# qq-character and geometric representation theory Note prepared for [Affine W-algebras and quantum groups](https://webusers.imj-prg.fr/~eric.vasserot/workshop2025/), September 2025, Paris Main reference: Kimura, *Higgsing qq-character and irreducibility*, [[arXiv:2205.08312]](https://arxiv.org/abs/2205.08312) ## Part 1 : Overview Motivations in representation theory - classification of (irreducible) representations - tensor product behavior - and more $\leadsto$ character theory is a useful tool for these purposes Let - $\mathfrak{g}$ : simple Lie algebra - $V$ : finite-dimensional $\mathfrak{g}$-module with weight space decomposition $$ V = \bigoplus_{\lambda \in P} V_\lambda, \quad V_\lambda = \{ v \mid h \cdot v = \lambda(h) v, h \in \mathfrak{h} \} $$ where we denote by $P$ the weight lattice, and $\mathfrak{h}$ is the Cartan subalgebra of $\mathfrak{g}$. Then, we have the character defined as a formal series, $$ \chi(V) = \sum_\lambda (\dim V_\lambda) e^\lambda $$ which is seen as a trace over $V$, $\chi(V) = \operatorname{tr}_V (e^\lambda)$. **Highest-weight module** We write $I = \{1,\ldots,\operatorname{rk}\mathfrak{g}\}$. Let - $w = (w_i)_{i \in I} \in \mathbb{Z}_{\ge 0}^{\operatorname{rk} \mathfrak{g}}$ - $v = (v_i)_{i \in I} \in \mathbb{Z}_{\ge 0}^{\operatorname{rk} \mathfrak{g}}$ - $\omega = (\omega_i)_{i \in I}$ : fundamental weights - $\alpha = (\alpha_i)_{i \in I}$ : simple roots For the highest-weight module associated with $w$, we may write $$ \lambda = \lambda_{w,v} = \sum_{i \in I} (w_i \omega_i - v_i \alpha_i) $$ Set $Y_i = e^{\omega_i}$ and $A_i = e^{\alpha_i}$. Recalling that each $A_i$ is a monomial of $(Y_i)$, we have $$ \chi(V) = \sum_{\lambda} (\dim V_\lambda) \prod_{i \in I} Y_i^{w_i} A_i^{-v_i} \in \mathbb{Z}[Y_i^{\pm 1}]_{i \in I} $$ Hence, the character is a ring homomorphism from the Grothendieck ring of the finite-dimensional $\mathfrak{g}$-modules to the Laurent polynomial ring of $Y$-variables, $$ \chi : \operatorname{Rep} \mathfrak{g} \to \mathbb{Z}[Y_i^{\pm 1}]_{i \in I} $$ ### q-character Let - $U_q(\widehat{\mathfrak{g}})$ : quantum affine algebra with $q^\mathbb{Z} \neq 1$ ($U_q(L{\mathfrak{g}})$ : quantum loop algebra) - $V_{w;x}$ : finite-dimensional $U_q(\widehat{\mathfrak{g}})$-module (type I module) of highest-weight $w$ with the spectral parameter $x$ with the decomposition [Chari--Pressley] $$ V_{w;x} = \bigoplus_\lambda V_{\lambda;x} $$ Then, the $q$-character is given as follows, $$ \chi_q(V_{w;x}) = \sum_\lambda (\dim V_{\lambda;x}) Y_{1,x_1}^{w_1} Y_{2,x_2}^{w_2} \cdots \in \mathbb{Z}[Y_{i,x}^{\pm 1}]_{i=1}^{\operatorname{rk} \mathfrak{g}} $$ where the monomial of formal $x$-dependent variables $(Y_{i,x})_{i=1}^{\operatorname{rk} \mathfrak{g}}$ is called the $\ell$-weight. **Theorem** [Frenkel--Reshetikhin] : $\chi_q$ is a ring homomorphism, $$ \chi_q : \operatorname{Rep} U_q(\widehat{\mathfrak{g}}) \to \mathbb{Z}[Y_{i,x}^{\pm 1}]_{i \in I} $$ **Theorem** [Frenkel--Mukhin, conjectured by Frenkel--Reshetikhin] : Let $(S_i)_{i \in I}$ be the set of screening operators. Then, we have $$ \operatorname{Im} \chi_q = \bigcap_{i \in I} \ker S_i $$ *Example* : For $\mathfrak{g}=\mathfrak{sl}_2$, $V_x$ : 2-dimensional $U_q(\widehat{\mathfrak{g}})$-module ($w=1$), we have \begin{align} \chi_q(V_x) & = Y_x + 1/Y_{xq} \\ \chi_q(V_x \otimes V_{x'}) & = \chi_q(V_x) \chi_q(V_{x'}) \\ & = Y_x Y_{x'} + \frac{Y_x}{Y_{x'q}} + \frac{Y_{x'}}{Y_{xq}} + \frac{1}{Y_{xq} Y_{x'q}} \end{align} Hence, the tensor product module $V_x \otimes V_{x'}$ is irreducible for generic $x$, $x'$: No dominant monomial $Y_{1,x_1}^{w_1} \cdots Y_{d,x_d}^{w_d}$ ($w_i \ge 0$) except for the top term (the highest weight monomial). *Specialization* : $x' = xq$ \begin{align} \chi_q(V_x \otimes V_{xq}) & = \chi_q(V_x) \chi_q(V_{xq}) \\ & = Y_{x} Y_{xq} + \frac{Y_x}{Y_{xq^2}} + \frac{1}{Y_{xq} Y_{xq^2}} + 1 \\ & = \chi_q(V_{2;x}^{(2)}) + \chi_q(V_x^{(0)}) \end{align} where we denote by $V_x^{(k)}$ the $(k+1)$-dimensional $U_q(\widehat{\mathfrak{sl}}_2)$-module. In this case, there are two dominant monomials, $Y_{x} Y_{xq}$ and $1 = Y_x^0$. ### qq-character - Geometric definition [Nekrasov] - Algebraic definition [Frenkel--Reshetikhin, K--Pestun] $$ \operatorname{Im} \chi_q = \bigcap_{i \in I} \ker S_i \leadsto \text{quantum version of } S_i $$ We write $\chi_{qq} = \chi_{q_1,q_2}$ and $q_{1} q_2 = q_{12}$ for simplicity. *Example* : $\mathfrak{g}=\mathfrak{sl}_2$, $V_x = V_x^{(1)}$ : 2-dimensional $U_q(\widehat{\mathfrak{g}})$-module \begin{align} \chi_{qq}(V_x) & = Y_x + 1/Y_{x q_{12}} \\ \chi_{qq}(V_x \otimes V_{x'}) & = Y_x Y_{x'} + \mathscr{S}(x'/x) \frac{Y_{x'}}{Y_{x q_{12}}} + \mathscr{S}(x/x') \frac{Y_x}{Y_{x' q_{12}}} + \frac{1}{Y_{x q_{12}} Y_{x' q_{12}}} \end{align} where $$ \mathscr{S}(x) = \frac{(1 - x/q_1)(1 - x/q_2)}{(1 - x)(1 - x / q_{12})} $$ For $x \neq 1, q_{12}$, we have $$ \mathscr{S}(x) = \mathscr{S}(x^{-1} q_{12}) $$ *Specializations* : - $x' = x q_1$ : \begin{align} \chi_{qq}(V_x \otimes V_{xq_1}) & = Y_x Y_{x q_1} + \underbrace{\mathscr{S}(q_1)}_{0} \frac{Y_{xq_1}}{Y_{x q_{12}}} + \mathscr{S}(q_1^{-1}) \frac{Y_x}{Y_{x q_{1}^2 q_2}} + \frac{1}{Y_{x q_{12}} Y_{x q_1^2 q_2} } \\ & = Y_x Y_{x q_1} + \mathscr{S}(q_1^{-1}) \frac{Y_x}{Y_{x q_{1}^2 q_2}} + \frac{1}{Y_{x q_{12}} Y_{x q_1^2 q_2}} \\ & = \chi_{qq}(V_x^{(2)}) \end{align} - $x' = x q_{12}$ : It hits a pole of $\mathscr{S}(x'/x)$ with the residue given by $$ \operatorname*{Res}_{x' = x q_{12}} \chi_{qq}(V_x \otimes V_{x'}) = \frac{(1-q_1)(1-q_2)}{1-q_{12}} \cdot 