1. Introduction
For each lattice polytope 
$P$ there is a divisor 
$D$ on a projective toric variety, so that the lattice points in 
$P$ are in natural bijection with a basis of the global sections of the line bundle 
$\mathscr{O}(D)$. This correspondence forms the foundation of a bridge relating algebraic geometry and combinatorics that has been exploited extensively in many contexts in the past several decades. Brion’s formula is a striking example: it expresses the generating function of lattice points in 
$P$ (a Laurent polynomial) as a sum of rational functions determined by the tangent cones at each vertex of 
$P$. The simplest case is when 
$P$ is the interval from 
$0$ to 
$m$, in which case Brion’s formula is the elementary identity
\begin{equation}
1+x+x^2+\cdots+x^m = \frac{1}{1-x} + \frac{x^m}{1-x^{-1}}.
\end{equation} In this article, we establish a 
$q$-analogue of Brion’s formula. To state the formula, we require the 
$q$-Pochhammer symbol
\begin{equation*}
(x;q)_d = \prod_{i=1}^d (1-xq^{i-1}),
\end{equation*} as well as its infinite limit 
$(x;q)_\infty$, which is an analytic function when 
$|q| \lt 1$. The 
$q$-multinomial coefficients are polynomials in 
$q$ defined by
\begin{equation*}
\left[\begin{array}{c} m \\ k_1,\ldots,k_r \end{array}\right]_q = \frac{(q;q)_m }{(q;q)_{k_1} \cdots (q;q)_{k_r}},
\end{equation*} where 
$m=k_1+\cdots+k_r$. They specialize to 
$1$ at 
$q=0$ and to the usual multinomial coefficients at 
$q=1$. When 
$r=2$, these are the 
$q$-binomial coefficients and one usually writes 
$\left[\begin{array}{c} m \\ k \end{array}\right]_q$ in place of 
$\left[\begin{array}{c} m \\ k,m-k \end{array}\right]_q$.
Here is our 
$q$-analogue of (*):
\begin{equation}\begin{aligned}
&\left[\begin{array}{c} m \\ 0 \end{array}\right]_q + \left[\begin{array}{c} m \\ 1 \end{array}\right]_qx + \left[\begin{array}{c} m \\ 2 \end{array}\right]_q x^2 + \cdots + \left[\begin{array}{c} m \\ m \end{array}\right]_qx^m \\
& \quad = \frac{(q;q)_m}{(q;q)_\infty}\left( \frac{1}{(x;q)_\infty} \sum_{d\geq 0} \frac{q^{md}}{(q^{-1};q^{-1})_d (xq^{-1};q^{-1})_d} \right.\\
&\quad \left.+ \frac{x^m}{(x^{-1};q)_\infty} \sum_{d\geq 0} \frac{q^{md}}{(q^{-1};q^{-1})_d (x^{-1}q^{-1};q^{-1})_d} \right) . \end{aligned}\end{equation} With some algebraic manipulation, the Jacobi triple product identity can be deduced from the case 
$m=0$.
Our main theorem is a generalization of (*q) to higher-dimensional polytopes. Let 
$M$ be a lattice of rank 
$n$, so 
$M\cong\mathbb{Z}^n$, and let 
$N=\operatorname{Hom}(M,\mathbb{Z})$ be the dual lattice. The corresponding real vector spaces are 
$M_{\mathbb{R}}$ and 
$N_{\mathbb{R}}$. We write a polytope 
$P \subseteq M_{\mathbb{R}}$ as an intersection of half-spaces, so
\begin{equation*}
P = \bigcap_{i=1}^{r} \{u\, |\, \langle u, v_i \rangle \geq -a_i \},
\end{equation*} where 
$v_i\in N$ is the primitive inward normal vector defining the 
$i^{\mathrm{th}}$ facet of 
$P$, and 
$a_i$ is an integer.
The inward normal vectors 
$v_1,\ldots,v_r$ determine a homomorphism 
$\mathbb{Z}^r \to N$. We denote the kernel by 
$A$, writing 
$\beta\colon A \to \mathbb{Z}^r$ for the inclusion and 
$A_+ = \beta^{-1}(\mathbb{Z}^r_{\geq 0})$ for the intersection with the positive orthant.
The result is cleanest under some assumptions on the polytope 
$P$. We assume 
$P$ is smooth: each vertex is contained in exactly 
$n$ facets, and the corresponding primitive normal vectors form a basis for 
$N$. We further assume the polytope is radially symmetric, meaning that 
$\sum_{i=1}^r v_i=0$. These conditions can be removed, at the expense of adding extra inequalities 
$\langle u, v \rangle \geq -a$; see Remark 4.6. On the other hand, there are well-known families of polytopes which satisfy the stated hypotheses—for example, the (generalized) permutahedra are smooth and radially symmetric.
For each vertex 
$p$ of 
$P$, let 
$I(p) \subseteq \{1,\ldots,r\}$ be the set of indices 
$i$ so that 
$v_i$ is an inward normal vector of a facet containing 
$p$, so 
$\{v_i \,|\, i\in I(p)\}$ is a basis for 
$N$. Let 
$\{u_i(p) \, |\, i\in I(p)\}$ be the dual basis of 
$M$. The vector 
$u_i(p)$ is the primitive vector along the edge of 
$P$ which is not contained in the facet defined by 
$v_i$. For each 
$d\in A_+$ and vertex 
$p$ of 
$P$, we define
\begin{equation*}
\mathsf{J}_{d,p} = \left(\prod_{i\in I(p)} \frac{1}{(x^{u_i(p)} q^{-1}; q^{-1})_{\beta(d)_i} }\right)\left(\prod_{j\not\in I(p)}\frac{1}{(q^{-1}; q^{-1})_{\beta(d)_j} }\right) .
\end{equation*}Theorem 1.
Let 
$P$ be a smooth, radially symmetric polytope, realized as an intersection of 
$r$ half-spaces 
$\{u\,|\, \langle u,v_i \rangle \geq -a_i \}$, with notation as above. Then
\begin{align*}
&\sum_{u \in P \cap M} \left[\begin{array}{c} |a| \\ \langle u,v_1 \rangle+a_1,\ldots,\langle u,v_r \rangle+a_r \end{array}\right]_q x^u\\
&\quad = \frac{(q;q)_{|a|}}{(q;q)_\infty^{r-n}} \sum_{p \in V(P)}\sum_{d\in A_+}\frac{ x^p\, q^{\sum a_i\, \beta(d)_i} \cdot \mathsf{J}_{d,p}}{\prod_{i\in I(p)}( x^{u_i(p)};q )_\infty },
\end{align*} where 
$|a|=a_1+\cdots+a_r$ and 
$V(P)$ is the set of vertices of 
$P$.
The left-hand side belongs to 
$\mathbb{Z}[q][M]$; i.e., it is a Laurent polynomial in 
$x$ and a polynomial in 
$q$. The right-hand side is a formal power series in both 
$x$ and 
$q$, and it seems (to us) surprising that a 
$q\to 1$ limit exists, because each term has a pole of infinite order at 
$q=1$. The 
$q\to 0$ limit, on the other hand, recovers Brion’s formula in the smooth, radially symmetric case. (To see this, note that 
$\mathsf{J}_{d,p}$ is 
$0$ at 
$q=0$ unless 
$d=0$, in which case it is 
$1$.)
The theorem follows from Theorem 4.4, which relaxes the requirement of radial symmetry and gives a similar formula for any smooth polytope 
$P$. As in [Reference Brion3], the basic proof technique is localization in the equivariant K-theory of a toric variety. Our main observation is that a canonical 
$q$-enumeration is provided by toric quasimap spaces, introduced by Morrison-Plesser and Givental in the context of Gromov–Witten theory [Reference Givental12, Reference Morrison and Plesser16]. These spaces fit together to form an ind-variety contained in the toric arc scheme studied by Arkhipov–Kapranov [Reference Arkhipov and Kapranov2]. This infinite-dimensional scheme provides the geometric context for the infinite products appearing in the theorem.
A different 
$q$-enumeration of lattice points was investigated by Chapoton, who introduced a 
$q$-analogue of the Ehrhart polynomial [Reference Chapoton7]. His 
$q$-enumeration arises by specializing each Laurent monomial 
$x^u$ to a power 
$q^{l(u)}$, for some linear function 
$l$ on the lattice. In fact, Chapoton’s 
$q$-analogue is a specialization of a weighted enumeration arising from the weighted Ehrhart theory introduced by Stapledon somewhat earlier [Reference Stapledon18].Footnote 1
The left-hand side of (*q) is the 
$n^{\mathrm{th}}$ Rogers–Szegő polynomial. This is a classical family of polynomials, which were shown to be orthogonal with respect to a certain measure on the circle by Szegő in 1926 [Reference Szego19]. Multivariate versions of them play a role in representation theory and combinatorics [Reference Cameron and Vinroot6, Reference Hikami15, Reference Vinroot20]. In 
$\S$ 5, we show how the generalized Rogers–Szegő polynomials, associated to more general polytopes, are related by 
$q$-difference operators.
We also consider a natural analogue of Duistermaat–Heckman (DH) measure for the quasimap ind-variety. Classical DH measure is the pushforward of Liouville measure along the moment map of a symplectic manifold with torus action, producing a measure supported on the moment polytope. (For a toric variety, the DH measure is just Lebesgue measure on the polytope.) This requires modification for infinite-dimensional spaces. Adapting the construction of DH measure from [Reference Brion and Procesi4] to the ind-variety, we define a loop-Duistermaat Heckman probability measure by changing the rescaling used in a limiting procedure. This yields a probability measure on the ambient vector space 
$M_{\mathbb{R}}$. (Finite approximations to this measure, for values of 
$q$ between 
$0$ and 
$1$, produce measures supported on dilations of the polytope; see Figure 1.) Our second result, which describes 
$q=1$ behavior of the coefficients of the generalized Rogers–Szegő polynomial of 
$kP$ as 
$k$ goes to infinity, is that the loop-DH measure is Gaussian. We denote by 
$m_D$ a particular distinguished point inside 
$P$ (see Definition 6.6).
Approximations to the loop-DH measure. The polytope 
$P$ is a lattice hexagon, the convex hull of 
$(0,0)$, 
$(1,0)$, 
$(0,1)$, 
$(2,1)$, 
$(1,2)$, and 
$(2,2)$. The three images depict the integer-point transforms of the LHS of the formula in Theorem 1, for the dilation 
$kP$ with 
$k=30$, evaluated at 
$q=0.2$, 
$q=0.6$, and 
$q=0.9$. (The vertical scales differ.) Images generated by Mathematica.
