{\langle\nu,\lambda\rangle}m$. We say $M$ is integrable, if for any $m\in M$, there exists $n_{0}\in{\mathbb{N}}$ such that for any $n>n_{0}$, $E_{i}^{(n)}m=0$ and $F_{i}^{(n)}m=0$. One can see details in [3] and [6].

Given a dominant weight $\lambda$ (i.e. $\langle i,\lambda\rangle\in{\mathbb{N}}$ for any $i\in I^{\prime}$), it is well-known that irreducible integrable highest weight module $\Lambda_{\lambda}$ is defined by

In [7, Chapter 11, 12 and 14], Lusztig consider a finite quiver $Q=(I,H,\Omega)$ with admissible automorphism $a$ for a given Cartan matrix.

A finite quiver $Q=(I,H,\Omega)$ consists of finite sets $I,H$ and $\Omega$, where $I$ is the set of vertices, $H$ is the set of all oriented arrows $s(h)\xrightarrow{h}t(h)$ and $\Omega\subset H$ is a subset such that $H$ is the disjoint union of $\Omega$ and $\bar{\Omega}$. Here $\bar{}:H\rightarrow H$ is the map taking the opposite orientation. We call such $\Omega$ an orientation of $Q$.

For a finite quiver $(I,H,\Omega)$ without loops, an admissible automorphism $a$ consists of two permutations $a:I\rightarrow I$ and $a:H\rightarrow H$ such that
(a) $a(s(h))=s(a(h))$ and $a(t(h))=t(a(h))$ for any $h\in H$;
(b) $s(h)$ and $t(h)\in I$ belong to different $a$-orbits for any $h\in H$.

Given a symmetrizable generalized Cartan matrix $C=(c_{ij})_{i,j\in I^{\prime}}=DB$, there is (not unique in general) a finite quiver $Q=(I,H,\Omega)$ with addmissible automorphism $a$ such that the $a$-orbits of $I$ are bijective to $I^{\prime}$, the order of an $a$-orbit corresponding to $i\in I^{\prime}$ equals to $s_{i}$, and the number of arrows between $a$-orbits corresponding to $i,j\in I^{\prime}$ equals to $c_{i,j}s_{i}=c_{j,i}s_{j}$. For such a quiver with automorphism, Lusztig has considered a full subcategory $\mathcal{Q}=\coprod\limits_{\mathbf{V}}\mathcal{Q}_{\mathbf{V}}$ of perverse sheaves on the moduli space $\coprod\limits_{\mathbf{V}}\mathbf{E}_{\mathbf{V},\Omega}$ of quiver representations. Via the periodic functor $a^{\ast}$, Lusztig has constructed a certain graded additive category $\tilde{\mathcal{Q}}$ from $\mathcal{Q}$, consisting of pairs $(B,\phi)$ of semisimple complexes $B$ and morphisms $\phi:a^{\ast}B\rightarrow B$. He has also defined the induction functor and restriction functors for the category $\tilde{\mathcal{Q}}$. (See details in [7, Chapter 9 and 12] or Section 2.1.) The (generalized) Grothendieck group of $\tilde{\mathcal{Q}}$ has a bialgebra structure, which is canonically isomorphic to the positive (or negative) part $\mathbf{U}^{\pm}$ of $\mathbf{U}$. Moreover, certain objects of $\tilde{\mathcal{Q}}$ provide an integral basis of $\mathbf{U}^{\pm}$, which is called the signed basis by Lusztig.

When the generalized Cartan matrix $C$ is symmetric, or equivalently, the admissible automorphism $a$ is trivial, the simple perverse sheaves in $\mathcal{Q}$ form a basis of $\mathbf{U}^{\pm}$, which is called the canonical basis and shares many remarkable properties.

When the generalized Cartan matrix $C$ is symmetric, the authors of [2] have considered certain localizations of Lusztig’s category $\mathcal{Q}$ to realize the irreducible integrable highest weight module $\Lambda_{\lambda}$.

