On a q-analog of a Sahi result

We obtain a $q$-analog of a well known Sahi result on the joint spectrum of $S(GL_n \times GL_n)$-invariant differential operators with polynomial coefficients on the vector space of complex $n \times n$-matrices.Comment: 9 pages, some improvements in exposition have been mad

Regular functions on the Shilov boundary

In this paper a quantum analog of the $*$-algebra of regular functions on the Shilov boundary $S(\mathbb D)$ of bounded symmetric domain $\mathbb D$ is constructed. The algebras of regular functions on $S(\mathbb D)$ are described in terms of generators and relations for two particular series of bounded symmetric domains. Also, the degenerate principal series of quantum Harich-Chandra modules related to $S(\mathbb D)=U_n$ is investigated.Comment: 17 page

Homomorphisms between different quantum toroidal and affine Yangian algebras

This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of $\mathfrak{sl}_n$, denoted by $\mathcal{U}^{(n)}_{q_1,q_2,q_3}$ and $\mathcal{Y}^{(n)}_{h_1,h_2,h_3}$, respectively. Our motivation arises from the milestone work of Gautam and Toledano Laredo, where a similar relation between the quantum loop algebra $U_q(L \mathfrak{g})$ and the Yangian $Y_h(\mathfrak{g})$ has been established by constructing an isomorphism of $\mathbb{C}[[\hbar]]$-algebras $\Phi:\widehat{U}_{\exp(\hbar)}(L\mathfrak{g})\to \widehat{Y}_\hbar(\mathfrak{g})$ (with $\ \widehat{}\$ standing for the appropriate completions). These two completions model the behavior of the algebras in the formal neighborhood of $h=0$. The same construction can be applied to the toroidal setting with $q_i=\exp(\hbar_i)$ for $i=1,2,3$. In the current paper, we are interested in the more general relation: $\mathrm{q}_1=\omega_{mn}e^{h_1/m}, \mathrm{q}_2=e^{h_2/m}, \mathrm{q}_3=\omega_{mn}^{-1}e^{h_3/m}$, where $m,n\in \mathbb{N}$ and $\omega_{mn}$ is an $mn$-th root of $1$. Assuming $\omega_{mn}^m$ is a primitive $n$-th root of unity, we construct a homomorphism $\Phi^{\omega_{mn}}_{m,n}$ from the completion of the formal version of $\mathcal{U}^{(m)}_{\mathrm{q}_1,\mathrm{q}_2,\mathrm{q}_3}$ to the completion of the formal version of $\mathcal{Y}^{(mn)}_{h_1/mn,h_2/mn,h_3/mn}$. We propose two proofs of this result: (1) by constructing the compatible isomorphism between the faithful representations of the algebras; (2) by combining the direct verification of Gautam and Toledano Laredo for the classical setting with the shuffle approach.Comment: v2: 30 pages, significant modifications from the previous version, minor mistakes corrected. v3: Published version, 30 pages, minor corrections, some details adde

A q-Analog of the Hua Equations

A necessary condition is established for a function to be in the image of a quantum Poisson integral operator associated to the Shilov boundary of the quantum matrix ball. A quantum analogue of the Hua equations is introduced.Comment: 22 pages, LaTeX2

AGT, Burge pairs and minimal models

We consider the AGT correspondence in the context of the conformal field theory $M^{\, p, p^{\prime}}$ $\otimes$ $M^{H}$, where $M^{\, p, p^{\prime}}$ is the minimal model based on the Virasoro algebra $V^{\, p, p^{\prime}}$ labeled by two co-prime integers $\{p, p^{\prime}\}$, $1 < p < p^{\prime}$, and $M^{H}$ is the free boson theory based on the Heisenberg algebra $H$. Using Nekrasov's instanton partition functions without modification to compute conformal blocks in $M^{\, p, p^{\prime}}$ $\otimes$ $M^{H}$ leads to ill-defined or incorrect expressions. Let $B^{\, p, p^{\prime}, H}_n$ be a conformal block in $M^{\, p, p^{\prime}}$ $\otimes$ $M^{H}$, with $n$ consecutive channels $\chi_{i}$, $i = 1, \cdots, n$, and let $\chi_{i}$ carry states from $H^{p, p^{\prime}}_{r_{i}, s_{i}}$ $\otimes$ $F$, where $H^{p, p^{\prime}}_{r_{i}, s_{i}}$ is an irreducible highest-weight $V^{\, p, p^{\prime}}$-representation, labeled by two integers $\{r_{i}, s_{i}\}$, $0 < r_{i} < p$, $0 < s_{i} < p^{\prime}$, and $F$ is the Fock space of $H$. We show that restricting the states that flow in $\chi_{i}$ to states labeled by a partition pair $\{Y_1^{i}, Y_2^{i}\}$ such that $Y^{i}_{2, {\tt R}} - Y^{i}_{1, {\tt R} + s_{i} - 1} \geq 1 - r_{i}$, and $Y^{i}_{1, {\tt R}} - Y^{i}_{2, {\tt R} + p^{\prime} - s_{i} - 1} \geq 1 - p + r_{i}$, where $Y^{i}_{j, {\tt R}}$ is row-${\tt R}$ of $Y^{i}_j, j \in \{1, 2\}$, we obtain a well-defined expression that we identify with $B^{\, p, p^{\prime}, H}_n$. We check the correctness of this expression for ${\bf 1.}$ Any 1-point $B^{\, p, p^{\prime}, H}_1$ on the torus, when the operator insertion is the identity, and ${\bf 2.}$ The 6-point $B^{\, 3, 4, H}_3$ on the sphere that involves six Ising magnetic operators.Comment: 22 pages. Simplified the presentatio