Theorem 2.
Let 
$P$ be smooth and radially symmetric, and let 
$X$ be the associated toric variety. The polytope 
$P$ determines very ample divisors 
$D$ on 
$X$ and 
$D^0$ on the quasimap ind-variety 
$\mathscr{Q}(X)_\infty$. The loop-DH probability measure of 
$\mathscr{Q}(X)_\infty$ with respect to 
$D^0$ is Gaussian with mean 
$m_D$, and with covariance matrix associated to the quadratic form
\begin{equation*}
u \mapsto \sum_{i=1}^r \frac{1}{\langle m_D, v_i\rangle+a_{i}} \langle u, v_i\rangle^2.
\end{equation*}In other words, the loop-DH measure has density function given by a multiple of
\begin{equation*}
u\mapsto \exp\left(-\frac{1}{2}\sum_{i=1}^r \frac{\langle u-m_D,v_i\rangle^2}{\langle m_D, v_i\rangle+a_{i}}\right).
\end{equation*}2. Toric varieties
By way of fixing notation and conventions, we quickly review basic facts about toric varieties. Our main references are [Reference Cox, Little and Schenck9] and [Reference Fulton11]. We work over the field 
$\mathbb{C}$ (which for our purposes may be replaced by any field). Let 
$T$ be an 
$n$-dimensional torus with character group 
$M=\operatorname{Hom}(T,\mathbb{C}^*)$ and co-character group 
$N=\operatorname{Hom}(\mathbb{C}^*,T)$, so 
$N=\operatorname{Hom}(M,\mathbb{Z})$ is dual to 
$M$, and both 
$N$ and 
$M$ are (non-canonically) isomorphic to 
$\mathbb{Z}^n$. As before we write 
$N_{\mathbb{R}} = N\otimes_{\mathbb{Z}} \mathbb{R}$ and 
$M_{\mathbb{R}} = M\otimes_{\mathbb{Z}} \mathbb{R}$ for the corresponding real vector spaces.
Let 
$\Delta$ be a rational polyhedral fan in 
$N_{\mathbb{R}}$, i.e. a collection of pointed rational polyhedral convex cones which fit together along their faces. We write 
$\Delta^{(k)} \subseteq \Delta$ for the subset of 
$k$-dimensional cones. Of particular importance are the rays 
$\Delta^{(1)} = \{\rho_1,\ldots,\rho_r\}$. Each ray 
$\rho_i$ has a primitive generator 
$v_i \in N$.
We assume 
$\Delta$ is smooth and complete. So there is an exact sequence
\begin{equation*}
0 \to A \xrightarrow{\beta} \mathbb{Z}^r \to N \to 0,
\end{equation*} where 
$\mathbb{Z}^r \to N$ sends the 
$i^{\mathrm{th}}$ standard basis vector to the primitive generator of the 
$i^{\mathrm{th}}$ ray, 
$e_i\mapsto v_i$. The kernel 
$A$ is isomorphic to 
$\mathbb{Z}^{n-r}$.
Dualizing, one has an exact sequence
\begin{equation*}
0 \to M \to \mathbb{Z}^r \to B \to 0,
\end{equation*} where 
$B=\operatorname{Hom}(A,\mathbb{Z})$. Let 
$G\subseteq (\mathbb{C}^*)^r$ be the subtorus corresponding to the surjection of lattices 
$\mathbb{Z}^r \to B$, so we have 
$G\cong (\mathbb{C}^*)^{n-r}$ and 
$T=(\mathbb{C}^*)^r/G$.
We will make use of the Cox construction, which realizes the toric variety 
$X(\Delta)$ as a GIT quotient of 
$\mathbb{C}^r$ by 
$G$. Each subset 
$I \subseteq \{1,\ldots,r\}$ determines a coordinate subspace 
$E_I = \{e_i^*=0 \,|\, i\in I\} \subseteq \mathbb{C}^r$, as well as a collection of rays. Let 
$Z(\Delta) \subseteq \mathbb{C}^r$ be the union of those coordinate subspaces 
$E_I$ such that the corresponding set of rays 
$\{\rho_i \,|\, i\in I\}$ is not contained in any cone of 
$\Delta$. Then
\begin{equation*}
X(\Delta) = (\mathbb{C}^r \smallsetminus Z(\Delta) )/G.
\end{equation*}Basic facts from toric geometry say that
\begin{equation*}
B \cong \operatorname{Pic}(X) \cong H^2(X,\mathbb{Z}),
\end{equation*} and the cone of effective divisors is the image of 
$\mathbb{Z}^r_{\geq0}$ in 
$B$. Dually,
\begin{equation*}
A\cong H_2(X,\mathbb{Z}),
\end{equation*} and the cone of nef curves is the preimage of 
$\mathbb{Z}^r_{\geq0}$ under the embedding 
$\beta\colon A \hookrightarrow \mathbb{Z}^r$. This cone is written 
$A_+\subseteq A$.
When 
$X(\Delta)$ is projective, an ample line bundle 
$\mathscr{O}(D)$ corresponds to a polytope 
$P$ with normal fan 
$\Delta$. Facets of 
$P$ correspond to 
$T$-invariant divisors of 
$X$, and vertices of 
$P$ correspond to fixed points. We sometimes abuse notation by identifying vertices and fixed points, writing both 
$p\in P$ and 
$p\in X^T$.
Sections of 
$\mathscr{O}(D)$ may be described in terms of the Cox construction as follows. Suppose 
$P$ is defined as the intersection of half-spaces 
$\{u\,|\,\langle u,v_i\rangle \geq -a_i \}$, for 
$i=1,\ldots,r$; then 
$D=\sum a_i D_i$, where 
$D_1,\ldots,D_r$ are the 
$T$-invariant divisors. Let 
$f_1,\ldots,f_r$ be standard coordinates on 
$\mathbb{C}^r$, generating the Cox ring of 
$X$. Then, for non-negative integers 
$b_i$, a Laurent monomial
\begin{equation*}
x^u = \prod_{i=1}^r f_i^{b_i-a_i},
\end{equation*} is a 
$T$-equivariant section of 
$\mathscr{O}(D)$ if and only if 
$\sum (b_i-a_i)D_i = 0$ in 
$\operatorname{Pic}(X)$; equivalently, 
$u= \sum (b_i-a_i)e_i$ lies in 
$M \subset \mathbb{Z}^r$, where 
$e_1,\ldots,e_r$ is the standard basis. By construction, 
$\langle u,v_i \rangle = b_i-a_i \geq -a_i$ for each 
$i$, so 
$u\in P$.
Example 2.1. The simple case where 
$X=\mathbb{P}^1$ will be useful for illustrating constructions later. Here the space 
$N_{\mathbb{R}}\cong \mathbb{R}$ is one-dimensional, and the fan 
$\Delta$ consists of the two rays 
$\rho_1 = \mathbb{R}_{\geq0}$ and 
$\rho_2 = \mathbb{R}_{\leq0}$ meeting at the origin 
$\{0\}$. The map 
$\mathbb{Z}^2 \to N\cong \mathbb{Z}$ is given by 
$e_1 \mapsto v_1 = 1$, 
$e_2\mapsto v_2 = -1$, with kernel 
$A$ generated by 
$e_1+e_2$. So the Cox construction is the usual construction of 
$\mathbb{P}^1 = (\mathbb{C}^2 \smallsetminus \{0\}) / \mathbb{C}^*$.
Writing 
$x$ for the coordinate on 
$T\cong \mathbb{C}^*$, the action on 
$\mathbb{P}^1$ is by 
$x\cdot [a,b] = [a,x\cdot b] = [x^{-1} a, b]$. The two 
$T$-invariant divisors are 
$D_1 = [0,1]$ and 
$D_2 = [1,0]$. The corresponding polytopes are 
$P_{1} = [-1,0]$ and 
$P_{2} = [0,1]$, respectively.
3. Quasimap spaces and arc schemes
Next we review the construction of the toric quasimap space; see [Reference Givental12, Reference Morrison and Plesser16] for proofs and details. We are interested in parametrizing maps 
$f\colon \mathbb{P}^1 \to X(\Delta)$ of degree 
$d$, that is, 
$f_*[\mathbb{P}^1] = d$ in 
$A=H_2(X)$. Let 
$\operatorname{Hom}_d(\mathbb{P}^1,X)$ be the space of such maps. To describe this space, one can lift such maps to 
$\mathbb{C}^r$, where they can be specified as an 
$r$-tuple of univariate polynomials. We consider only degrees 
$d$ lying in 
$A_+$. In general, 
$A_+$ is properly contained in the semigroup of all effective curves.
For any 
$r$-tuple of non-negative integers 
$\delta=(\delta_1,\ldots,\delta_r)$, let
\begin{equation*}
\mathbb{C}^r_{\delta} = \mathbb{C}[t]_{\leq \delta_1} \oplus \cdots \oplus \mathbb{C}[t]_{\leq \delta_r},
\end{equation*} where 
$\mathbb{C}[t]_{\leq k} = \{f(t) = f^{(0)} + f^{(1)} t + \cdots + f^{(k)} t^k \}$ is the 
$(k+1)$-dimensional space of polynomials of degree at most 
$k$.
The vector space 
$\mathbb{C}^r_{\delta}$ has dimension 
$r+\delta_1+\cdots+\delta_r$, so the torus 
$(\mathbb{C}^*)^{r+\sum \delta_i}$ acts coordinate-wise. The torus 
$(\mathbb{C}^*)^r$ embeds “diagonally”, so that
\begin{equation*}
(z_1,\ldots,z_r)\cdot (f_1(t),\ldots,f_r(t)) = (z_1 f_1(t),\ldots,z_r f_r(t)).
\end{equation*} So the subtorus 
$G \subseteq (\mathbb{C}^*)^r$ also embeds in 
$(\mathbb{C}^*)^{r+\sum \delta_i}$. The quasimap space is constructed as a certain GIT quotient of 
$\mathbb{C}^r_{\delta}$ by this action of 
$G$.
For any subset 
$I \subseteq \{1,\ldots,r\}$, let 
$E_{I,\delta} \subseteq \mathbb{C}^r_\delta$ be the locus where 
$f_i\equiv0$ for each 
$i\in I$. Viewing 
$E_I = \mathbb{C}^{r-\#I}$ as a coordinate subspace and writing 
$\delta(\widehat{I})$ for the subsequence of 
$\delta$ omitting 
$i\in I$, this is the same as 
$\mathbb{C}^{r-\#I}_{\delta(\widehat{I})}$.
