Exact power series in the asymptotic expansion of the matrix coefficients with the corner $K$-type of $P_J$-principal series representations of $Sp(2,R)$

摘要

Let$G$beasymplecticLiegroupofrank$2$,$Sp(2,mathbb{R})$and$P_J$beitsmaximalparabolicsubgroupcalledtheJacobiparabolicsubgroupwithnon-abelianunipotentradica.Theradialpartofmatrixcoefficientsofthe$P_J$-principalseriesrepresentationsof$G$werestudiedinrelationtotheAppell'shypergeometricfunction.Theleadingtermsoftheexpansionofthefunctionaroundtheinfinitywerewellinvestigatedingeneralcases(semisimpleLiegroupsandrepresentations).Inthispaper,wedeterminethepowerseriesexpansionotherthanleadingtermsfortheabovespecialcase.