The Capelli problem for and the spectrum of invariant differential operators

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• 作者：Siddhartha Sahi^a ; ^{sahi@math.rutgers.edu" class="auth_mail" title="E-mail the corresponding author} ; Hadi Salmasian^b ; ^{hsalmasi@uottawa.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：17B10 ; 05E05
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：5 November 2016
• 年：2016
• 卷：303
• 期：Complete
• 页码：1-38
• 全文大小：687 K

文摘

The “Capelli problem” for the symmetric pairs (gl×gl,gl)$(\mathfrak{gl}\times \mathfrak{gl},\mathfrak{gl})$, (gl,o)$(\mathfrak{gl},\mathfrak{o})$, and (gl,sp)$(\mathfrak{gl},\mathfrak{sp})$ is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter (see , ,  and ). In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs (g,k):=(gl(m|2n),osp(m|2n))$(\mathfrak{g},\mathfrak{k}):=(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$ and (gl(m|n)×gl(m|n),gl(m|n))$(\mathfrak{gl}(m|n)\times \mathfrak{gl}(m|n),\mathfrak{gl}(m|n))$, acting on W:=S²(C^m|2n)$W:={\mathcal{S}}^2({\mathbb{C}}^{m|2n})$ and C^m|n⊗(C^m|n)^⁎${\mathbb{C}}^{m|n}\otimes {({\mathbb{C}}^{m|n})}^{⁎}$.

To achieve this goal, we first prove that the center of the universal enveloping algebra of the Lie superalgebra g$\mathfrak{g}$ maps surjectively onto the algebra PD(W)^g$\mathcal{PD}{(W)}^{\mathfrak{g}}$ of g$\mathfrak{g}$-invariant differential operators on the superspace W, thereby providing an affirmative answer to the “abstract” Capelli problem for W  . Our proof works more generally for gl(m|n)$\mathfrak{gl}(m|n)$ acting on S²(C^m|n)${\mathcal{S}}^2({\mathbb{C}}^{m|n})$ and is new even for the “ordinary” cases (m=0$m=0$ or n=0$n=0$) considered by Howe and Umeda in [9].

We next describe a natural basis {D_λ}$\left\{D_{\lambda }\right\}$ of PD(W)^g$\mathcal{PD}{(W)}^{\mathfrak{g}}$, that we call the Capelli basis. Using the above result on the abstract Capelli problem, we generalize the work of Kostant and Sahi ,  and  by showing that the spectrum of D_λ$D_{\lambda }$ is given by a polynomial 20e1ff907dc" title="Click to view the MathML source">c_λ$c_{\lambda }$, which is characterized uniquely by certain vanishing and symmetry properties.

We further show that the top homogeneous parts of the eigenvalue polynomials 20e1ff907dc" title="Click to view the MathML source">c_λ$c_{\lambda }$ coincide with the spherical polynomials d_λ$d_{\lambda }$, which arise as radial parts of k$\mathfrak{k}$-spherical vectors of finite dimensional g$\mathfrak{g}$-modules, and which are super-analogues of Jack polynomials. This generalizes results of Knop and Sahi [14].

Finally, we make a precise connection between the polynomials 20e1ff907dc" title="Click to view the MathML source">c_λ$c_{\lambda }$ and the shifted super Jack polynomials of Sergeev and Veselov [25] for special values of the parameter. We show that the two families are related by a change of coordinates that we call the “Frobenius transform”.