nLab relative Langlands program

Idea

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.

Recent ongoing work by Ben-Zvi, Sakellaridis, and Venkatesh attempt to relate it to topological quantum field theory (as inspired by both the Kapustin-Witten TQFT as well as the analogies of arithmetic topology). This is expected as well to illuminate topics in special values of L-functions.

Quantization in the relative Langlands program

The reference for this section is #Sakellaridis2021.

Let $F$ be the global base 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.

References

• David Ben-Zvi, Yiannis Sakellaridis, Akshay Venkatesh, Relative Langlands duality (pdf).

• David Ben-Zvi, Relative Geometric Langlands Duality (MSRI lectures)

• David Ben-Zvi, Relative Langlands (pdf)

• Yiannis Sakellaridis, Spherical Varieties, Functoriality, and Quantization (arXiv:2111.03004)

• Yiannis Sakellaridis, Hamiltonian Spaces and Periods of Automorphic Forms (slides)

• Raphael Beuzart-Plessis, The Relative Langlands Program (YouTube) (Part 2) (Part 3)