Spherical indecomposable representations of Lie superalgebras

Abstract

We present a classification of all spherical indecomposable representations of classical and exceptional Lie superalgebras. We also include information about stabilizers, symmetric algebras, and Borels for which sphericity is achieved. In one such computation, the symmetric algebra of the standard module of osp(m|2n) is computed, which in particular gives the representation-theoretic structure of polynomials on the complex supersphere.Lemma 3.10. If g is basic, then an irreducible representation V is spherical if and only if V * is. If (V, g) is spherical, then (V * , g) is equivalent to it. 9

Capelli operators for spherical superharmonics and the Dougall-Ramanujan identity

“…If t ∈ 2Z the result follows from the analogous result for osp(2m|2n) with 2m − 2n = t, where m, n ∈ N (see for instance [26]) using the functor F m|n defined in Proposition 8.1(vi). (iii) Note that γ At (A t ) is the centralizer of E t inside γ Bt (B t ).…”

“…As will be seen in Proposition 3.1, the indecomposable components of P(V ) can be characterized as generalized eigenspaces of the restriction of the Casimir operator to each homogeneous component. A proof of Proposition 3.1 is given in A. Sherman's PhD thesis [26] (see also [3]).…”

“…Set d i := dim V i for i ∈ 0, 1 . It is known that P(V ) is a semisimple and multiplicity-free g-module unless d 0 , d 1 ∈ 2Z + and d 0 ≤ d 1 (see [26,3]). Let PD (V ) denote the superalgebra of superpolynomial-coefficient superdifferential operators on V , equipped with the natural g-module structure defined by x • D := xD − (−1) |D|•|x| Dx for homogeneous x ∈ g and D ∈ PD(V ) (for further details see for example [19,Sec.…”

“…In the cases that P(V ) is a semisimple and multiplicity-free g-module, the irreducible components of P(V ) are naturally indexed by elements of P (see [8,3,26]). Then by a general algebraic construction (see [18,21] or Definition 1.5 below) one obtains a distinguished basis {D λ } λ∈P of Capelli operators for the algebra PD(V ) g of g-invariant differential operators.…”

Branching laws for spherical harmonics on superspaces in exceptional cases

“…In the exceptional cases, the Fischer decomposition is rather an indecomposable (but not necessarily irreducible) decomposition of homogeneous polynomials under the action of osp(m|2n), see theorems A and B below. In the general case, the osp(m|2n)-module of polynomials on R m|2n was also described recently in [16].…”