1 = \frac{(1-q_1)(1-q_2)}{1-q_{12}} \chi_{qq}(V_x^{(0)}) $$ - $x' = x$ $$ \chi_{qq}(V_x \otimes V_{x}) = Y_x^2 + \left(c + \frac{(1-q_1)(1-q_2)}{1-q_{12}} x \partial_x \log (Y_x Y_{x q_{12}}) \right) \frac{Y_x}{Y_{x q_{12}}} + \frac{1}{Y_{x q_{12}}^2} $$ where $c = \lim_{x \to 1} (\mathscr{S}(x) + \mathscr{S}(x^{-1})) < \infty$. **Conjecture** (Theorem for $\mathfrak{g} = \mathfrak{sl}_2$) [K] : - The $qq$-character of the standard $U_q(\widehat{\mathfrak{g}})$-module is irreducible for any $x$, i.e., it contains a unique dominant monomial, except at singular loci. - The residue gives "lower" $qq$-characters. **Remark** : - $\chi_{qq}$ is not a ring homomorphism (non-commutative). - $\chi_{qq} \in \mathbb{Q}(q_{1,2},x)[\partial^k Y_{i,x}^{\pm 1}]_{i \in I, k \ge 0}$ (differential polynomial ring) ### Historical remark - '97, Frenkel--Reshetikhin : deformed W-algebra $\leadsto$ $(q,t)$-character $=$ $qq$-character - '98, --- : $q$-character ('95 : Knight : $q$-character for Yangian) - '00, Nakajima : $t$-analogue of $q$-character - '13, Nekrasov--Pestun--Shatashvili : $q$-character from quantum Seiberg--Witten theory - '15, Nekrasov : $qq$-character from double quantization of Seiberg--Witten theory - '15, K--Pestun : $qq$-character and W-algebra (physics perspective of Frenkel--Reshetikhin) ### Geometric perspective Quiver variety $\mathfrak{M}_{w,v}$ $\leadsto$ graded quiver variety $\mathfrak{M}_{w,v}^\text{gr}$: fixed point set under $\mathbb{S}^1$ action Euler characteristic of $\mathfrak{M}_{w,v}^\text{gr}$ agrees with the weight space dimension, $$ \chi(\mathfrak{M}_{w,v}^\text{gr}) = \dim V_{\lambda_{w,v}} $$ Two possible generalizations of $\chi$ : - Poincaré polynomial $\leadsto$ $t$-analogue of $q$-character - $\chi_{y}$-genus $\leadsto$ $qq$-character ($q_1$ identified with the equivariant parameter of $\mathbb{S}^1$ action; $y \to q_2$) ## Part 2 : Vertex operator formalism of $qq$-character and W-algebras **Proposition** [Frenkel--Reshetikhin, K--Pestun] : Operator-valued (fundamental) $qq$-character is the generating current of the deformed W-algebras denoted by W$_{q_{1,2}}(\mathfrak{g})$, i.e., $$ [\chi_{qq}^{(\text{op})}, Q_i] = 0 $$ where $(Q_i)_{i=1}^{\operatorname{rk}\mathfrak{g}}$ are the screening charges of W$_{q_{1,2}}(\mathfrak{g})$. **Remark** : For the $q$-character, we have $\operatorname{Im} \chi_q = \bigcap_{i \in I} \ker S_i$. **Remark** : The double affine W-algebra can be constructed by replacing $\mathfrak{g} \to \widehat{\mathfrak{g}}$, which corresponds to the uplift from the quantum affine algebra to the quantum toroidal algebra, $U_q(\widehat{\mathfrak{g}}) \to U_q(\widehat{\widehat{\mathfrak{g}}})$. For this purpose, we shall consider the $qq$-character of the Fock/MacMahon module of