For a dominant weight $\lambda$, they have considered the moduli space $\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega}$ of a framed quiver and certain thick subcategory $\mathcal{N}_{\mathbf{V}}$ of the $G_{\mathbf{V}}$-equivariant derived category $\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega})$ of constructible $\overline{\mathbb{Q}}_{l}$-sheaves. They have defined functors $\mathcal{E}_{i}$ and $\mathcal{F}_{i},i\in I$ for the Verdier quotient $\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega})/\mathcal{N}_{\mathbf{V}}$ and showed that Lusztig’s sheaves $\mathcal{Q}_{\mathbf{V},\mathbf{W}}$ for the framed quiver are preserved by these functors $\mathcal{E}_{i}$ and $\mathcal{F}_{i}$ (up to isomorphisms in localizations). (See details in [2] or Section 2.2.) The Grothendieck group of $\coprod\limits_{\mathbf{V}}\mathcal{Q}_{\mathbf{V},\mathbf{W}}/\mathcal{N}_{\mathbf{V}}$ becomes a $\mathbf{U}$-module with the action of functors $\mathcal{E}_{i}$ and $\mathcal{F}_{i}$, and is isomorphic to $\Lambda_{\lambda}$ canonically. Moreover, the nonzero simple perverse sheaves in $\mathcal{Q}_{\mathbf{V},\mathbf{W}}/\mathcal{N}_{\mathbf{V}}$ form the canonical basis of $\Lambda_{\lambda}$.

It is natural to ask how to generalize the construction of [2] to symmetrizable cases. More precisely, given a symmetrizable generalized Cartan matrix $C$, our goal is to realize the irreducible highest weight module $\Lambda_{\lambda}$ of $\mathbf{U}$ associated with $C$ via sheaves on moduli spaces of a quiver with automorphism.

In this article, we apply Lusztig’s method of periodic functors to localizations, then we obtain a category $\widetilde{\mathcal{Q}/\mathcal{N}}$. Its Grothendieck group is a $\mathbf{U}$-module, which is canonically isomorphic to $\Lambda_{\lambda}$ of the symmetrizable quantum group associated to $C$. See Theorem 4.6 and 4.10 for details.

Moreover, the signed basis of $\Lambda_{\lambda}$ is naturally provided by those $(B,\phi)$ in $\widetilde{\mathcal{Q}/\mathcal{N}}$ under the canonical isomorphism, which is almost orthogonal with respect to a contravariant geometric pairing. See Proposition 4.8 for details.

As an application, we also deduce a symmetrizable crystal structure on Nakajima’s quiver varieties and Lusztig’s nilpotent varieties of preprojective algebras. See Theorem 4.15 and its corollary for details.

In section 2, we introduce the notations and results of [2]. In section 3, we define our category $\widetilde{\mathcal{Q}/\mathcal{N}}$ and its functors, and prove the commutative relations of these functors. There is a great discrepancy between the proofs of commutative relations in symmetric cases and those in symmetrizable cases. In section 4, we determine the module structure of the Grothendieck group of $\widetilde{\mathcal{Q}/\mathcal{N}}$, construct the signed basis of $\Lambda_{\lambda}$ and deduce the crystal structure on quiver varieties and nilpotent varieties.

Y. Lan is partially supported by the National Natural Science Foundation of China [Grant No. 1288201], Yumeng Wu and J. Xiao are partially supported by Natural Science Foundation of China [Grant No. 12031007]. This paper is a continuation of a collaborative work [2] with Jiepeng Fang, we are very grateful to his detailed discussion.

In this subsection, we introduce the category of Lusztig’s sheaves for quivers. Assume $\mathbf{k}$ is a algebraic closed field with $char(\mathbf{k})=p>0$. Given a quiver $Q=(I,H,\Omega)$ and an $I$-graded $\mathbf{k}$-space $\mathbf{V}=\bigoplus\limits_{i\in I}\mathbf{V}_{i}$ of dimension vector $\nu$, the affine variety $\mathbf{E}_{\mathbf{V},\Omega}$ is defined by

The algebraic group $G_{\mathbf{V}}=\prod\limits_{i\in I}GL(\mathbf{V}_{i})$ acts on $\mathbf{E}_{\mathbf{V},\Omega}$ by composition naturally. We denote the $G_{\mathbf{V}}$-equivariant derived category of constructible $\overline{{\mathbb{Q}}}_{l}$-sheaves on $\mathbf{E}_{\mathbf{V},\Omega}$ by $\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\Omega})$. We denote $[n]$ the $n$-times shift functor and $\mathbf{D}$ the Verdier duality.

Let $\mathcal{S}$ be the set of finite sequences $\boldsymbol{\nu}=(\nu^{1},\nu^{2},\cdots,\nu^{s})$ of dimension vectors such that each $\nu^{l}=a_{l}i_{l}$ for some $a_{l}\in{\mathbb{N}}_{\geqslant 1}$ and $i_{l}\in I$. If $\sum\limits_{1\leqslant l\leqslant s}\nu^{l}=\nu$, we say $\boldsymbol{\nu}$ is a flag type of $\nu$ or $\mathbf{V}$.

