[Paper Review] The Capelli eigenvalue problem for Lie superalgebras

Siddhartha Sahi, Hadi Salmasian+1 more|arXiv (Cornell University)|Jul 19, 2018

Advanced Topics in Algebra25 references4 citations

TL;DR

This paper completes the solution to the Capelli eigenvalue problem for Lie superalgebras by determining the eigenvalues $ c_{ ho}( u) $ of Capelli operators $ D_{ ho} $ acting on irreducible representations $ V_{ u} $ of the Lie superalgebra $ g $, using restricted root systems of type $ \mathsf{A}(m,n) $ or $ \mathsf{Q}(n) $. It provides explicit formulas in terms of Sergeev-Veselov's shifted super Jack polynomials and Okounkov-Ivanov's factorial Schur $ Q $-polynomials, and proves the surjectivity of the center map into the algebra of $ \fg $-invariant differential operators except for the exceptional Jordan superalgebra $ \mathfrak{F} $.

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 $\mathcal P(V)$ of superpolynomials on $V$ is a completely reducible and multiplicity-free representation of $\mathfrak g$, with a decomposition $\mathcal P(V):=\bigoplus_{λ\inΩ}V_λ$, where $\left(V_λ ight)_{λ\inΩ}$ is a family of irreducible $\mathfrak g$-modules parametrized by a set of partitions $Ω$. In these cases, one can define a natural basis $\left(D_λ ight)_{λ\inΩ}$ of "Capelli operators" for the algebra $\mathcal{PD}(V)^{\mathfrak g}$. In this paper we complete the solution to the Capelli eigenvalue problem, which is to determine the scalar $c_μ(λ)$ by which $D_μ$ acts on $V_λ$. We associate a restricted root system $\mathitΣ$ 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 $\mathitΣ$ for $\mathcal{P}(V)$ to be completely reducible and multiplicity-free. When $\mathitΣ$ satisfies the latter condition we obtain an explicit formula for the eigenvalue $c_μ(λ)$, in terms of Sergeev-Veselov's shifted super Jack polynomials when $\mathitΣ$ is of type $\mathsf{A}(m,n)$, and Okounkov-Ivanov's factorial Schur $Q$-polynomials when $\mathitΣ$ is of type $\mathsf{Q}(n)$.

Motivation & Objective

• Address the long-standing Capelli eigenvalue problem in the superalgebra setting, extending classical results to Lie superalgebras.
• Characterize when the space of superpolynomials $ \mathscr{P}(V) $ is completely reducible and multiplicity-free as a representation of $ \fg $.
• Provide an explicit formula for the eigenvalues $ c_{\mu}(\lambda) $ of Capelli operators $ D_{\mu} $ acting on irreducible $ \fg $-modules $ V_{\lambda} $.
• Establish a necessary and sufficient condition on the restricted root system $ \Sigma $ for the multiplicity-free property of $ \mathscr{P}(V) $.
• Prove that the natural map from the center of the universal enveloping algebra of $ \fg $ to $ \mathscr{PD}(V)^{\fg} $ is surjective, except when $ J \cong \mathfrak{F} $, the 10-dimensional exceptional Jordan superalgebra.

Proposed method

• Use the Tits-Kantor-Koecher construction to associate a 3-graded Lie superalgebra $ \fg^{\flat} $ to a finite-dimensional unital complex simple Jordan superalgebra $ J $, with $ \fg^{\flat}(-1) \cong J $.
• Define $ V := \fg^{\flat}(-1)^* $ and $ \fg := \fg^{\flat}(0) $, and study the action of $ \fg $ on the superpolynomial algebra $ \mathscr{P}(V) $.
• Introduce a restricted root system $ \Sigma $ associated with the symmetric pair $ (\fg, \fk) $, which is either of type $ \mathsf{A}(m,n) $ or $ \mathsf{Q}(n) $.
• Establish a necessary and sufficient condition on $ \Sigma $ for $ \mathscr{P}(V) $ to be completely reducible and multiplicity-free, using representation-theoretic and root system analysis.
• Construct a basis $ (D_{\lambda})_{\lambda \in \Omega} $ of $ \mathfrak{g} $-invariant superpolynomial differential operators, called Capelli operators, and determine their eigenvalues via highest weight theory.
• Use the connection between $ \mathscr{PD}(V)^{\fg} $ and the algebra of differential operators on the supermanifold $ \mathcal{G}/\mathcal{K} $, and apply the Harish-Chandra homomorphism to relate eigenvalues to symmetric polynomials.

Experimental results

Research questions

• RQ1 What conditions on the restricted root system $ \Sigma $ ensure that $ \mathscr{P}(V) $ is completely reducible and multiplicity-free as a $ \fg $-module?
• RQ2 How can the eigenvalues $ c_{\mu}(\lambda) $ of the Capelli operators $ D_{\mu} $ acting on irreducible $ \fg $-modules $ V_{\lambda} $ be explicitly computed?
• RQ3 What is the role of the restricted root system $ \Sigma $ in classifying the structure of $ \mathscr{P}(V) $ and the algebra $ \mathscr{PD}(V)^{\fg} $?
• RQ4 Under what conditions is the natural map from the center of $ \mathbf{U}(\fg) $ to $ \mathscr{PD}(V)^{\fg} $ surjective?
• RQ5 How do shifted super Jack polynomials and factorial Schur $ Q $-polynomials arise naturally in the eigenvalue formula for $ c_{\mu}(\lambda) $?
• RQ6 Is the eigenvalue formula uniform across all types of Jordan superalgebras, and how does the exceptional case $ J \cong \mathfrak{F} $ differ?

Key findings

• The space $ \mathscr{P}(V) $ of superpolynomials on $ V = \mathfrak{g}^{\flat}(-1)^* $ is completely reducible and multiplicity-free if and only if the associated restricted root system $ \Sigma $ satisfies a specific structural condition.
• When $ \Sigma $ is of type $ \mathsf{A}(m,n) $, the eigenvalue $ c_{\mu}(\lambda) $ is given explicitly by the evaluation of Sergeev-Veselov's shifted super Jack polynomials.
• When $ \Sigma $ is of type $ \mathsf{Q}(n) $, the eigenvalue $ c_{\mu}(\lambda) $ is given by the evaluation of Okounkov-Ivanov's factorial Schur $ Q $-polynomials.
• The natural map from the center of the universal enveloping algebra $ \mathbf{U}(\fg) $ to the algebra $ \mathscr{PD}(V)^{\fg} $ of $ \fg $-invariant superpolynomial differential operators is surjective in all cases except when $ J \cong \mathfrak{F} $, the 10-dimensional exceptional Jordan superalgebra.
• A necessary and sufficient condition on $ \Sigma $ for the multiplicity-free decomposition of $ \mathscr{P}(V) $ is derived, based on the root system's structure and the action of the Weyl group.
• The eigenvalue $ c_{\mu}(\lambda) $ is computed via the Harish-Chandra homomorphism, linking the action of $ D_{\mu} $ on $ V_{\lambda} $ to the highest weight $ \tilde{\lambda} $ and symmetric polynomials in $ \tilde{\mathfrak{h}} $.