the corresponding toroidal algebra. Set $\mathfrak{g} = \mathfrak{sl}_2$, $q_1, q_2 \in \mathbb{C}^\times$. **Definition** : We define the following vertex operators. - Y-operator (fundamental weight) : $$ Y(x) = {:\exp\left( \sum_{n \neq 0} y_n x^{-n} \right):} , \quad [y_n,y_m] = - \frac{1}{n} \frac{(1-q_1^{-n})(1-q_2^{-n})}{1 + q_{12}^{-n}} \delta_{n+m,0} $$ - A-operator (simple root) : $$ A(x) = {:\exp\left( \sum_{n \neq 0} a_n x^{-n} \right):} , \quad [a_n,a_m] = - \frac{1}{n} (1-q_1^{-n})(1-q_2^{-n})(1 + q_{12}^{n}) \delta_{n+m,0} $$ where $A(x) = {: Y(x) Y(x q_{12}^{-1}) :}$ - Screening current : $$ S(x) = {:\exp\left(\sum_{n \neq 0} s_n x^{-n} \right):} $$ where $A(x) = q_1^{-1} {: S(x) / S(xq_2^{-1}) :}$. We define also the screening charge, $$ Q(x) = \sum_{k \in \mathbb{Z}} S(x q_2^k) $$ which is a difference analogue of the standard form, $Q = \oint S(z) dz$. More precisely, we need to incorpolate the zero modes. In the following, we do not write the normal ordering symbol when no confusion arises. **Lemma** : Operator product relation \begin{align} Y(x) A(x') & = \mathscr{S}(x'/x)^{-1} :Y(x)A(x'): , \quad |x'/x| < 1 \\ A(x') Y(x) & = \mathscr{S}(q_{12} x/x')^{-1} :Y(x)A(x'): , \quad |x/x'| < 1 \\ Y(x) S(x') & = \frac{1 - x'/x}{1-q_1^{-1}x'/x} :Y(x)S(x'): , \quad |x'/x| < 1 \\ S(x') Y(x) & = q_1 \frac{1 - x/x'}{1-q_1 x/x'} :Y(x)S(x'): , \quad |x/x'| < 1 \end{align} Here, the rational functions are interpreted as formal series of either $x'/x$ or $x/x'$ depending on the radial ordering. **Proposition** : We have $$ [Y(x),S(x')] = (1-q_1) \delta(q_1^{-1}x'/x) :Y(x)S(x'): $$ where $\delta(z) = \sum_{n\in\mathbb{Z}} z^n$ the formal series $\delta$-function. Then, \begin{align} & [Y(x) + :Y(x)A^{-1}(x q_{12}):, S(x')] \\ & = (1-q_1) \left( \delta(q_1^{-1}x'/x) :Y(x)S(x'): - \delta(q_{12}^{-1}x'/x) :Y(x)S(x'q_2^{-1}):\right) \\ & =: X(x,x') - X(x,x'q_2^{-1}) \end{align} is a $q_2$-difference, implying $$ [Y(x) + :Y(x)A^{-1}(x q_{12}):, Q(x')] = 0 $$ *Proof* : Direct computation based on the previous Lemma. **Corollary** : The operator analogue of the $qq$-character for the two-dimensional module of $U_q(\widehat{\mathfrak{sl}}_2)$ (fundamental $qq$-character) is given by $$ \chi_{qq}^{\text{(op)}}(V_x^{(1)}) = Y(x) + {:Y(x)A^{-1}(xq_{12}):} = Y(x) + Y(xq_{12})^{-1} $$ which commutes with the screening charge, $[\chi_{qq}^{\text{(op)}}(V_x^{(1)}),Q(\cdot)] = 0$, i.e., $\chi_{qq}^{\text{(op)}}(V_x^{(1)}) \in \ker (\operatorname{ad}Q)$. The operator $qq$-character agrees with the deformed Virasoro generating current ($q$-Virasoro algebra) [Shiraishi--Awata--Kubo--Odake]: Set $q_i = e^{\hbar \epsilon_i}$ ($i=1,2$). Then, we have $$ \chi_{qq}^{\text{(op)}}(V_x^{(1)}) = T_{q\text{Vir}}(x) = 2 - \frac{\epsilon_1}{\epsilon_2} \left(x^2 