The Capelli problem for \mathfrakgl(m|n) and the spectrum of invariant differential operators

2015/06/09 by Sahi, Siddhartha, Salmasian, Hadi
#05E05 #17B10 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1506.03107

Abstract

The "Capelli problem" for the symmetric pairs (\mathfrakgl× \mathfrakgl,\mathfrakgl) (\mathfrakgl,\mathfrako), and (\mathfrakgl,\mathfraksp) is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter. In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs (\mathfrakg,\mathfrakg):=(\mathfrakgl(m|2n),\mathfrakosp(m|2n)) and (\mathfrakgl(m|n)×\mathfrakgl(m|n),\mathfrakgl(m|n)), acting on W:=S²(\mathbb C^m|2n) and \mathbb C^m|n⊗(\mathbb C^m|n)^*. We also give an affirmative answer to the abstract Capelli problem for these cases.

Related

• The Capelli eigenvalue problem for Lie superalgebras
• Intersection of harmonics and Capelli identities for symmetric pairs
• Explicit formulas for Eigenvalues of Capelli operators for the Lie\n superalgebra frakosp(1|2n)
• General Capelli-type identities
• Schur Q-functions and the Capelli eigenvalue problem for the Lie superalgebra \mathfrak q(n)