Let
\begin{equation*}
\quad Z(\Delta)_{\delta} = \bigcup_I E_{I,\delta},
\end{equation*} the union over 
$I$ such that 
$\{\rho_i\,|\, i\in I\}$ is not contained in a cone of 
$\Delta$.
The space of (toric) quasimaps is defined as
\begin{equation*}
\mathscr{Q}(\Delta)_{d} = (\mathbb{C}^r_{\beta(d)} \smallsetminus Z(\Delta)_{\beta(d)})/G.
\end{equation*} By construction, it contains 
$\operatorname{Hom}_d(\mathbb{P}^1,X)$ as a dense open subset.
Proposition 3.1. The toric quasimap space 
$\mathscr{Q}(\Delta)_{d}$ is a toric variety of dimension 
$n+\sum \beta(d)_i$, smooth and complete whenever 
$X(\Delta)$ is.
Let us write 
$T_d = (\mathbb{C}^*)^{r+\sum \beta(d)_i}/G$ for the torus acting on 
$\mathscr{Q}(\Delta)_d$. We are also interested in actions by various subtori. Using 
$(\mathbb{C}^*)^r \hookrightarrow (\mathbb{C}^*)^{r+\sum \beta(d)_i}$ as above, we have an inclusion of 
$T=(\mathbb{C}^*)^r/G$ in 
$T_d$, so the same torus acting on 
$X(\Delta)$ also acts on 
$\mathscr{Q}(\Delta)_d$. On the other hand, there is a loop rotation action of 
$\mathbb{C}^*$ on 
$\mathbb{C}^r_{\beta(d)}$ by
\begin{equation*}
q\cdot (f_1(t),\ldots,f_r(t)) = (f_1(q^{-1} t),\ldots,f_r(q^{-1} t)),
\end{equation*} and this descends to an action of 
$\mathbb{C}^*$ on 
$\mathscr{Q}(\Delta)_d$. We consider the induced homomorphism 
$\mathbb{T} = T \times \mathbb{C}^* \to T_d$, defining an action on 
$\mathscr{Q}(\Delta)_d$. (For 
$d \gt 0$, this is a subtorus 
$\mathbb{T} \subseteq T_d$, but for 
$d=0$, we have 
$T_d = T$ and the 
$\mathbb{C}^*$ factor of 
$\mathbb{T}$ acts trivially.) Localization with respect to 
$\mathbb{T}$ will be the primary tool in proving the main theorem.
Proposition 3.2. The fixed locus 
$(\mathscr{Q}(\Delta)_d)^{\mathbb{C}^*}$ by the loop rotation action is a union of components 
$\mathscr{Q}(\Delta)_d^{(d')}$, for each decomposition 
$d'\leq d$ (meaning 
$d=d'+d^{\prime\prime}$ is a decomposition as a sum of nef classes). Furthermore, there is a 
$T$-equivariant isomorphism 
$\mathscr{Q}(\Delta)_d^{(d')} \cong X(\Delta)$ for all 
$d'$.
Example 3.3. Continuing our example with 
$X=\mathbb{P}^1$, for any 
$d\in \mathbb{Z}_{\geq0}$, we have 
$\beta(d) = (d,d)$ and 
$\mathbb{C}^2_{\beta(d)} = \mathbb{C}[t]_{\leq d} \oplus \mathbb{C}[t]_{\leq d}$, the space of pairs of polynomials of degree at most 
$d$. Parametrizing by coefficients, this is isomorphic to 
$\mathbb{C}^{2d+2}$. The subtorus 
$G\cong\mathbb{C}^*$ acts by scaling, 
$z\cdot (f_1(t),f_2(t)) = (z f_1(t), z f_2(t) )$, and the subset 
$Z(\Delta)_\delta = \{0\}$. Put together, we see the quasimap space is 
$\mathscr{Q}(\Delta)_d \cong \mathbb{P}^{1+2d}$.
A pair 
$(f_1(t),f_2(t))$ defines a degree 
$d$ morphism 
$\mathbb{P}^1 \to \mathbb{P}^1$ if and only if the polynomials 
$f_1$ and 
$f_2$ share no common factor—including at 
$t=\infty$, so 
$t^d$ must appear in at least one of 
$f_1,f_2$ with non-zero coefficient. Thus 
$\operatorname{Hom}_d(\mathbb{P}^1,\mathbb{P}^1)$ arises as the open set defined by the non-vanishing of the resultant 
$\mathrm{Res}(f_1,f_2)$.
Writing 
$f_i = f_{i}^{(0)} + f_{i}^{(1)}t + \cdots + f_{i}^{(d)} t^d$, the action of 
$\mathbb{T} = T\times \mathbb{C}^*$ on 
$\mathscr{Q}_d \cong \mathbb{P}^{1+2d}$ is represented as
\begin{equation*}
(x,q)\cdot
\left[\begin{array}{cccc}
f_{1}^{(0)} & f_{1}^{(1)} & \cdots & f_{1}^{(d)} \\
f_{2}^{(0)} & f_{2}^{(1)} & \cdots & f_{2}^{(d)}
\end{array}\right]
=
\left[\begin{array}{cccc}
f_{1}^{(0)} & q^{-1} f_{1}^{(1)} & \cdots & q^{-d} f_{1}^{(d)} \\
x f_{2}^{(0)} & xq^{-1} f_{2}^{(1)} & \cdots & xq^{-d} f_{2}^{(d)}
\end{array}\right]
\end{equation*} (Since the coordinates 
$f_i^{(j)}$ are a basis of functions on the space 
$\mathbb{C}^2_{\beta(d)}$, they are the Cox variables for 
$\mathscr{Q}_d$, considered as a toric variety.)
From this description, restricting to the loop rotation action by setting 
$x=1$, the 
$\mathbb{C}^*$-fixed loci consists of 
$d+1$ components
\begin{equation*}
\mathscr{Q}_d^{(d')} =
\left[\begin{array}{cccccc}
0 & \cdots &0 & f_{1}^{(d')} &0& \cdots \\
0 & \cdots &0 & f_{2}^{(d')} &0& \cdots
\end{array}\right]
\cong \mathbb{P}^1,
\end{equation*} for 
$0\leq d'\leq d$.
The quasimap spaces 
$\mathscr{Q}(\Delta)_d$ form a system of closed embeddings for 
$d$ varying over 
$A_+\subset A = H_2(X,\mathbb{Z})$. Namely, given two such curve classes 
$d'\leq d$, there is a closed embedding 
$\mathscr{Q}(\Delta)_{d'}\hookrightarrow\mathscr{Q}(\Delta)_{d}$ induced by the inclusion 
$\mathbb{C}^r_{\beta(d')}\hookrightarrow\mathbb{C}^r_{\beta(d)}$. We refer to the limiting ind-variety as the toric polynomial space 
$\mathscr{Q}(\Delta)_\infty$.
The ind-variety 
$\mathscr{Q}(\Delta)_\infty$ sits inside an even larger space, the toric arc scheme of Arkhipov–Kapranov [Reference Arkhipov and Kapranov2]. The toric arc scheme is constructed in nearly the same manner as 
$\mathscr{Q}(\Delta)_d$. Let
\begin{equation*}
\mathbb{C}_\infty^r = \mathbb{C}[[t]]\oplus\cdots\oplus\mathbb{C}[[t]].
\end{equation*} This space has an infinite-dimensional torus action given by scaling coordinates of the tuple of power series. For 
$I\subset\{1,\ldots,r\}$, let the locus 
$E_{I,\infty}\subseteq \mathbb{C}^r_\infty$ be defined as the tuples where all the coefficients of 
$f_i$ vanish for all 
$i\in I$, and
\begin{equation*}
Z(\Delta)_{\infty}=\bigcup_I E_{I,\infty},
\end{equation*} where once again the union is over 
$I$ such that 
$\{\rho_i| i\in I\}$ is not contained in a cone of 
$\Delta$. The Arkhipov–Kapranov toric arc scheme is
\begin{equation*}
\Lambda^0 X = (\mathbb{C}_\infty^r \smallsetminus Z(\Delta)_\infty) / G.
\end{equation*} Just as with quasimap spaces, the torus 
$\mathbb{T}$ acts on 
$\Lambda^0 X$. The quasimap space 
$\mathscr{Q}(\Delta)_d$ embeds as a finite-dimensional invariant subvariety in 
$\Lambda^0 X$, and the following diagram commutes:
The closed points of the ind-variety 
$\mathscr{Q}(\Delta)_\infty$ (or equivalently, the union of closed points of 
$\mathscr{Q}(\Delta)_d$ over all 
$d\in A_+$) correspond to the tuples of power series inside 
$\Lambda^0 X$ where only finitely many coefficients of any given power series are non-zero. In other words, the toric polynomial space 
$\mathscr{Q}(\Delta)_\infty$ embeds into 
$\Lambda^0 X$ as the subset consisting of power series which are in fact polynomial.
Definition 3.4. Let 
$D_i$ be the 
$T$-invariant divisor corresponding to the ray 
$\rho_i$ in 
$X$. For each 
$k$ such that 
$\beta_i(d)\geq k\geq 0$ there are divisors 
$D_i^k$ in 
$\mathscr{Q}(\Delta)_d$, 
$\Lambda^0 X$, and 
$\mathscr{Q}(\Delta)_\infty$ defined by the vanishing of the 
$k^{\mathrm{th}}$ coefficient of 
$f_i(t)$. For each 
$d\in A_+$ and each 
$T$-invariant divisor 
$D=\sum_i a_i D_i$ on 
$X$, we let 
$D^d$ be the divisor 
$\sum_{i}a_i D_i^{\beta_i(d)}$ on 
$\mathscr{Q}(\Delta)_\infty$. We use the same notation for analogously defined divisors on 
$\mathscr{Q}(\Delta)_d$ and on 
$\Lambda^0 X$, the base space being clear from context.
Arkhipov and Kapranov observed that 
$\Lambda^0 X$ admits a family of self-embeddings. Recall 
$\beta$ is the inclusion 
$H_2(X,\mathbb{Z})\hookrightarrow\mathbb{Z}^r$ defined in 
$\S$ 2. An element 
$d$ in the semigroup 
$A_+$ corresponds to a one-parameter subgroup of 
$G$, and by composing with the inclusion 
$G\hookrightarrow (\mathbb{C}^*)^r$, we can write the image of 
$d$ in 
$\mathbb{Z}^r$ explicitly as the cocharacter 
$(t^{\beta(d)_1},\ldots,t^{\beta(d)_r})$ of 
$(\mathbb{C}^*)^r$.