For a flag type $\boldsymbol{\nu}$ of $\mathbf{V}$, the flag variety $\mathcal{F}_{\boldsymbol{\nu},\Omega}$ is the smooth variety which consists of pairs $(x,f)$, where $x\in\mathbf{E}_{\mathbf{V},\Omega}$ and $f=(\mathbf{V}=\mathbf{V}^{s}\subseteq\mathbf{V}^{s-1}\subseteq\cdots\subseteq\mathbf{V}^{0}=0)$ is a filtration of $I$-graded space such that $x(\mathbf{V}^{l})\subseteq\mathbf{V}^{l}$ and the dimension vector of $\mathbf{V}^{l-1}/\mathbf{V}^{l}=\nu^{l}$ for any $l$. There is a proper map $\pi_{\boldsymbol{\nu}}:\mathcal{F}_{\boldsymbol{\nu},\Omega}\rightarrow\mathbf{E}_{\mathbf{V},\Omega};(x,f)\mapsto x.$ Hence by the decomposition theorem in [1], the complex $L_{\boldsymbol{\nu}}=(\pi_{\boldsymbol{\nu},\Omega})_{!}\bar{\mathbb{Q}}_{l}[\mbox{\rm dim}\mathcal{F}_{\boldsymbol{\nu},\Omega}]$ is a semisimple complex on $\mathbf{E}_{\mathbf{V},\Omega}$, where $\bar{\mathbb{Q}}_{l}$ is the constant sheaf on $\mathcal{F}_{\boldsymbol{\nu},\Omega}$.

The framed quiver $\tilde{Q}=(I\cup I^{*},\tilde{H},\tilde{\Omega})$ of $Q=(I,H,\Omega)$ is defined by letting $I^{*}$ be a copy of $I$, $\tilde{\Omega}$ be $\Omega\cup\{i\rightarrow i^{*},i\in I\}$, and $\tilde{H}=\tilde{\Omega}\cup\bar{\tilde{\Omega}}$. Given $I$-graded spaces $\mathbf{V}$ and $\mathbf{W}$, the moduli space of the framed quiver is defined by $\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega}:=\bigoplus\limits_{i\in I}\mbox{\rm Hom}(\mathbf{V}_{i},\mathbf{W}_{i})\oplus\mathbf{E}_{\mathbf{V},\Omega}$. The algebraic group $G_{\mathbf{V}}$ acts canonically on $\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega}$ and we can consider the $G_{\mathbf{V}}$-equivariant derived category $\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega})$.

Assume $I=\{i_{1},i_{2},\cdots,i_{t}\}$ and $d_{i_{l}}=\mbox{\rm dim}\mathbf{W}_{i_{l}}$ for each $l$, let $\boldsymbol{d}=(d_{i_{1}}i_{1},d_{i_{2}}i_{2},\cdots d_{i_{t}}i_{t})$ be a flag type of $\mathbf{W}$. For any flag type $\boldsymbol{\nu}=(\nu^{1},\nu^{2},\cdots,\nu^{s})$, the flag type

is a flag type of $\mathbf{V}\oplus\mathbf{W}$. Then by [2, Proposition 3.1] simple objects in ${\mathcal{Q}}_{\mathbf{V},\mathbf{W}}$ are exactly simple direct summands (up to shifts) of those $L_{\boldsymbol{\nu}\boldsymbol{d}}$, where $\boldsymbol{\nu}$ runs over all flag types of $\mathbf{V}$.

In this subsection, we introduce the localization of Lusztig’s sheaves in [2]. Choose an orientation $\Omega_{i}$ such that $i\in I$ is a source in $\Omega_{i}$. Let $\mathbf{E}_{\mathbf{V},\mathbf{W},i}^{0}$ be the open subset of $\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega_{i}}$ defined by

and $\mathbf{E}_{\mathbf{V},\mathbf{W},i}^{\geqslant 1}$ be its complement. Let $\mathcal{N}_{\mathbf{V},i}$ be the thick subcategory of $\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega_{i}})$, consisting of complexes supported on $\mathbf{E}_{\mathbf{V},\mathbf{W},i}^{\geqslant 1}$.

Recall that the Fourier-Deligne transform $\mathcal{F}_{\Omega_{i},\Omega}:\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega_{i}})\rightarrow\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega})$ induces derived equivalence between different orientations. (See details in [2, Section 2.2] or [7, Chapter 10].) We can consider the thick subcategory $\mathcal{F}_{\Omega_{i},\Omega}(\mathcal{N}_{\mathbf{V},i})$ of $\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega})$ for each $i$, and they generate a thick subcategory denoted by ${\mathcal{N}}_{\mathbf{V}}$.

