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。单李超代数的所有对合自同构都是由Serganova分类的，所以所有对称单李超代数的列表是已知的。与李代数的情况相比，对称李超代数的球函数理论还处于非常早期的阶段。这个困难的方案的建议方法是基于量子可积系统的理论。Sergeev（2001）的一个重要发现，发现了一个经典级数的球函数与Chalykh，Feigin和Veselov发展的变形量子Calogero-Moser系统理论之间的关系，球函数的一个特殊情况是有限维不可约表示的特征标，它们生成相应李超代数的Grothendieck环.对于基本经典对称李超代数，最近用Serganova的广义根系的概念明确地描述了这些环。变形CM系理论提供了这些Grothendieck环在变形CM算子及其量子积分作用下的某些变形。猜想是基本经典对称李超代数的球函数代数可以被描述为相应的变形CM算子及其量子积分的特殊化这一方法在正交辛李超代数的表示理论中已经取得了很大的成功：证明了超Jacobi多项式（对应的变形CM算子的本征函数）的一个合适的极限就是Penkov和Serganova研究的Euler特征标，很自然地可以期望类似的现象也发生在球函数上.

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.