T_\text{Vir}(x) - \frac{(\epsilon_1+\epsilon_2)^2}{4\epsilon_1\epsilon_2} \right) \hbar^2 + O(\hbar^4) $$ Their product gives \begin{align} \chi_{qq}^{\text{(op)}}(V_x^{(1)}) \chi_{qq}^{\text{(op)}}(V_{x'}^{(1)}) & = \left( Y(x) + Y(xq_{12})^{-1} \right)\left( Y(x') + Y(x'q_{12})^{-1} \right) \\ & = f(x'/x)^{-1} :\left( Y(x) Y(x') + \mathscr{S}(x'/x) \frac{Y({x'})}{Y(x q_{12})} + \mathscr{S}(x/x') \frac{Y(x)}{Y(x' q_{12})} + \frac{1}{Y(x q_{12}) Y(x' q_{12})} \right): \\ & = f(x'/x)^{-1} \chi_{qq}^{(\text{op})}(V_x^{(1)} \otimes V_{x'}^{(1)}) \end{align} where $$ f(z) = \exp \left( \sum_{n \ge 0} \frac{(1-q_1^{-n})(1-q_2^{-n})}{n(1+q_{12}^{-n})} z^n \right) $$ **Proposition** : The higher-order $qq$-character is given by \begin{align} \chi_{qq}^{(\text{op})}( V_{x_1}^{(1)} \otimes \cdots \otimes V_{x_w}^{(1)}) & = {:Y(x_1) \cdots Y(x_w):} + \cdots \\ & = \sum_{\mathsf{I} \sqcup \mathsf{J} = \{1,\ldots,w\} } \prod_{\alpha \in \mathsf{I}, \beta \in \mathsf{J}} \mathscr{S}(x_\alpha/x_\beta) :\prod_{\alpha \in \mathsf{I}} Y(x_\alpha) \prod_{\beta \in \mathsf{J}} Y(x_\beta q_{12})^{-1}: \end{align} which contains $2^w$ monomials. This $qq$-character is irreducible for generic $x$ (unique dominant monomial $Y(x_1) \cdots Y(x_w)$). Meanwhile, imposing the *$q$-segment condition* [Chari--Pressley], $$ (x_1,\ldots,x_w) = (x,x q_1,\ldots,x q_1^{w-1}) \qquad (*) $$ several $\mathscr{S}$-factors vanish since $\mathscr{S}(q_1) = 0$, and the remaining $w+1$ monomials are interpreted as the $qq$-character of $V_x^{(w)}$ $$ \left. \chi_{qq}^{(\text{op})}( V_{x_1}^{(1)} \otimes \cdots \otimes V_{x_w}^{(1)}) \right|_{(*)} = \chi_{qq}^{(\text{op})} (V_x^{(w)}) $$ **Proposition** : We have the following classical limit of the $qq$-character, \begin{align} \chi_{qq}(V_x^{(w)}) \to \begin{cases} \chi_{q_1} (V_x^{(w)}) & (q_2 \to 1)\\ \chi_{q_2} (V_x^{(1)})^w & (q_1 \to 1) \end{cases} \end{align} *Example* : $w = 2$ $$ \chi_{qq}(V_x^{(2)}) = Y(x) Y(x q_1) + \mathscr{S}(q_1^{-1}) \frac{Y(x)}{Y(x q_1^2 q_2)} + \frac{1}{Y(x q_{12}) Y(x q_1^2 q_2)} $$ Recalling $$ \mathscr{S}(q_1^{-1}) \to \begin{cases} 1 & (q_2 \to 1) \\ 2 & (q_1 \to 1) \end{cases} $$ we have $$ \chi_{qq}(V_x^{(2)}) \to \begin{cases} \displaystyle Y(x) Y(x q_1) + \frac{Y(x)}{Y(x q_1^2)} + \frac{1}{Y(x q_{1}) Y(x q_1^2)} & (q_2 \to 1) \\ \displaystyle \left( Y(x) + \frac{1}{Y(xq_2)} \right)^2 & (q_1 \to 1) \end{cases} $$ ### General case $U_q(\widehat{\mathfrak{g}})$ - Weights and roots : $Y_i(x)$, $A_i(x)$, $i \in I$ - Highest-weight monomial of weight $w = (w_i)_{i \in I}$ with spectral parameters $(x_{i,\alpha})_{i \in I,\alpha = 1,\ldots,w_i}$: $$ Y_w(x) = {: \prod_{i \in I} \prod_{\alpha=1}^{w_i} Y_i(x_{i,\alpha}) :} $$ - $qq$-character of weight $w$ is given by $$ \chi_{qq} = Y_w(x) + \cdots $$ The lower terms written as $Y_w(x) A^{-1}(\cdot) \cdots A^{-1}(\cdot)$ are generated by iWeyl reflection in order that it commutes with the screening charges, $$ \text{iWeyl} : \quad Y_i(x) \mapsto {:Y_i(x) A_{i}^{-1}(x q_{12}):} $$ *Example* : Quantum toroidal algebra of $\mathfrak{gl}_1$ : $U_q(\widehat{\widehat{\mathfrak{gl}}}_1)$ Let $\mathscr{F}_x$ be the Fock module with spectral parameter $x$ and $q_3$ is another parameter of the algebra. We also introduce $q_4 = (q_1 q_2 q_3)^{-1}$ to simplify the notation. Then, the $qq$-character is given by an infinite series parametrized by partitions, \begin{align} \chi_{qq}( \mathscr{F}_x ) & = Y(x) + \mathscr{S}(q_3^{-1}) \frac{Y(x q_3^{-1}) Y(x q_4^{-1})}{Y(x q_{12})} + \cdots \\ & = \sum_{\lambda} Z_\lambda(q_{1,2,3,4}) \prod_{x \in \partial^+ \lambda} Y(x) \prod_{x \in \partial^- \lambda} Y(x q_{12})^{-1} \end{align} where $Z_\lambda(q_{1,2,3,4})$ is a rational function of $q_{1,2,3,4}$ (K-theoretic Nekrasov function of rank one; equivariant $\chi_{q_3}$ genus of the Hilbert scheme of points on $\mathbb{C}^2$). ## Part 3 : Geometric realization of $qq$-character and quiver variety Here is a strategy to obtain geometric realization of $qq$-character: 1. Vertex operator formalism to the integral formula of the $qq$-character 2. Identification with the equivariant integral over the quiver variety ### Vertex operator formalism to the integral formula The $qq$-character is generated from the highest-weight monomial by the iWeyl reflections, $$ \text{iWeyl} : Y_i(x) \mapsto :Y_i(x)A_i^{-1}(x q_{12}): $$ **Lemma** : We have $$ {:Y_i(x)A_i^{-1}(x q_{12}):} = \frac{1 - q_{12}}{(1-q_1)(1-q_2)} \oint A_i^{-1}(z) Y_i(x) \frac{dz}{2\pi \iota z} $$ where the integration contour is taken aroung the pole of the $\mathscr{S}$-factor appearing from the operator product of $Y_i$ and $A_i$ at $z = x q_{12}$. This motivates us to define the following operator. **Definition** : We define the $R$-operator, which generates the iWeyl reflection, $$ R_i = \frac{1 - q_{12}}{(1-q_1)(1-q_2)} \oint A_{i}^{-1}(z) \frac{dz}{2\pi \iota z} $$ **Remark** : The $q$-character is generated by the screening operator. **Proposition** : For the $qq$-character of the module $V_{w}$, which allows the following weight decomposition, $$ \chi_{qq}^{(\text{op})}(V_{w;x}) = \sum_{v} \chi_{qq}^{(\text{op})}(V_{w,v;x}) $$ each contribution associated with the weight $\lambda_{w,v} = \sum_{i \in I} (w_i \omega_i - v_i \alpha_i)$ is given as follows, \begin{align} \chi_{qq}^{(\text{op})}(V_{w,v;x}) & = \frac{1}{v!} \left( \prod_{i \in I} R_i^{v_i} \right) {:Y_w(x):} \\ & = (\text{const.}) \times \oint \prod_{i \in I} \prod_{\alpha=1}^{v_i} A_i^{-1}(z_{i,\alpha}) :Y_w(x): \prod_{i,\alpha} \frac{dz_{i,\alpha}}{2\pi\iota z_{i,\alpha}} \end{align} where we write $v! = \prod_{i \in I} v_i!