Definition 3.5. For 
$d\in A_+$, let 
$\epsilon_d:\Lambda^0 X\rightarrow \Lambda^0 X$ be the self-embedding
\begin{equation*}
(f_1(t),\ldots,f_r(t))\mapsto (t^{\beta(d)_1} f_1(t),\ldots,t^{\beta(d)_r} f_r(t)).
\end{equation*} This restricts to a self-embedding on the polynomial space 
$\mathscr{Q}(\Delta)_\infty$ which we also denote by 
$\epsilon_d$.
These self-embeddings commute: in fact, 
$\epsilon_{d}\circ\epsilon_{d'}=\epsilon_{d+d'}$. They are evidently equivariant with respect to the 
$\mathbb{T}$-action.
Lemma 3.6. Let 
$D = \sum_i a_i D_i$ be nef on 
$X$. Then 
$D^0 = \sum_i a_i D_i^0$ restricts to a nef divisor on each 
$\mathscr{Q}(\Delta)_d$.
Proof. We will show that 
$D^0$ pairs positively with any 
$T\times \mathbb{C}^*$-invariant effective irreducible curve 
$C$ in 
$\mathscr{Q}(\Delta)_d$.
As a toric variety, 
$\mathscr{Q}(\Delta)_d$ has Cox ring variables 
$f_i^{(j)},$ for 
$1\leq i\leq r$ and 
$0\leq j\leq\beta(d)_i$, and for any 
$k, k'$, 
$\frac{f_{i}^{(k)}}{f_{i}^{(k')}}$ defines a rational function on 
$X$ with divisor 
$D_i^k-D_i^{k'}$. Thus, 
$D^0$ is equivalent to 
$D^{d'}=\sum_i a_i D_i^{\beta(d')_i}$ for any 
$d'\leq d$ in 
$A_+$.
Let 
$\epsilon_{d'}(p)$ be a 
$T\times\mathbb{C}^*$-fixed point of 
$C$, for some 
$d'\leq d$ in 
$A_+$. If 
$E$ is an ample divisor on 
$X$, then for 
$s \gt 0$, 
$D+sE$ can be moved to an effective 
$\mathbb{Q}$-divisor 
$\sum_i b_i D_i$ whose support does not contain 
$p$. Similarly, the divisor 
$D^0+sE^0$ on 
$\mathscr{Q}(\Delta)_d$ can be moved to 
$\sum_i b_i D^{\beta(d')_i}_i$, whose support does not contain 
$\epsilon_{d'}(p)$—consequently, its support does not contain 
$C$. So 
$C\cap (D^0+sE^0)\geq 0$ for all 
$s \gt 0$, and therefore 
$C\cap D^0\geq 0$ as well.
4. Lattice points and localization on the arc scheme
In this section, we study the geometry of the 
$\mathbb{T}$-action on 
$\mathscr{Q}(\Delta)_\infty$, leading to the proof of our analogue of Brion’s identity for 
$q$-weighted lattice point enumeration. First we review some of the ingredients we require, from equivariant K-theory and toric varieties.
Let 
$X$ be a complete non-singular variety equipped with the action of a torus 
$T$ with character group 
$M$. Let 
$L$ be a 
$T$-equivariant line bundle on 
$X$. Then each cohomology group 
$H^i(X,L)$ is naturally a 
$T$-representation, and the equivariant Euler characteristic of 
$L$ is defined to be the element
\begin{equation*}
\chi_T(X,L) = \sum_{i\geq 0} (-1)^i [ H^i(X,L) ],
\end{equation*} of the representation ring 
$R(T) = \mathbb{Z}[M]$. When 
$X$ has finitely many fixed points, the Atiyah-Bott localization formula says
\begin{equation}
\chi_T(X,L) = \sum_{p\in X^T} \frac{L|_p}{(1-x^{-u_1(p)})\cdots (1-x^{-u_n(p)})},
\end{equation} where 
$u_1(p),\ldots,u_d(p)$ are weights on the 
$T$-module 
$T_pX$. See [Reference Chriss and Ginzburg8, Chapter 5] or [Reference Anderson and Can1] for further details on equivariant K-theory in this context.
When 
$X$ is a complete toric variety and 
$L$ is a nef line bundle, Demazure vanishing says that all higher cohomology groups are zero:
\begin{equation}
H^i(X,L) = 0 \quad \text{for } \quad i \gt 0.
\end{equation}See [Reference Fulton11, §3.5] or [Reference Cox, Little and Schenck9, Theorem 9.2.3].
Brion uses (1) together with (2) to deduce his formula for lattice points in a polytope [Reference Brion3]. Here we will employ a variation to establish our 
$q$-weighted enumeration.
Returning to our context, we define a weighting of lattice points as follows. Let 
$\mathscr{A}=\{(H_i,v_i,a_i)\}$ be an oriented hyperplane arrangement in 
$M_{\mathbb{R}}$. Explicitly, this means that for each 
$i$ we specify a primitive vector 
$v_i$ and integer 
$a_i$ such that the hyperplane 
$H_i$ is defined by
\begin{equation}
\langle u,v_i \rangle = -a_i.
\end{equation} These choices determine positive and negative half-spaces 
$H_{i,+}$ and 
$H_{i,-}$, by replacing the “
$=$” in (3) with “
$\geq$” or “
$\leq$.”
Given such an arrangement 
$\mathscr{A}$ and any 
$u\in M$, we let
\begin{equation}
g_{\mathscr{A},u}=\prod_{i=1}^r\frac{1}{(q;q)_{\langle u,v_i \rangle + a_i }} = \prod_{i=1}^r \prod_{k=1}^{\langle u,v_i \rangle + a_i } \frac{1}{1-q^k} .
\end{equation} Now, let 
$X=X(\Delta)$ be a smooth and projective toric variety, so 
$\Delta$ is the inward normal fan of a polytope, and let 
$v_1,\ldots,v_r$ once again refer to the primitive vectors in the rays of 
$\Delta$. Let 
$a_i$ be a tuple of integers further satisfying that the intersection of positive half-spaces 
$P = \bigcap_i H_{i,+} = \bigcap_i \{u\,|\,\langle u, v_i \rangle \geq -a_i\}$ is non-empty, and each hyperplane 
$H_i$ touches 
$P$. We write 
$\mathscr{A}_P$ for the corresponding oriented hyperplane arrangement, noting that this depends not only on 
$P$ but also on the rays of 
$\Delta$. This data determines a nef line bundle 
$\mathscr{O}(D)=\mathscr{O}(\sum_i a_i D_i)$ on 
$X$ whose global sections are in natural bijection with the lattice points in 
$P$. The same data also determines a line bundle 
$\mathscr{O}(D^0) = \mathscr{O}(\sum_i a_i D^0_i)$ on each 
$\mathscr{Q}(\Delta)_d$, and on 
$\mathscr{Q}(\Delta)_\infty$.
Theorem 4.1. The equivariant Euler characteristic is given by
\begin{equation}
\chi_{\mathbb{T}}\left(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(D^0) \right) = \sum_{u\in P\cap M} g_{\mathscr{A}_P,u}\, x^u.
\end{equation}Proof. Since 
$\mathscr{O}(D)$ is nef on 
$X$, the line bundle 
$\mathscr{O}(D^0)|_{\mathscr{Q}(\Delta)_d}$ is nef on 
$\mathscr{Q}(\Delta)_d$ by Lemma 3.6, so Demazure vanishing implies that
\begin{equation*}
\chi_{\mathbb{T}}\left(\mathscr{Q}(\Delta)_d,\, \mathscr{O}(D^0) \right) = H^0\left(\mathscr{Q}(\Delta)_d,\, \mathscr{O}(D^0)\right),
\end{equation*} as 
$\mathbb{T}$-modules, for each 
$d$. There are inclusions of 
$\mathbb{T}$-modules 
$H^0\left(\mathscr{Q}(\Delta)_d,\, \mathscr{O}(D^0)\right) \hookrightarrow H^0\left(\mathscr{Q}(\Delta)_{d'},\, \mathscr{O}(D^0)\right)$ for 
$d\leq d'$, corresponding to the distinguished embeddings 
$\mathscr{Q}(\Delta)_d \hookrightarrow \mathscr{Q}(\Delta)_{d'}$. The infinite-dimensional 
$\mathbb{T}$-module 
$H^0\left(\mathscr{Q}(\Delta)_\infty,\, \mathscr{O}(D^0)\right)$, by definition, is the union of these, and 
$\chi_{\mathbb{T}}\left(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(D^0) \right)$ is its graded character. To compute it, we determine the characters of each 
$H^0\left(\mathscr{Q}(\Delta)_d,\, \mathscr{O}(D^0)\right)$ and take the limit as 
$d\to \infty$.
As a toric variety, 
$\mathscr{Q}(\Delta)_d$ has Cox ring variables 
$f^{(j)}_i$, for 
$1\leq i \leq r$ and 
$0 \leq j \leq \beta(d)_i$. A basis for the sections of 
$\mathscr{O}(D^0)$ consists of monomials 
$\left(\prod_{i} (f_i^{(0)})^{-a_i}\right)\cdot\left(\prod_{i}\prod_{j=0}^{\beta(d)_i} (f_i^{(j)})^{b_{i,j}}\right)$ such that the 
$b_{i,j}$ are non-negative, and 
$\sum_{i}\left((\sum_{j=0}^{\beta(d)_i} b_{i,j})-a_i\right)D_i=0$ in 
$\mathrm{Pic}(X)$.
We must compute the 
$\mathbb{T}$-character of the span of these sections. On 
$X$, an element of the distinguished basis of sections of 
$\mathscr{O}(D)$ corresponds to a 
$T$-character 
$x^u$ where 
$u$ is a lattice point in 
$P$, or equivalently a choice of 
$b_i\geq 0$ such that 
$\sum_i (b_i-a_i)e_i\in \mathbb{Z}^r$ is the image of some 
$u\in M$. These choices are related via the identity 
$b_i=\langle u,v_i \rangle +a_i$.
