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The Capelli eigenvalue problem for Lie superalgebras

  • Siddhartha Sahi
  • Hadi SalmasianEmail author
  • Vera Serganova
Article
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Abstract

For a finite dimensional unital complex simple Jordan superalgebra J, the Tits–Kantor–Koecher construction yields a 3-graded Lie superalgebra \(\mathfrak {g}^\flat \cong \mathfrak {g}^\flat (-1)\oplus \mathfrak {g}^\flat (0)\oplus \mathfrak {g}^\flat (1)\), such that \(\mathfrak {g}^\flat (-1)\cong J\). Set \(V:=\mathfrak {g}^\flat (-1)^*\) and \(\mathfrak {g}:=\mathfrak {g}^\flat (0)\). In most cases, the space \(\mathscr {P}(V)\) of superpolynomials on V is a completely reducible and multiplicity-free representation of \(\mathfrak {g}\), and there exists a direct sum decomposition \(\mathscr {P}(V):=\bigoplus _{\lambda \in \Omega }V_\lambda \), where \(\left( V_\lambda \right) _{\lambda \in \Omega }\) is a family of irreducible \(\mathfrak {g}\)-modules parametrized by a set of partitions \(\Omega \). In these cases, one can define a natural basis \(\left( D_\lambda \right) _{\lambda \in \Omega }\) of “Capelli operators” for the algebra \(\mathscr {PD}(V)^{\mathfrak {g}}\) of \(\mathfrak {g}\)-invariant superpolynomial differential operators on V. In this paper we complete the solution to the Capelli eigenvalue problem, which asks for the determination of the scalar \(c_\mu (\lambda )\) by which \(D_\mu \) acts on \(V_\lambda \). We associate a restricted root system \(\varSigma \) to the symmetric pair \((\mathfrak {g},\mathfrak {k})\) that corresponds to J, which is either a deformed root system of type \(\mathsf {A}(m,n)\) or a root system of type \(\mathsf {Q}(n)\). We prove a necessary and sufficient condition on the structure of \(\varSigma \) for \(\mathscr {P}(V)\) to be completely reducible and multiplicity-free. When \(\varSigma \) satisfies the latter condition we obtain an explicit formula for the eigenvalue \(c_\mu (\lambda )\), in terms of Sergeev–Veselov’s shifted super Jack polynomials when \(\varSigma \) is of type \(\mathsf {A}(m,n)\), and Okounkov-Ivanov’s factorial Schur Q-polynomials when \(\varSigma \) is of type \(\mathsf {Q}(n)\). Along the way, we prove that the natural map from the centre of the enveloping algebra of \(\mathfrak {g}\) into \(\mathscr {PD}(V)^{\mathfrak {g}}\) is surjective in all cases except when \(J\cong F \), where \( F \) is the 10-dimensional exceptional Jordan superalgebra.

Notes

Acknowledgements

The research of Siddhartha Sahi was partially supported by a Simons Foundation grant (509766), of Hadi Salmasian by NSERC Discovery Grants (RGPIN-2013-355464 and RGPIN-2018-04044), and of Vera Serganova by an NSF Grant (1701532). This work was initiated during the Workshop on Hecke Algebras and Lie Theory, which was held at the University of Ottawa. The first and the second named authors thank the National Science Foundation (DMS-162350), the Fields Institute, and the University of Ottawa for funding this workshop.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Siddhartha Sahi
    • 1
  • Hadi Salmasian
    • 2
    Email author
  • Vera Serganova
    • 3
  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA
  2. 2.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  3. 3.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA

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