$. *Example* : $\mathfrak{g} = \mathfrak{sl}_2$ \begin{align} \chi_{qq}^{(\text{op})}(V_{w,v;x}) & = (\text{const.}) \times \oint A^{-1}(z_1) \cdots A^{-1}(z_v) {:Y_w(x):} \\ & = (\text{const.}) \times \oint \prod_{1 \le I < J \le v} \mathscr{S}(z_I/z_J)^{-1} \prod_{I,\alpha} \mathscr{S}(q_{12}x_\alpha/z_I) {:Y_w(x) A^{-1}(z_1) \cdots A^{-1}(z_v):} \prod_{I=1}^v \frac{dz_I}{2\pi\iota z_I} \end{align} ### Identification with the equivariant integral over the quiver variety The contour integral formula discussed in the vertex operator formalism agrees with the equivariant integral over the corresponding quiver variety. Let - $\Gamma = (\Gamma_0,\Gamma_1)$ : a quiver with sets of vertices and edges, $\Gamma_0$ and $\Gamma_1$ - $\mathfrak{g}_\Gamma$ : Lie algebra associated with quiver $\Gamma$ (the set $I$ identified with $\Gamma_0$) - $V_{w}=V_{w;x}$ : finite-dimensional standard $U_q(\widehat{\mathfrak{g}}_\Gamma)$-module with spectral parameter $x$ associated with the highest-weight $w$ having the decomposition $V_{w;x} = \oplus_v V_{w,v;x}$ - $\Gamma_0$-graded vector spaces $W = (W_i)_{i\in \Gamma_0}$, $V = (V_i)_{i \in \Gamma_0}$ with $W_i = \mathbb{C}^{w_i}$, $V_i = \mathbb{C}^{v_i}$ - $\Gamma_1$-graded vector spaces $M = (M_e)_{e \in \Gamma_1}$ with $\operatorname{ch} M_e = \nu_e$ - $Q = \mathbb{C}^2 = Q_1 \oplus Q_2$ with $\operatorname{ch} Q_i = q_i$ for $i = 1, 2$, and $\operatorname{ch} \det Q = q_{12}$. We define the equivarint K-theory class valued Cartan matrix, $$ c_{ji} = (1 + \det Q) \delta_{i,j} - \sum_{e:i \to j} M_e - \sum_{e:j \to i} (\det Q) M_e^{\vee} , \quad i, j \in \Gamma_0 $$ and also define $Y_{w,v} = (W_i - \sum_{j \in I} c_{ij} V_j)_{i \in \Gamma_0}$. Let $\mathscr{M}_{w,v}$ be the quiver variety constructed from $W$ and $V$, and we define a $\Gamma_0$-graded formal bundle $Y = (Y_i)_{i \in \Gamma_0}$, suth that we identity $$ Y_{i,x} = \operatorname{ch} \wedge_x Y_i = \sum_k (-x)^k \operatorname{ch} \wedge^k Y_i $$ Then, we have the following. **Theorem** [Nekrasov,K--Pestun] : The contribution of the weight $V_{w,v;x}$ to the $qq$-character is given by the equivariant integral over $\mathscr{M}_{w,v}$, $$ \chi_{qq}(V_{w,v}) = \int_{\mathscr{M}_{w,v}} \operatorname{ch} \wedge Y \otimes Y_{w,v} \operatorname{ch} \wedge_{q_2^{-1}} T\mathscr{M}_{w,v} \operatorname{td} \mathscr{M}_{w,v} $$ We remark that, in the absence of the insertion $\operatorname{ch} \wedge Y \otimes Y_{w,v}$, it computes the $\chi_{q_2^{-1}}$-genus of $\mathscr{M}_{w,v}$. *Example* : $\mathfrak{g}=\mathfrak{sl}_2$ ($A_1$ quiver). We have $W = \mathbb{C}^w$ and $V = \mathbb{C}^v$ and linear maps, $I \in \operatorname{Hom}(W,V)$, $J \in \operatorname{Hom}(V,W)$. The quiver variety is given by $$ \mathscr{M}_{w,v} = \{ (I,J) \mid IJ = 0, \text{stability condition} \}/\mathrm{GL}(V) \cong T^\vee\operatorname{Gr}(v,w) $$ We have the equivariant actions, $T = T_W \times T_V \times T_Q$, where $$ T : (I,J) \mapsto (g I h^{-1}, q_{12} h J g^{-1}), \quad g \in T_V, h \in T_W, (q_1,q_2) \in T_Q $$ We have the following complex $$ 0 \to \operatorname{Hom}(V,V) \to \operatorname{Hom}(W,V) \otimes \operatorname{Hom} (V,W) \otimes \det Q \to \operatorname{Hom}(V,V) \otimes \det Q \to 0 $$ from which we obtain $$ T\mathscr{M}_{w,v} = V W^\vee + W V^\vee \det Q - V V^\vee - V V^\vee \det Q $$ We also have $$ Y_{w,v} = W - (1 + \det Q) V $$ Set $\operatorname{ch} W = \sum_{\alpha=1}^w x_\alpha$, $\operatorname{ch} V = \sum_{i = 1}^v z_i$. Then, the equivariant integral is given by \begin{align} & \int_{\mathscr{M}_{w,v}} \operatorname{ch} \wedge Y \otimes Y_{w,v} \operatorname{ch} \wedge_{q_2^{-1}} T\mathscr{M}_{w,v} \operatorname{td} \mathscr{M}_{w,v} \\ & = \frac{1}{v!} \oint_{T_V} \frac{\prod_{\alpha=1}^w Y_{x_\alpha}}{\prod_{i=1}^v Y_{z_i} Y_{z_i q_{12}^{-1}}} \prod_{1 \le i < j \le v} \mathscr{S}(z_j/z_i)^{-1} \prod_{i,\alpha} \mathscr{S}(q_{12} x_\alpha/z_i) \prod_{i=1}^v \frac{dz_i}{2\pi \iota z_i} \end{align} The equivariant fixed points are in one-to-one correspondence to poles of the integrand classified by the *Jefferey--Kirwan formula*. In this context, the stability condition corresponds to the choice of the selection vector. We may have degenerate fixed points, corresponding to higher order poles, yielding derivative terms. *Example* : Jordan quiver and beyond In this case, the quiver variety is identified with the Hilbert scheme of points on $\mathbb{C}^2$ for $w = 1$ denoted by $\operatorname{Hilb}^v(\mathbb{C}^2)$. In thic case, we have the $qq$-character of the Fock module of quantum toroidal algebra of $\mathfrak{gl}_1$. We can similarly construct the $qq$-character for the MacMahon module based on $\operatorname{Hilb}^v(\mathbb{C}^3)$. Since it is not given by an usual quiver variety, we need to modify the integral formula (taking into account the *potential* of the quiver), $$ \chi_{qq} = \int_{[\operatorname{Hilb}^v(\mathbb{C}^3)]^\text{vir}} \operatorname{ch} \wedge Y \otimes Y_{w=1,v} \operatorname{td}(\operatorname{Hilb}^v(\mathbb{C}^3)) $$ For the higher-rank case $w > 1$, it will be a Quot-scheme. One may also consider $\operatorname{Hilb}^v(\mathbb{C}^4)$, which gives rise to the magnificent four theory of Nekrasov [[Nekrasov](https://arxiv.org/abs/1712.08128)], formulated based on K-theoretic equivariant DT4 invariants, and the associated $qq$-character is also studied (implying a new module?) [[K--Noshita](https://arxiv.org/abs/2310.08545)].