With these 
$b_i$ fixed, consider the sections 
$\left(\prod_{i} (f_i^{(0)})^{-a_i}\right)\cdot\left(\prod_{i}\prod_{j=0}^{\beta(d)_i} (f_i^{(j)})^{b_{i,j}}\right)$ such that 
$\sum_{j=0}^{\beta(d)_i} b_{i,j}=b_i$. The character of 
$\prod_{i} f_i^{b_i-a_i}$ is 
$x^u$, as is that of 
$\prod_{i} (f^{(0)}_i)^{b_i-a_i}$. So 
$\left(\prod_{i} (f_i^{(0)})^{-a_i}\right)\cdot\left(\prod_{i}\prod_{j=0}^{\beta(d)_i} (f_i^{(j)})^{b_{i,j}}\right)$ has character 
$x^u\prod_{i}\prod_{j=0}^{\beta(d)_i} (q^j)^{b_{i,j}}$. So the coefficient of 
$x^u$ in the graded character of 
$H^0\left(\mathscr{Q}(\Delta)_d,\, \mathscr{O}(D^0)\right)$ is
\begin{equation*}
\sum \prod_i \prod_{j=0}^{\beta(d)_i} (q^j)^{b_{i,j}},
\end{equation*} the sum over 
$b_{i,j}\geq 0$ such that 
$\sum_{j=0}^{\beta(d)_i} b_{i,j} = b_i$.
In the limit as 
$d\rightarrow\infty$, the upper bound disappears for the indices 
$j$ of 
$b_{i,j}$. The resulting sum is over all choices of weakly increasing sequences 
$0\leq c_{i,1}\leq c_{i,2} \leq \cdots \leq c_{i,b_i}$ for each 
$i$ from 
$1$ to 
$r$, where the summand is the statistic
\begin{equation*}
\prod_{i=1}^r\prod_{j=1}^{b_i} q^{c_{i,j}}.
\end{equation*} Holding all but one weakly increasing sequence fixed, we see that the whole sum must factor into a product over 
$i$:
\begin{equation*}
\prod_{i=1}^r \sum_{0\leq c_{i,1}\leq c_{i,2} \leq \ldots \leq c_{i,b_i}} \prod_{j=1}^{b_i} q^{c_{i,j}}.
\end{equation*}But
\begin{equation*}
\sum_{0\leq c_{i,1}\leq c_{i,2} \leq \ldots \leq c_{i,b_i}} \prod_{j=1}^{b_i} q^{c_{i,j}}=\frac{1}{(q;q)_{b_i}}=\frac{1}{(q;q)_{\langle u,v_i\rangle + a_i}}.
\end{equation*} Thus the coefficient of 
$x^u$ in the character of 
$H^0(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(D^0))$ is
\begin{equation*}
\prod_{i=1}^r\frac{1}{(q;q)_{\langle u,v_i\rangle + a_i}}= g_{\mathscr{A}_P,u}.
\end{equation*}This proves the theorem.
Next we describe the 
$\mathbb{T}$-fixed points of 
$\mathscr{Q}(\Delta)_\infty$, and decompose the corresponding tangent spaces into characters of 
$\mathbb{T}$.
Proposition 4.2. The 
$\mathbb{T}$-fixed points of 
$\mathscr{Q}(\Delta)_\infty$ are in bijection with pairs (
$p,d)$, with 
$p\in X^T$ and 
$d\in A_+$.
Proof. The locus fixed by loop rotation is easy to determine. By definition 
$\mathscr{Q}(\Delta)_\infty$ is a union of 
$\mathscr{Q}(\Delta)_d$ as 
$d$ varies over 
$A_+$, and the subvariety 
$\mathscr{Q}(\Delta)_0\hookrightarrow\mathscr{Q}(\Delta)_\infty$ is certainly 
$\mathbb{C}^*$-fixed. To exhaust the 
$\mathbb{C}^*$-fixed components in 
$\mathscr{Q}(\Delta)_d$ for higher 
$d$, it is enough to take the disjoint union of 
$\epsilon_d(\mathscr{Q}(\Delta)_0)$ over 
$d\in A_+$, because 
$\epsilon_d(\mathscr{Q}(\Delta)_0)=\mathscr{Q}(\Delta)_d^{(d)}\subset\mathscr{Q}(\Delta)_d\subset\mathscr{Q}(\Delta)_\infty$. The subvariety 
$\mathscr{Q}(\Delta)_0$ is 
$T$-equivariantly isomorphic to 
$X$, so its 
$T$-fixed points are in natural bijection with those of 
$X$. The same holds for each 
$\epsilon_d(\mathscr{Q}(\Delta)_0)$, so the fixed points are simply 
$\epsilon_d(p)$ for 
$d\in A_+$ and 
$p\in X^T=\mathscr{Q}(\Delta)_0$.
Recall from the introduction that for each vertex 
$p$ of a smooth polytope 
$P$, there is a subset 
$I(p)\subseteq \{1,\ldots,r\}$ so that the 
$v_i$ for 
$i\in I(p)$ are the inward normal vectors of the facets containing 
$p$, and 
$\{u_i(p)\,|\, i\in I(p)\}$ is the basis of 
$M$ dual to 
$\{v_i\,|\, i\in I(p)\}$. We are using the notation
\begin{equation}
\mathsf{J}_{d,p} = \left(\prod_{i\in I(p)} \frac{1}{(x^{u_i(p)} q^{-1}; q^{-1})_{\beta(d)_i} }\right)\left(\prod_{j\not\in I(p)}\frac{1}{(q^{-1}; q^{-1})_{\beta(d)_j} }\right).
\end{equation}Proposition 4.3. Let 
$p$ be a fixed point in 
$X$, corresponding to a vertex of 
$P$. The equivariant multiplicity to 
$\mathscr{Q}(\Delta)_\infty$ at 
$\epsilon_d(p)$ is equal to
\begin{equation*}
\left(\frac{1}{(q;q)_\infty}\right)^{r-n}\frac{\mathsf{J}_{d,p}}{\prod_{i\in I(p)}( x^{u_i(p)};q )_\infty}.
\end{equation*} In our usage of the term, “equivariant multiplicity” is a synonym for “Bott denominator”: it is the product of factors 
$(1-\chi)^{-1}$ over characters 
$\chi$ occurring in the cotangent space to a variety at a fixed point, as appears in the summands of the Atiyah–Bott formula (1).
Proof. The characters appearing in the denominator of 
$\mathsf{J}_{d,p}$ are the conormal weights to 
$\epsilon_d(\Lambda^0 X)\subseteq \Lambda^0X$ at the fixed point 
$\epsilon_d(p)$. The other characters in the formula are cotangent weights at 
$p \in X =\mathscr{Q}(\Delta)_0 \subset \mathscr{Q}(\Delta)_\infty$, as one sees from the Cox description of 
$\mathscr{Q}(\Delta)_d$ for large enough 
$d$.
Now we can state and prove our main theorem:
Theorem 4.4. Let 
$P$ be a smooth polytope, with notation as above. Then
\begin{equation}
\sum_{u \in P \cap M} g_{\mathscr{A}_P,u}\, x^u = \frac{1}{(q;q)_\infty^{r-n}} \sum_{p \in V(P)} x^p\sum_{d\in A_+}\frac{q^{\sum a_i\, \beta(d)_i} \cdot \mathsf{J}_{d,p}}{\prod_{i\in I(p)}( x^{u_i(p)};q )_\infty },
\end{equation} where 
$g_{\mathscr{A}_P,u}$ is the rational function defined in (4), and 
$V(P)$ is the set of vertices of 
$P$.
In the radially symmetric case, so 
$\sum v_i = 0$, the theorem from the introduction follows immediately by multiplying both sides by 
$(q;q)_{|a|}$, where 
$|a|=a_1+\cdots+a_r$ as before.
Proof. The polytope 
$P$ determines a toric variety 
$X$, normal fan 
$\Delta$ and a line bundle 
$\mathscr{O}(\sum_i a_i D_i)$. The left-hand side is simply 
$\chi_{\mathbb{T}}(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(\sum_i a_i D_i^0))$, by Theorem 4.1.
The right-hand side comes by applying the Atiyah-Bott formula to compute the Euler characteristic of 
$\mathscr{O}(\sum_i a_i D^0_i)$. This is a sum over the 
$\mathbb{T}$-fixed points 
$\epsilon_d(p)$ of the product of the 
$\mathbb{T}$-character of the line 
$i^*_{\epsilon_d(p)}\mathscr{O}(\sum_i a_i D^0_i)$ and the equivariant multiplicity at 
$\epsilon_d(p)$. The latter is computed by Proposition 4.3. The former is equal to 
$x^p \cdot q^{\sum a_i\, \beta(d)_i}$, as can be seen from the Cox construction of 
$\mathscr{Q}(\Delta)_{d'}$ for 
$d'\gg d$.
Example 4.5. Continuing our example with 
$\mathbb{P}^1$. Using coordinates 
$[f_i^{(j)}]$ as before, the fixed points for action of 
$\mathbb{T} = T\times\mathbb{C}^*$ on 
$\mathscr{Q}_\infty$ are 
$p_k^{(d)}$, for 
$k=1,2$ and 
$d\geq 0$, given by
\begin{align*}
p_k^{(d)} &= \{f_i^{(j)}=0 \text{for all } i\neq k, j\neq d\}.
\end{align*} For 
$k=1$, the tangent characters are
\begin{align*}
q^{d}, \ldots, q, q^{-1}, q^{-2}, \ldots; \\
xq^{d}, \ldots, xq, x, xq^{-1}, xq^{-2}, \ldots;
\end{align*} they are similar but with 
$x$ replaced by 
$x^{-1}$ if 
$k=2$. For instance, the action on the tangent space to 
$p=p_1^{(2)}$ is by
\begin{align*}
&(x,q)\cdot
\left[\begin{array}{ccccccc}
* & * & 1 & * & * & * & \cdots\\
* & * & * & * & * & * & \cdots
\end{array}\right]\\
&\quad =
\left[\begin{array}{ccccccc}
* & q^{-1} * & q^{-2} & q^{-3} * & q^{-4} * & q^{-5} * & \cdots \\
x* & xq^{-1} * & xq^{-2} * & xq^{-3} * & xq^{-4} * & xq^{-5} * & \cdots
\end{array}\right] \\
&\quad = \left[\begin{array}{ccccccc}
q^{2}* & q * & 1 & q^{-1} * & q^{-2} * & q^{-3} * & \cdots \\
xq^{2}* & xq * & x * & xq^{-1} * & xq^{-2} * & xq^{-3} * & \cdots
\end{array}\right].
