The induced-representation multiplicity formula for Bessel and Fourier–Jacobi models

Let

$$\sigma_V=|\cdot|^{s_{V,1}}\sigma_{V,1}\times\cdots\times|\cdot|^{s_{V,l_V}}\sigma_{V,l_V},\qquad \sigma_W=|\cdot|^{s_{W,1}}\sigma_{W,1}\times\cdots\times|\cdot|^{s_{W,l_W}}\sigma_{W,l_W},$$

where $\sigma_{V,i}$ and $\sigma_{W,i}$ are irreducible tempered representations and $s_{V,i},s_{W,i}\in\mathbb C$ satisfy

$$\operatorname{Re}(s_{V,1})\geq\cdots\geq\operatorname{Re}(s_{V,l_V})>0,\qquad \operatorname{Re}(s_{W,1})\geq\cdots\geq\operatorname{Re}(s_{W,l_W})>0.$$

Let $\pi_{V_0}$ and $\pi_{W_0}$ be irreducible tempered representations in the Bessel cases, and let $\widetilde\pi_{W_0}$ be an irreducible tempered representation in the Fourier–Jacobi cases.

Induced-representation multiplicity formula. In the Bessel cases, for possibly reducible induced representations

$$\pi_V=\sigma_V\rtimes\pi_{V_0},\qquad \pi_W=\sigma_W\rtimes\pi_{W_0},$$

one has

$$m(\pi_V\boxtimes\pi_W)=m(\pi_{V_0}\boxtimes\pi_{W_0}).$$

In the Fourier–Jacobi cases, for possibly reducible induced representations

$$\pi_V=\sigma_V\rtimes\pi_{V_0},\qquad \widetilde\pi_W=\sigma_W\rtimes\widetilde\pi_{W_0},$$

one has

$$m\bigl(\pi_V\boxtimes(\widetilde\pi_W\otimes\omega_{W,\psi_F})\bigr) =m\bigl(\pi_{V_0}\boxtimes(\widetilde\pi_{W_0}\otimes\omega_{W_0,\psi_F})\bigr).$$

The formula is the uniform reduction underlying the paper's treatment of Archimedean Bessel and Fourier–Jacobi multiplicities. Its proof is the main result of the article in the Archimedean setting, while the corresponding general non-Archimedean uniform approach is described as forthcoming.