Radial restriction of spherical functions on supergroups
Mitra Mansouri, Hadi Salmasian
TL;DR
The paper develops a purely algebraic framework for spherical functions on Lie supergroups by introducing the commutative dual algebra $\\mathcal{A}(\\mathfrak g,\\mathfrak k) = (U(\\mathfrak g)/\\mathcal I)^*$ with $\\mathcal I = \\mathfrak kU(\\mathfrak g) + U(\\mathfrak g)\\mathfrak k$. It proves that the radial restriction map from $\\mathcal{A}(\\mathfrak g,\\mathfrak k)$ to $S(\\mathfrak a)^*$ is injective, providing a powerful tool to study K-biinvariant functions purely algebraically for a broad class of symmetric pairs. For the explicit pair $(\\mathfrak{gl}(1|2),\\mathfrak{osp}(1|2))$, the paper computes a basis for the ideal $\\mathcal I$, revealing connections to Bernoulli numbers and Euler zigzag numbers, and highlighting intricate combinatorial structures in the super setting. Overall, the work extends spherical-function theory to Lie supergroups through dual Hopf-algebraic methods, offering a framework that does not rely on non-degenerate invariant forms and linking harmonic analysis on superspaces to rich algebraic and combinatorial phenomena.
Abstract
Using the Hopf superalgebra structure of the enveloping algebra $U(\mathfrak g)$ of a Lie superalgebra $\mathfrak=\mathrm{Lie}(G)$, we give a purely algebraic treatment of $K$-bi-invariant functions on a Lie supergroup $G$, where $K$ is a sub-supergroup of $G$. We realize $K$-bi-invariant functions as a subalgebra $\mathcal A(\mathfrak g,\mathfrak k)$ of the dual of $U(\mathfrak g)$ whose elements vanish on the coideal $\mathcal I=\mathfrak kU(\mathfrak g)+U(\mathfrak g)\mathfrak k$, where $\mathfrak k=\mathrm{Lie}(K)$. Next, for a general class of supersymmetric pairs $(\mathfrak g,\mathfrak k)$, we define the radial restriction of elements of $\mathcal A(\mathfrak g,\mathfrak k)$ and prove that it is an injection into $S(\mathfrak a)^*$, where $\mathfrak a$ is the Cartan subspace of $(\mathfrak g,\mathfrak k)$. Finally, we compute a basis for $\mathcal I$ in the case of the pair $(\mathfrak{gl}(1|2), \mathfrak{osp}(1|2))$, and uncover a connection with the Bernoulli and Euler zigzag numbers.