The category $\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega})/{\mathcal{N}}_{\mathbf{V}}$ inherits a perverse $t$-structure from $\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega})$, and the Verdier Duality $\mathbf{D}$ of $\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega})$ also acts on $\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega})/{\mathcal{N}}_{\mathbf{V}}$ and ${\mathcal{Q}}_{\mathbf{V},\mathbf{W}}/{\mathcal{N}}_{\mathbf{V}}$.

If we denote the localization functor by $L:\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega})\rightarrow\mathcal{D}^{b}_{G_{\mathbf{V}}}(\mathbf{E}_{\mathbf{V},\mathbf{W},\Omega})/{\mathcal{N}}_{\mathbf{V}}$, then $L$ restricts to an additive functor $L:{\mathcal{Q}}_{\mathbf{V},\mathbf{W}}\rightarrow{\mathcal{Q}}_{\mathbf{V},\mathbf{W}}/{\mathcal{N}}_{\mathbf{V}}$.

In this subsection, we recall the definition of ${\mathcal{E}}^{(n)}_{i}$ and ${\mathcal{F}}^{(n)}_{i}$ in [2].

In this subsection, we review the definition of the periodic functor and refer [7, Chapter 11] for more details. Let $o$ be a fixed positive integer.

Let ${\mathcal{C}}$ be a $\overline{{\mathbb{Q}}}_{l}$-linear additive category, a periodic functor on ${\mathcal{C}}$ is a linear functor $a^{*}:{\mathcal{C}}\rightarrow{\mathcal{C}}$ such that $(a^{*})^{o}$ is the identity functor on ${\mathcal{C}}$.

For a given generalized Cartan matrix $C=DB$, we construct a finite quiver $Q=(I,H,\Omega)$ with an admissible automorphism $a$ by using [7, Proposition 14.1.2] and let $o$ be the order of $a$. The automorphism $a$ induces naturally an admissible automorphism of the framed quiver $\tilde{Q}$. We denote the set of $a$-orbits of $I$ by $\underline{I}=I/\langle a\rangle$, and denote the set of $a$-invariant dimension vectors by ${\mathbb{N}}I^{a}$. Then any $\underline{i}\in\underline{I}$ determines a dimension vector $\underline{i}=\sum\limits_{i\in\underline{i}}i$ in ${\mathbb{N}}I^{a}$.

Let $\mathcal{S}^{\prime}$ be the set of finite sequences $\boldsymbol{\nu}=(\nu^{1},\nu^{2},\cdots,\nu^{s})$ of dimension vectors such that each $\nu^{l}=a_{l}\underline{i}_{l}$ for some $a_{l}\in{\mathbb{N}}_{\geqslant 1}$ and $\underline{i}_{l}\in\underline{I}\subset{\mathbb{N}}I^{a}$. For $\nu\in{\mathbb{N}}I^{a}$, we say $\boldsymbol{\nu}\in\mathcal{S}^{\prime}$ is an $a$-flag type of $\nu$ if $\sum\limits_{1\leqslant l\leqslant s}\nu^{l}=\nu$. We can also define semisimple complex $L_{\boldsymbol{\nu}}$ for $\boldsymbol{\nu}\in\mathcal{S}^{\prime}$ like Section 2.1.

Given a $I$-graded space $\mathbf{W}$ with dimension vectors $\omega\in{\mathbb{N}}I^{a}$, we assume $\omega=\sum\limits_{\underline{i}\in\underline{I}}e_{\underline{i}}\underline{i}$. Assume $I^{\prime}=\{\underline{i}_{1},\underline{i}_{2},\cdots,\underline{i}_{m}\}$, we fix an $a$-flag type $\boldsymbol{e}=(e_{\underline{i}_{1}}\underline{i}_{1},e_{\underline{i}_{2}}\underline{i}_{2},\cdots,e_{\underline{i}_{m}}\underline{i}_{m})$. If $\boldsymbol{\nu}$ in $\mathcal{S}^{\prime}$ is an $a$-flag type of $\mathbf{V}$, then $\boldsymbol{\nu}\boldsymbol{e}$ is an $a$-flag type of $\mathbf{V}\oplus\bf W$.

Since the automorphism $a$ is addmissible, there are no arrows between vertices in $\underline{i}$, hence the functors ${\mathcal{E}}^{(n)}_{i}$ (or ${\mathcal{F}}^{(n)}_{i}$), $i\in\underline{I}$ commute with each other. The compositions of functors above are well-defined.