Symmetric Lie superalgebras and quantum integrability

对称李超代数是具有对合自同构的复李超代数。利用Serganova对单李超代数的所有对合自同构进行了分类，从而得到了所有对称单李超代数的列表。与李代数的情况相反，对称李超代数的球函数理论还处于非常早期的阶段。提出的方法来解决这个困难的程序是基于量子可积系统的理论。这可以追溯到Sergeev（2001）的一个重要观察，他发现了一个经典级数的球面函数与早期由Chalykh， Feigin和Veselov发展的变形量子Calogero-Moser系统理论之间的关系。球函数的一种特殊情况是有限维不可约表示的特征，它产生了相应李超代数的Grothendieck环。对于基本经典李超代数，这些环最近用Serganova的广义根的概念明确地描述了。变形CM系统的理论给出了在变形CM算子及其量子积分作用下这些Grothendieck环的某些变形。假设基本经典对称李超代数的球函数代数可以被描述为对应族的一个专门化，并且可以使用变形CM算子的谱分解来研究。该方法在正辛李超代数的表示理论中已经非常成功：证明了超雅可比多项式（即相应变形CM算子的本征函数）的合适极限不是别的，而是Penkov和Serganova研究的欧拉特征。很自然地，对于球函数也会发生类似的现象。

Symmetric Lie superalgebra is a complex Lie superalgebra with an involutive automorphism. All involutive automorphisms of simple Lie superalgebras were classified by Serganova, so the list of all symmetric simple Lie superalgebras is known.In contrast to the Lie algebra case the theory of spherical functions for symmetric Lie superalgebras is at a very early stage. The proposed approach to this difficult programme is based on the theory of quantum integrable systems. It goes back to an important observation of Sergeev (2001), who discovered a relation of spherical functions of one of the classical series with the theory of deformed quantum Calogero-Moser systems developed earlier by Chalykh, Feigin and Veselov.A particular case of spherical functions are the characters of finite-dimensional irreducible representations, which generate the Grothendieck ring of the corresponding Lie superalgebra. For basic classical Lie superalgebras these rings were recently explicitly described using Serganova's notion of generalised root systems.The theory of the deformed CM systems provides certain deformations of these Grothendieck rings with the action of the deformed CM operators and their quantum integrals.The conjecture is that the algebra of spherical functions for basic classical symmetric Lie superalgebras can be described as a specialisation of the corresponding family and can be studied using the spectral decomposition of the deformed CM operators.The approach was already very successful in the representation theory of orthosymplectic Lie superalgebras:it was shown that a suitable limit of the super Jacobi polynomials (which are the eigenfunctions of the corresponding deformed CM operators) are nothing else but the Euler characters studied by Penkov and Serganova.It is natural to expect that a similar phenomenon happens for the spherical functions as well.