Journal of Lie Theory 35 (2025), No. 4, 861--878
Copyright Heldermann Verlag 2025
Radial Restriction of Spherical Functions on Supergroups
Mitra Mansouri
Department of Mathematics and Statistics, University of Ottawa, Canada
mmans053@uottawa.ca
Hadi Salmasian
Department of Mathematics and Statistics, University of Ottawa, Canada
hsalmasi@uottawa.ca
[Abstract-pdf]
\newcommand{\g}{\mathfrak} \newcommand{\cA}{\mathcal A} \newcommand{\cI}{\mathcal I} Using the Hopf superalgebra structure of the enveloping algebra $U(\g g)$ of a Lie superalgebra $\g g=\mathrm{Lie}(G)$, 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 $\cA(\g g,\g k)$ of the dual of $U(\g g)$ whose elements vanish on the coideal $\cI=\g kU(\g g)+U(\g g)\g k$, where $\g k=\mathrm{Lie}(K)$. Next, for a general class of supersymmetric pairs $(\g g,\g k)$, we define the radial restriction of elements of $\cA(\g g,\g k)$ and prove that it is an injection into $S(\g a)^*$, where $\g a$ is the Cartan subspace of $(\g g,\g k)$. Finally, we compute a basis for $\cI$ in the case of the pair $(\g{gl}(1|2)$, $\g{osp}(1|2))$, and uncover a connection with the Bernoulli and Euler zigzag numbers.
Keywords: Lie superalgebras, spherical functions, enveloping algebras, coideals, Bernoulli numbers, Euler zigzag numbers.
MSC: 17B10, 43A90, 11B68.
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