Radial restriction of spherical functions on supergroups

Using the Hopf superalgebra structure of the enveloping algebra $U(\mathfrak{g})$ of a Lie superalgebra $=\mathrm{L}\mathrm{i}\mathrm{e}(G)$, 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 $\mathcal{I}=\mathfrak{k}U(\mathfrak{g})+U(\mathfrak{g})\mathfrak{k}$, where $\mathfrak{k}=\mathrm{L}\mathrm{i}\mathrm{e}(K)$. 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 $S(\mathfrak{a}{)}^{\ast }$, 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 $(\mathfrak{g}\mathfrak{l}(1\mathrm{\mid }2),\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\mathrm{\mid }2))$, and uncover a connection with the Bernoulli and Euler zigzag numbers.