\end{align*}So the equivariant multiplicity at this fixed point is
\begin{align*}
&\frac{1}{(1-q^{-2})(1-q^{-1})}\cdot \frac{1}{(1-x^{-1}q^{-2})(1-x^{-1}q^{-1})} \\
&\times \cdot \frac{1}{(1-q)(1-q^{2})(1-q^{3}) \cdots} \cdot \frac{1}{(1-x^{-1})(1-x^{-1}q)(1-x^{-1}q^{2})(1-x^{-1}q^{3}) \cdots} \\
&= \frac{1}{(q^{-1};q^{-1})_2}\cdot \frac{1}{(x^{-1};q^{-1})_2} \cdot \frac{1}{(q;q)_\infty}\cdot\frac{1}{(x^{-1};q)_{\infty}}.
\end{align*} The first two factors come from the conormal weights to 
$\epsilon_d(\mathscr{Q}_\infty)$, and agree with 
$\mathsf{J}_{d,p}$. The other two factors are the same as the cotangent weights at 
$p_1^{(0)}$.
To apply Theorem 4.4 to the polytope 
$P = [0,2]$, which corresponds to the divisor 
$2D_2$ (where 
$D_2$ is the point 
$[1,0]$), recall that the defining inequalities are 
$\langle u,v_i \rangle \geq -a_i$, with 
$v_1=1$, 
$a_1=0$, 
$v_2=-1$, 
$a_2=2$. (See Example 2.1.) The LHS of (7) is
\begin{equation*}
\frac{1}{(1-q)(1-q^{2})} + \frac{x}{(1-q)^2} + \frac{x^2}{(1-q)(1-q^2)}.
\end{equation*} The RHS is the sum over 
$d\geq 0$ of terms
\begin{align*}
&\frac{1}{(q;q)_\infty}\cdot \frac{q^{2d}}{(q^{-1};q^{-1})_d\cdot (x;q^{-1})_d \cdot(x;q)_{\infty}} \\
&+ \frac{x^2}{(q;q)_\infty}\cdot\frac{q^{2d}}{(q^{-1};q^{-1})_d \cdot (x^{-1};q^{-1})_d \cdot (x^{-1};q)_{\infty}}.
\end{align*} Multiplying both sides by 
$(q;q)_2 = (1-q)(1-q^2)$, we recover the 
$m=2$ case of (*q) from the Introduction. Sending 
$q\to0$, only the 
$d=0$ terms survive on the RHS, and we recover (*).
Remark 4.6. Say a fan 
$\Delta$ is radially symmetric if the primitive generators of its rays sum to 
$0$. The normal fan of any polytope 
$P$ can be refined to a smooth, radially symmetric fan 
$\Delta$, as follows. Given any polytope 
$P$, let 
$\Delta_0$ be its normal fan. Refine 
$\Delta_0$ to obtain a fan 
$\Delta_1$ such that for every cone 
$\sigma\in\Delta_1$, 
$-\sigma$ is also in 
$\Delta_1$; in particular its primitive ray generators sum to 
$0$. Further refine 
$\Delta_1$ to obtain a smooth fan 
$\Delta$ with the same property. Now one can write 
$P=\bigcap \{u \,|\, \langle u,v_i\rangle \geq -a_i \}$ with some redundant inequalities.
This lets us remove the radial symmetry and smoothness hypotheses on 
$P$. Theorem 4.4 applies to the fan 
$\Delta$, as does the formula of Theorem 1. However, only for smooth, radially symmetric 
$P$ are those formulas canonical.
Remark 4.7. When 
$X$ is a product of projective spaces, the rational function 
$\mathsf{J}_{d,p}$ is equal to the restriction at 
$p$ of the 
$d^{\mathrm{th}}$ term of the J-function for quantum K-theory. More generally, it is a localization of the K-theoretic I-function; see [Reference Givental13, Reference Givental and Tonita14].
5. Generalized Rogers–Szegő polynomials and Jackson partial derivatives
In this section, we study the action of difference operators on 
$q$-series appearing on the left-hand side of Theorem 4.4.
Let 
$\Delta$ be a smooth fan, with primitive ray generators 
$v_1,\ldots, v_r$ summing to 
$0$. Let 
$D=\sum_i a_i D_i$ be an effective 
$T$-divisor on the associated toric variety 
$X$. The condition that 
$D$ is effective is equivalent to the statement that the Newton polytope 
$P=\bigcap_i \{u \,|\, \langle u, v_i\rangle \geq -a_i \}$ contains an element of 
$M$. Let 
$|a| = \sum_i a_i$. We define the Rogers–Szegő polynomial of 
$D$ by
\begin{equation}
RS_D(x;q) = \sum_{u\in P\cap M}\left[\begin{array}{c} |a| \\ \langle u, v_1\rangle + a_1,\ldots,\langle u, v_r\rangle + a_r \end{array}\right]_qx^u.
\end{equation}The following comes from Theorem 4.1.
Proposition 5.1. Let 
$D=\sum_i a_i D_i$ be a 
$T$-invariant nef divisor on a smooth complete toric variety with fan 
$\Delta$ and rays 
$v_1,\ldots,v_r$ summing to 
$0$. Then
\begin{equation*}
(q;q)_{|a|}\cdot \chi_{\mathbb{T}}\left( \mathscr{Q}(\Delta)_\infty,\mathscr{O}(D^0)\right)=RS_D(x;q).
\end{equation*} For generic values of 
$c$, the polytope 
$P$ corresponding to the toric divisor 
$D$ is exactly the Newton polytope of 
$RS_D(x;c)$. If the 
$a_i$ satisfy the further condition that each hyperplane 
$H_i = \{u\,|\, \langle u, v_i\rangle = -a_i \}$ touches 
$P$, we write 
$RS_P(x;q)$ for 
$RS_D(x;q)$. (Given 
$P$, the 
$a_i$ satisfying this further condition are unique.) Though it is not reflected in the notation, the polynomial 
$RS_P(x;q)$ depends on both 
$P$ and the rays of 
$\Delta$.
Next we prove some facts about the action of difference operators on 
$RS_D(x;q)$. For explicitness, we henceforth identify 
$N$ and 
$M$ with 
$\mathbb{Z}^n$. The 
$i^{th}$ 
$q$-shift operator 
$T_{i,q}$ acts on 
$f(x_1,\ldots,x_n)$ by
\begin{equation*}
T_{i,q}:f(x_1,\ldots,x_i,\ldots,x_n)\mapsto f(x_1,\ldots,qx_i,\ldots,x_n),
\end{equation*} and the 
$i$th partial 
$q$-derivative (or Jackson derivative) acts on 
$f(x_1,\ldots,x_n)$ by
\begin{equation*}
\left( \frac{d}{dx_i}\right)_q:f \mapsto \frac{f-T_{i,q}f}{(1-q)x_i}.
\end{equation*}The behavior of these operators on Rogers–Szegő polynomials is best with some conditions on the polytopes and fans involved:
Definition 5.2. Given a fan 
$\Delta$ in 
$\mathbb{R}^n$, fix the standard basis of 
$\mathbb{R}^n$ and order the rays of 
$\Delta$ so that 
$v_1=e_1,\ldots,v_n=e_n$. We say an effective divisor 
$D=\sum_i a_i D_i$ on 
$X$ is first-orthant if its corresponding polytope 
$P$ is contained in 
$(\mathbb{R}_{\geq 0})^n$, with 
$a_1=\cdots=a_n=0$.
For each 
$1\leq i\leq n$, we denote by 
$P_i$ the (possibly empty) polytope in 
$(\mathbb{R}_{\geq 0})^n$ obtained by first replacing the condition 
$\{u\,|\, \langle u, v_i\rangle \geq 0 \}$ with 
$\{u \,|\, \langle u, v_i\rangle \geq 1 \}$, and then translating the result by 
$-e_i$. So 
$P_i$ is defined by 
$\left(\bigcap_{j=1}^n \{u\, |\, \langle u, v_j\rangle \geq 0 \}\right)\cap\left(\bigcap_{j=n+1}^r \{u \,|\, \langle u+e_i, v_j\rangle \geq -a_j \}\right)$.
Proposition 5.3. The partial 
$q$-derivatives of 
$RS_D(x;q)$ for 
$D$ a first-orthant divisor are given by
\begin{equation}
\left( \frac{d}{dx_i}\right)_q RS_D(x;q) = \left[|a|\right]_q RS_{D+\sum_{j=n+1}^{r}\langle e_i, v_j\rangle D_j}(x;q),
\end{equation} where 
$[b]_q = \frac{1-q^b}{1-q} = 1+\cdots+q^{b-1}$ for an integer 
$b$.
Proof. This is a straightforward computation:
\begin{align*}
\left( \frac{d}{dx_i}\right)_qRS_D(x;q) = & \sum_{u\in P\cap M}\left[\begin{array}{c} |a| \\ \langle u,v_1\rangle+a_1,\ldots,\langle u,v_r\rangle + a_r \end{array}\right]_q \frac{1-T_{i,q}}{(1-q)x_i}x^u\\
= & \sum_{\substack{u\in P\cap M,\\ \langle u,v_i\rangle \geq 1}}\left[\begin{array}{c} |a| \\ \langle u,v_1\rangle+a_1,\ldots,\langle u,v_r \rangle + a_r \end{array}\right]_q \frac{(1-q^{\langle u,v_i\rangle})}{(1-q)}\frac{x^u}{x_i}.
\end{align*} Replacing 
$u$ with 
$u+e_i$, this becomes
\begin{align*}
& \sum_{u\in P_i\cap M}\left[\begin{array}{c} |a| \\ \ldots,\langle u,v_i+e_i\rangle,\ldots \end{array}\right]_q \frac{(1-q^{\langle u+e_i, v_i \rangle})}{(1-q)}x^u,\\
= & \sum_{u\in P_i\cap M}\left[\begin{array}{c} |a| \\ \ldots,\langle u, v_i\rangle+1,\ldots \end{array}\right]_q \frac{(1-q^{\langle u, v_i\rangle+1})}{(1-q)}x^u,
\end{align*} noting that 
$a_i=0$ by assumption. We have
\begin{align*}
& \left[\begin{array}{c} |a| \\ \ldots,\langle u, v_i\rangle+1,\ldots \end{array}\right]_q (1-q^{\langle u,v_i\rangle+1})\\
= & \left[\begin{array}{c} |a|-1 \\ \ldots,\langle u, v_i\rangle,\ldots \end{array}\right]_q (1-q^{|a|}).
\end{align*}Thus,
\begin{equation*}
\left( \frac{d}{dx_i}\right)_qRS_D(x;q) = \sum_{u\in P_i\cap M}\left[\begin{array}{c} |a|-1 \\ \langle u, v_1\rangle,\ldots,\langle u,v_n\rangle,\ldots,\langle u+e_i, v_r\rangle + a_r \end{array}\right]_q \frac{(1-q^{|a|})}{(1-q)}x^u,
\end{equation*} which is exactly 
$[|a|]_q RS_{D+\sum_{j=n+1}^{r}\langle e_i,v_j\rangle D_j}(x;q)$.
