Givental Integral Representation for Classical Groups

A mirror symmetric solution to the quantum Toda lattice

We use representation theory to construct integral formulas for solutions to the quantum Toda lattice in general type. This result generalizes work of Givental for SL(n)/B in a uniform way to arbitrary type and can be interpreted as a kind of mirror theorem for the full flag variety G/B. We also prove the existence of a totally positive critical point of the 'superpotential' in every mirror fiber.

BC Toda链I：反射算符与特征函数

BC Toda chain I: reflection operator and eigenfunctions

本文研究BC型Toda链的特征函数，引入满足反射方程的反射算符和Baxter算符，证明其与哈密顿量的可交换性，并推导Baxter方程。

On parabolic Whittaker functions II

We derive a Givental-type stationary phase integral representation for the specified $\Gr_{m,N}$-Whittaker function introduced in \cite{GLO2}, which presumably describes the $S^1\times U_N$-equivariant Gromov-Witten invariants of Grassmann variety $\Gr_{m,N}$. Our main tool is a generalization of Whittaker model for principal series $\CU(\frak{gl}_N)$-modules. In particular, our construction includes a representation theory interpretation of the Batyrev--Ciocan-Fontanine--Kim--van Straten toric degeneration of Grassmannian, providing a direct connection between this toric degeneration of $\Gr_…

BC Toda链II：对称性。双视图

BC Toda chain II: symmetries. Dual picture

本文研究BC Toda链的Baxter算子对易性及波函数对谱参数的对称性，推导Mellin-Barnes积分表示，并证明波函数满足双差分方程，与超立方体Whittaker函数一致。

On $b$-Whittaker functions

The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed $\mathfrak{gl}_n$ open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a Mellin-Barnes integral representation for these eigenfunctions. In the present paper, we develop the analytic theory of the $b$-Whittaker functions from the perspective of quantum cluster algebras. We obtain a formula for the modular open Toda system's Baxter operator as a sequence of quantum cluster transformations, and thereby derive a new modular $b$-a…

The Bloch-Okounkov correlation functions, a classical half-integral case

Bloch and Okounkov's correlation function on the infinite wedge space has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, and certain character functions of $\hgl_\infty$-modules of level one. Recent works have calculated these character functions for higher levels for $\hgl_\infty$ and its Lie subalgebras of classical type. Here we obtain these functions for the subalgebra of type $D$ of half-integral levels and as a byproduct, obtain $q$-dimension formulas for integral modules of type $D$ at half-integral level.

Automorphic Integral Transforms for Classical Groups I: Endoscopy Correspondences

A general framework of constructions of endoscopy correspondences via automorphic integral transforms for classical groups is formulated in terms of the Arthur classification of the discrete spectrum of square-integrable automorphic forms. This suggests a principle, which is called the $(τ,b)$-theory of automorphic forms of classical groups, to reorganize and extend the series of work of Piatetski-Shapiro, Rallis, Kudla and others on standard $L$-functions of classical groups and theta correspondence.

Twisted doubling integrals for Brylinski-Deligne extensions of classical groups

We explain how to develop the twisted doubling integrals for Brylinski-Deligne extensions of connected classical groups. This gives a family of global integrals which represent Euler products for this class of non-linear extensions.

Twisted doubling integrals for classical groups

We describe the twisted doubling integrals of Cai-Friedberg-Ginzburg-Kaplan in a conceptual way. This also extends the construction to the quaternionic unitary groups. We carry out the unfolding argument uniformly in this article. To do so, we define a family of degenerate Whittaker coefficients that are suitable in this setup and study some of their properties. We also prove certain related global and local results that use the same tools.

Geometry of canonical bases and mirror symmetry

A decorated surface S is a surface with a finite set of special points on the boundary, considered modulo isotopy. Let G be a split reductive group. A pair (G, S) gives rise to a moduli space A(G, S), closely related to the space of G-local systems on S. It has a positive structure. So the set of its integral tropical points is defined. We introduce a rational positive function W on A(G, S), the potential. The condition that its tropicalisation is non-negative determines its subset. For SL(2), we recover the set of integral laminations on S. We prove that when S is a disc with n special points…

经典李群的部分旗概形上线丛的半整性

Half-integrality of line bundles on partial flag schemes of classical Lie groups

本文发展部分旗概形上等变线丛的 Galois 下降理论，考察抛物子群特征所附下降数据分类的计算问题，并据此分类经典李群标准 $\mathbb{Z}[1/2]$-形式上的等变线丛。

Monomial integrals on the classical groups

This paper presents a powerfull method to integrate general monomials on the classical groups with respect to their invariant (Haar) measure. The method has first been applied to the orthogonal group in [J. Math. Phys. 43, 3342 (2002)], and is here used to obtain similar integration formulas for the unitary and the unitary symplectic group. The integration formulas turn out to be of similar form. They are all recursive, where the recursion parameter is the number of column (row) vectors from which the elements in the monomial are taken. This is an important difference to other integration meth…