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For the special orthogonal group G = SO (2 n + 1) over a p -adic field, we consider a discrete series representation of a standard Levi subgroup of G . We prove that the Arthur R -group and the classical R -group of $\pi$ are isomorphic. If $\pi$ is generic, we consider the Aubert involution $\hat{\pi}$ . Under the assumption that $\hat{\pi}$ is unitary, we prove that the Arthur R -group of $\hat{\pi}$ is isomorphic to the R -group of $\hat{\pi}$ defined by Ban (Ann. Sci. Ecole Norm. Sup. 35 (2002), 673–693; J. Algebra 271 (2004), 749–767). This is done by establishing the connection between the A -parameters of $\pi$ and $\hat{\pi}$ . We prove that the A -parameter of $\hat{\pi}$ is obtained from the A -parameter of $\pi$ by interchanging the two $\textit{SL}(2,\mathbb{C})$ components.

In this paper we define a kind of generalized spherical functions on Sp(2, R). We call it ‘Fourier–Jacobi type’, since it can be considered as a generalized Whittaker model associated with the Jacobi maximal parabolic subgroup. Also we give the multiplicity theorem and an explicit formula of these functions for discrete series representations of Sp(2, R).