Theorem 5.4 Let 
$X$ be a smooth complete toric variety. The 
$\mathbb{Q}(q)$-linear span of 
$RS_D(x;q)$ for D a first-orthant divisor on 
$X$ is an indecomposable representation of the (commutative) algebra spanned by 
$\left( \frac{d}{dx_1}\right)_q,\ldots,\left( \frac{d}{dx_n}\right)_q$.
Proof. Let 
$D$ be any first-orthant divisor, and 
$P$ the Newton polytope of 
$D$. Let 
$(i_1,\ldots,i_n)\in P\cap M$ maximize the value of 
$x_1+\cdots+x_n$ on 
$P$. Then we can calculate that 
$\left( \frac{d}{dx_1}\right)^{i_1}_q\cdots\left( \frac{d}{dx_n}\right)_q^{i_n}RS_{D}(x;q)$ is a non-zero element of 
$\mathbb{C}(q)$: For any other point 
$(j_1,\ldots,j_n)$ in 
$P\cap M$, there is some index 
$k$ such that 
$i_k \gt j_k$, so 
$\left( \frac{d}{dx_k}\right)_q^{i_k} x_1^{j_1}\cdots x_n^{j_n}=0$. On the other hand, 
$\left( \frac{d}{dx_1}\right)^{i_1}_q\cdots\left( \frac{d}{dx_n}\right)_q^{i_n}x_1^{i_1}\cdots x_n^{i_n} = [i_1]_q\cdots [i_n]_q$.
Example 5.5. Let 
$X$ be 
$\mathbb{P}^n$, with fan given by 
$v_i=e_i$ for 
$i=1,\ldots,n$, and 
$ v_{n+1}=-e_1-\cdots-e_n$. The generalized Rogers–Szegő polynomial 
$RS_{kD_{n+1}}(x;q)$ is the classical multivariate Rogers–Szegő polynomial
\begin{equation}
RS_{k,n}=\sum_{i_0+i_1+\cdots+i_n = k} \left[\begin{array}{c} k \\ i_0,i_1,\ldots,i_n \end{array}\right]_qx_1^{i_1}\cdots x_n^{i_n},
\end{equation} which is a specialization of a single-row Macdonald polynomial [Reference Hikami15, Reference Szego19]. In the algebra generated by the 
$x_i$ and 
$\left(\frac{d}{dx_i} \right)_q$, there are operators
\begin{equation*}
R_i := \sum_{l=0}^{n} e_{l+1}(x)(q-1)^l\left(\frac{d}{dx_i}\right)_q^{l}.
\end{equation*} Setting 
$L_i:=\left(\frac{d}{dx_i} \right)_q$, we have the identities
\begin{equation*}
R_i(RS_{k-1,n})=RS_{k,n}, \quad L_i (RS_{k,n}) = [k]_q RS_{k-1,n}, \quad \text{and } \quad [L_i,R_i](RS_{k,n}) =q^{k}RS_{k,n}.
\end{equation*}6. Asymptotics and an analogue of DH measure
We now prove some results about measures derived from 
$\chi_{\mathbb{T}}(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(D^0))$. We start outside of the radially symmetric case, but specialize to it later. In particular, we will end with a limit theorem describing the asymptotic behavior of 
$RS_{kD}(x;q)$ as 
$k$ goes to infinity.
We again identify 
$M$ with 
$\mathbb{Z}^n$. For a Laurent polynomial in 
$x_i$ with 
$q$-series coefficients, 
$f(x) = \sum_u a_u(q)x^u$, its Fourier transform is defined as follows:
Definition 6.1. Let 
$\delta_u(y)$ be the Dirac measure at 
$u\in \mathbb{R}^n$. Let the measure 
$FT(f(x))$ be
\begin{equation*}
FT(f(x)) = \sum_u a_u(q)\delta_u(y).
\end{equation*} Note that if 
$g(x)$ is the function 
$\sum_u a_u(q)e^{-iu\cdot x}$ for 
$x\in \mathbb{R}^n$, and 
$q$ is chosen to that the 
$a_u(q)$ are real numbers, then 
$FT(f(x))$ is indeed the Fourier transform of 
$g(x)$.
Let 
$\tau_c:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be dilation by 
$c$. For a 
$T=(\mathbb{C}^*)^n$-equivariant line bundle on a projective 
$T$-variety 
$Y$, DH measures can be defined algebraically, following [Reference Brion and Procesi4], as
\begin{equation}
DH(Y,L) : = \lim_{k\rightarrow \infty} (\tau_k)_*\left( \frac{FT\left(\chi_T(Y,L^{\otimes k})\right)}{k^{\dim X}}\right).
\end{equation} If 
$Y$ is instead an ind-projective ind-variety such as 
$\mathscr{Q}(\Delta)_\infty$, the measure defined above no longer makes sense. The factor 
$k^{\dim X}$ is not well-defined, and 
$FT(\chi_T(Y,L^{\otimes k}))$ may no longer properly define a distribution.
We let 
$Y=\mathscr{Q}(\Delta)_\infty$ be the quasimap ind-variety of a toric variety 
$X=X(\Delta)$, which has an additional 
$\mathbb{C}^*$-action that can be used to grade 
$\chi_T(Y,L^{\otimes k})$. Then, we define the following measure for a nef divisor 
$D$ on 
$X$.
Definition 6.2. Let 
$D$ be a 
$T$-invariant nef divisor on a smooth complete toric variety 
$X$. The probability measure associated to 
$D$ is
\begin{equation}
\mu_D = \lim_{q\rightarrow 1} \frac{FT(\chi_{\mathbb{T}}(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(D^0)))}{\chi_{\mathbb{T}}(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(D^0))|_{x=1}}.
\end{equation}In the radially symmetric case, this becomes
\begin{equation*}
\mu_D = \lim_{q\rightarrow 1}\frac{FT(RS_D(x;q))}{RS_D(1;q)},
\end{equation*}by Proposition 5.1.
It is easy to check that 
$\mu_D$ is a probability measure. The following proposition describes the support of 
$\mu_D$. We use the notation 
$v_\Delta$ for the sum of all primitive ray generators:
\begin{equation*}
v_\Delta = \sum_i v_i.
\end{equation*}Proposition 6.3. Let 
$D=\sum_i a_i D_i$, and let 
$F$ be the face of the corresponding polytope 
$P=\bigcap_i \{u\,|\, \langle u, v_i\rangle \geq -a_i \}$ where the function 
$u\mapsto \langle u,v_\Delta \rangle$ is maximized. Then the measure 
$\mu_D$ is expressed by the formula
\begin{equation}
\mu_D := \frac{\sum_{F\cap M}\binom{\langle u, v_\Delta\rangle + |a|}{\langle u, v_1\rangle + a_1,\ldots,\langle u, v_r\rangle +a_r}\delta_u}{\sum_{F\cap M}\binom{\langle u, v_\Delta\rangle + |a|}{\langle u, v_1\rangle + a_1,\ldots,\langle u, v_r\rangle +a_r}}.
\end{equation} In particular, 
$\mu_D$ is supported on 
$F$.
Proof. By Theorem 4.1, we can rewrite 
$\mu_D$ as
\begin{equation*}
\mu_D = \lim_{q\rightarrow 1} \sum_{u\in P\cap M} \frac{g_{\mathscr{A}_{P,u}}}{\sum_{u'\in P\cap M}g_{\mathscr{A}_{P,u'}}}\delta_u.
\end{equation*} Let 
$u$ be in 
$P\cap M$. From (4),
\begin{equation*}
g_{\mathscr{A}_{P,u}}=\prod_{i=1}^r\frac{1}{(q;q)_{\langle u, v_i\rangle+a_i}}.
\end{equation*} If 
$u_F$ is any element of 
$F$, we have 
$c_F:=\langle u_F, v_{\Delta}\rangle+|a| \geq \langle u, v_{\Delta}\rangle + |a|$, with equality precisely when 
$u\in F$. Thus,
\begin{equation*}
(q;q)_{c_F}\cdot g_{\mathscr{A}_{P,u}} = \left( \prod^{c_F}_{l = 1+ \langle u, v_\Delta\rangle + |a|}(1-q^l)\right)\cdot \binom{\langle u, v_\Delta\rangle + |a|}{\langle u, v_1\rangle + a_1, \ldots, \langle u, v_r\rangle + a_r}_q.
\end{equation*} The first factor on the right-hand side is 
$1$ if 
$u\in F$, and vanishes at 
$q=1$ otherwise, so
\begin{equation*}
\lim_{q\rightarrow 1} (q;q)_{c_F}\cdot g_{\mathscr{A}_P,u}(q) =
\begin{cases}
\binom{\langle u, v_\Delta\rangle + |a|}{\langle u, v_1\rangle + a_1, \ldots, \langle u, v_r\rangle + a_r} & u \text{in } F\cap M,\\
0 & \text{otherwise.}
\end{cases}
\end{equation*}So,
\begin{equation*}
\mu_D = \lim_{q\rightarrow 1} \sum_{u\in P\cap M}\frac{g_{\mathscr{A}_P,u}}{\sum_{u'\in P\cap M} g_{\mathscr{A}_P,u'}}\delta_u = \lim_{q\rightarrow 1} \sum_{u\in P\cap M}\frac{(q;q)_{c_F}}{(q;q)_{c_F}}\cdot \frac{g_{\mathscr{A}_P,u}}{\sum_{u'\in P\cap M} g_{\mathscr{A}_P,u'}}\delta_u,
\end{equation*}which simplifies to the proposition.
Note that in the radially symmetric case, 
$F=P$.
Example 6.4. Let 
$X=\mathbb{P}^1$, with conventions as in our running example, and let 
$D=D_2$ be the point 
$[1,0]$, so 
$P=\bigcap_i \{u\,|\, \langle u, v_i\rangle \geq -a_i \}$ is the interval 
$[0,1]$ in 
$\mathbb{R}$. Then
\begin{equation*}
\mu_{kD} = \frac{1}{2^k}\sum_{l=0}^k\binom{k}{l}\delta_l.
\end{equation*} It is tempting to use 
$\mu_{kD}$ as a replacement of 
$\frac{FT(\chi_T(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(kD^0)))}{k^{\dim \mathscr{Q}(\Delta)_{\infty}}}$ in the definition of DH measure. This produces a measure whose support must be contained in 
$P$, as for the usual DH measure. However, in the limit 
$\lim_{k\rightarrow\infty}(\tau_k)_*\mu_{kD}$, the measure concentrates too much to retain useful information about 
$P$, as illustrated in the following example.
