The relative Langlands program is a generalization of the Langlands program that is phrased in terms of spherical varieties? instead of directly in terms of reductive groups. It is expected to recover the usual Langlands program in the group case. In more recent work, instead of focusing on a spherical variety $X$, the main focus is on a Hamiltonian$G$-space$M$, which is often related to $X$ via its cotangent bundle$M=T^{*}X$. The tools of geometric quantization can then be used, with $M$ playing the role of the classical phase space to be quantized.

Let $F$ be the globalbase field and let $G$ be a reductive group. Locally, the idea of “quantization” in the relative Langlands program refers to obtaining a unitary representation$\omega_{v}$ of $G(F_{v})$ out of the Hamiltonian$G$-space$M$, analogous to the theory of geometric quantization. We also want a “base vector” $\Phi_{v}^{0}\in \omega_{v}$ for almost all $v$. Globally, we want to put together these local representations and have an automorphic realization $\otimes'\omega_{v}\to C^{\infty}\big(G(F)\backslash G(\mathbb{A}_{F})\big)$. An example of this is given by the Segal-Shale-Weil representation and the formation of theta series?.

Once one has obtained the theta series, one may investigate the following:

The integral of the theta series against an automorphic form. This is related to periods of L-functions.

The projection of the $L^{2}$-norm of the theta series to an automorphic representation. This is related to the relative trace formula.