Radial restriction of spherical functions on supergroups

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.