Example 6.5. Let 
$D$ be as in Example 6.4. Then, let 
$\chi_{(\tau_k)_*\mu_{kD}}(x) = \int_{\mathbb{R}} e^{iy x}(\tau_k)_*\mu_{kD}(y)$ be the characteristic function (see e.g. [Reference Durrett10]) of the measure 
$(\tau_k)_*\mu_{kD}$. Explicitly,
\begin{align*}
\chi_{(\tau_k)_*\mu_{kD}}(x) &= \frac{1}{2^k}\sum_{l=0}^k\binom{k}{l} e^{ilx/k} = \left(\frac{1+e^{ix/k}}{2}\right)^k\\
& = e^{ix/2}\left(\frac{e^{-ix/2k}+e^{ix/2k}}{2}\right)^k = e^{ix/2}\cos^k(x/2k).
\end{align*} So 
$\lim_{k\rightarrow\infty}\chi_{(\tau_k)_*\mu_{kD}}(x)=e^{ix/2}$, and correspondingly 
$\lim_{k\rightarrow\infty}(\tau_k)_*\mu_{kD}(y) = \delta_{1/2}(y)$.
This motivates the following definition. Once again, 
$X=X(\Delta)$ is a smooth projective toric variety, 
$D$ a 
$T$-invariant nef divisor on 
$X$, and 
$D^0$ the corresponding divisor on the quasimap ind-variety 
$\mathscr{Q}(\Delta)_\infty$. We assume now that the generators 
$v_i$ of the rays of 
$\Delta$ sum to 
$0$; in other words, 
$X$ has an ample divisor whose corresponding polytope is radially symmetric. For a measure 
$\mu$ on 
$M_{\mathbb{R}}\cong \mathbb{R}^n$, let 
$E_{\mu} =(\int_{\mathbb{R}^n}y_id\mu)$ denote the corresponding expected value of the vector 
$(y_1,\ldots,y_n)$.
Definition 6.6. The centered loop-
$DH$ measure associated to 
$D$ is
\begin{equation*}
\nu_{D,\textrm{cent}} = \lim_{k\rightarrow\infty}(\tau_{\sqrt k})_*\left( \mu_{kD} \ast \delta_{-E_{\mu_{kD}}}\right).
\end{equation*} We define the potential of 
$D$ as
\begin{equation*}
\varphi_D(m) = \prod_{i=1}^r \left(\langle m, v_i\rangle + a_i\right)^{\left(\langle m, v_i\rangle + a_i\right)}.
\end{equation*} On the Newton polytope of 
$D$, 
$\varphi_D$ has a unique minimum, which we call 
$m_D$. We define the loop-
$DH$ measure to be
\begin{equation*}
\nu_D = \nu_{D,\textrm{cent}}\ast \delta_{m_D}.
\end{equation*} The measures 
$\nu_{D,\textrm{cent}}$ and 
$\nu_D$ are Gaussian and can be expressed naturally in terms of the fan of 
$X$. Let 
$L_D$ be the linear subspace of 
$M_{\mathbb{R}}$ generated by differences of vectors in the Newton polytope of 
$D$.
Theorem 6.7 Let 
$X$ be a smooth complete toric variety with a radially symmetric fan, and 
$D=\sum_i a_i D_i$ a 
$T$-invariant nef divisor. The probability measure 
$\nu_{D,\textrm{cent}}(u)$ on 
$M_{\mathbb{R}}$ is given by a Gaussian density function with mean 
$0$ and covariance matrix associated to the quadratic form
\begin{equation*}
u \mapsto \sum_{i\in I_D} \frac{1}{\langle m_D, v_i\rangle+a_{i}} \langle u, v_i\rangle^2,
\end{equation*} times the Dirac measure of 
$L_D$; that is, 
$\nu_{D,\textrm{cent}}(u)$ is a scalar multiple of
\begin{equation}
\exp\left(-\frac{1}{2}\sum_{i\in I_D} \frac{\langle u, v_i\rangle^2}{\langle m_D, v_i\rangle + a_i}\right)\cdot \delta_{L_D}(u).
\end{equation} Here 
$I_D$ is the set of indices 
$i$ such that 
$\langle u, v_i\rangle + a_i$ is not uniformly 
$0$ on all of 
$P$.
The proof is purely analytic and is given in the companion note [Reference Shah17]. Theorem 2 from the introduction follows, since convolving with 
$\delta_{m_D}$ replaces 
$u$ with 
$u-m_D$, and when 
$D$ is ample, 
$L_D=M_{\mathbb{R}}$.
To conclude, we consider a special case in which the proof that 
$\nu_D$ is Gaussian is very simple. It is inspired by results showing the pushforward in quantum 
$K$-theory of a homogeneous space becomes a ring homomorphism at 
$q=1$ [Reference Buch and Chung5], although one should be warned that the 
$q$ considered by those authors plays a rather different role from the 
$q$ appearing here.
Let 
$\mathscr{M}$ denote the set of probability measures on 
$M_{\mathbb{R}}$, which is a semigroup with respect to convolution.
Theorem 6.8 Suppose that 
$\mathcal{N}\subset Div_T(X)$ is a semigroup of divisors such that for 
$D,D'\in \mathcal{N}$,
\begin{align*}
&\frac{\chi_{\mathbb{T}}(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(D^0+D'^0))}{\chi_{\mathbb{T}}(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(D^0+D'^0))|_{x=1}}|_{q=1} = \frac{\chi_{\mathbb{T}}(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(D^0))}{\chi_{\mathbb{T}}(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(D^0))|_{x=1}}|_{q=1}\\
&\quad \cdot\frac{\chi_{\mathbb{T}}(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(D'^0))}{\chi_{\mathbb{T}}(\mathscr{Q}(\Delta)_\infty,\mathscr{O}(D'^0))|_{x=1}}|_{q=1},
\end{align*} or equivalently, that 
$\mu_{(-)}:\mathcal{N}\rightarrow\mathscr{M}$ is a homomorphism. Then for each 
$D\in \mathcal{N}$, 
$\nu_D$ is also Gaussian, and
\begin{equation*}
\nu_{(-)}:\mathcal{N}\rightarrow\mathscr{M},
\end{equation*}is also a homomorphism, landing in the subset of Gaussian measures.
Proof. We first verify that 
$\nu_{D,\textrm{cent}}$ is Gaussian. By definition,
\begin{equation*}
\nu_{D,\textrm{cent}} = \lim_{k\rightarrow\infty} (\tau_{\sqrt{k}})_*\left(\mu_{kD}\ast\delta_{-E_{\mu_{kD}}}\right).
\end{equation*} Since we have assumed that 
$\mu_{(-)}$ is a homomorphism, 
$\mu_{kD} = (\mu_D)^{\ast k}$. If 
$V_D$ is the random vector in 
$M_{\mathbb{R}}$ represented by the measure 
$\mu_D$, then the equation above means that 
$\mu_{kD}$ is the measure corresponding to a sum 
$\sum_{i=1}^k V_i$ of independent random vectors 
$V_i$, each with the same distribution as 
$V_D$. Thus 
$E_{\mu_{kD}}=k E(V_D)$ and the measure 
$\mu_{D,\textrm{cent}}$ corresponds to the random variable
\begin{equation*}
\lim_{k\rightarrow\infty} \frac{\sum_{i=1}^k (V_i- E(V_D))}{\sqrt{k}},
\end{equation*} which is a Gaussian with covariance equal to 
$Cov(V_D)$, by the central limit theorem as stated in [Reference Durrett10, Section 3.10].
We must also verify that 
$\nu_{D+D'}= \nu_{D}\ast\nu_{D'}$. First we check that 
$m_{D+D'}=m_D+m_{D'}$ under our assumptions on 
$D$ and 
$D'$. (It is not true in general!) In [Reference Shah17], it is shown that 
$\frac{E_{\mu_{kD}}}{k}\rightarrow m_D$. So 
$\frac{E_{\mu_{k(D+D')}}}{k}\rightarrow m_{D+D'}$. By assumption 
$\mu_{D+D'}=\mu_D\ast \mu_{D'}$, so we also have that 
$\frac{E_{\mu_{k(D+D')}}}{k} = \frac{E_{\mu_{kD}\ast \mu_{kD'}}}{k} = \frac{E_{\mu_{kD}} + E_{\mu_{kD'}}}{k} \rightarrow m_D+m_{D'}$. Thus 
$m_{D+D'} = m_D+m_{D'}$.
Finally, we calculate
\begin{align*}
\nu_{D+D'}
&= \lim_{k\rightarrow\infty} (\tau_{\sqrt k})_*\left( \mu_{k(D+D')} \ast \delta_{-E_{\mu_{k(D+D')}}}\right)\ast\delta_{m_{D+D'}} \\
&=\lim_{k\rightarrow\infty} (\tau_{\sqrt k})_*\left( \mu_{kD} \ast \delta_{-kE_{\mu_{D}}}\ast \mu_{kD'} \ast\delta_{-kE_{\mu_{D'}}}\right) \ast\delta_{m_D}\ast\delta_{m_{D'}}\\
&=\nu_{D}\ast\nu_{D'},
\end{align*} as desired, using 
$m_{D+D'} = m_D+m_{D'}$ in the second line.
Remark 6.9. The above theorem applies for instance when 
$X=\mathbb{P}^n$. The primitive elements in the rays of the fan of 
$X$ are 
$v_i = e_i$ for 
$i=1,\ldots,n$ and 
$v_{n+1} = -\sum_i e_i$. Then
\begin{equation*}
\mu_{kD_{n+1}} = \frac{1}{(n+1)^k} \sum_{i_0+\cdots+i_n=k}\binom{k}{i_0,\ldots,i_n}\delta_{(i_1,\ldots,i_n)},
\end{equation*} so 
$\mu_{kD_{n+1}}\ast\mu_{lD_{n+1}}=\mu_{(k+l)D_{n+1}}$.
Acknowledgements
We thank Hsian-Hua Tseng for helpful conversations about quasimaps and the J-function, as well as Frédéric Chapoton, Christian Haase, and Benjamin Nill for correspondence about lattice points in polytopes. We are grateful to the referee for a careful reading and numerous helpful suggestions for improving the exposition.
Funding
The authors were partially supported by NSF CAREER DMS-1945212. The second author was also supported by Charles University project PRIMUS/21/SCI/014.
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