Symmetric Lie superalgebras and deformed quantum Calogero–Moser problems

Abstract

The representation theory of symmetric Lie superalgebras and corresponding spherical functions are studied in relation with the theory of the deformed quantum Calogero-Moser systems. In the special case of symmetric pair g = gl(n, 2m), k = osp(n, 2m) we establish a natural bijection between projective covers of spherically typical irreducible g-modules and the finite dimensional generalised eigenspaces of the algebra of Calogero-Moser integrals Dn,m acting on the corresponding Laurent quasi-invariants An,m.

Quantum elliptic Calogero-Moser systems from gauge origami

“…Unlike the gauge theories with massive fundamental hypermultiplets [13], there is no natural truncation condition on the Young diagrams labeling instanton partition function. Instead, the equation of motion for the functional W A0 is given by 20) explicitly one obtains:…”

“…(5.19) In large x limit, we have 20) which is the desired degree. We will next show that X(x) indeed reproduces the commuting Hamiltonians of the edCM system.…”

“…As mentioned in this paper, we need to consider the orbifolded version of the X-function in order to extract the commuting Hamiltonians of the eCM system. 2 The trigonometric version is studied, e.g., in [20,21]. 3 ゲージ折紙 (日本語); 規範摺紙 (中文 繁體字); 规范折纸 (中文 简体字).(2.12)where the symbol PV means the principal value integral.…”

Spherical indecomposable representations of Lie superalgebras

“…The representation theory of the space of functions as well as the algebra of invariant differential operators has been studied in recent years (see e.g. [SS16], [All12], [AS15], [SV17], [SSS18]). In [SSS18], the authors associate to each simple Jordan superalgebra J a supersymmetric pair (g, k) via the TKK construction.…”

Intertwining operator for AG2 Calogero–Moser–Sutherland system

“…Examples of such configurations include deformations of the root systems A n and C n . These examples are related to symmetric superspaces, [14]- [17], and to special representations of Cherednik algebras [18]. The corresponding configurations of vectors have to satisfy so-called locus conditions [19].…”