Asymptotics of Spherical Superfunctions on Rank One Riemannian Symmetric Superspaces

We compute the Harish-Chandra $c$-function for a generic class of rank-one purely non-compact Riemannian symmetric superspaces $X=G/K$ in terms of Euler $\\Gamma$ functions, proving that it is meromorphic. Compared to the even case, the poles of the $c$-function are shifted into the right half-space. We derive the full asymptotic Harish-Chandra series expansion of the spherical superfunctions on $X$. In the case where the multiplicity of the simple root is an even negative number, they have a closed expression as Jacobi polynomials for an unusual choice of parameters.

2010 Mathematics Subject Classification: Primary 22E45, 17B15; Secondary 58A50

Keywords and Phrases: Harish-Chandra c-function, Lie supergroup, Riemannian symmetric superspace, spherical superfunction

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