Intersection cohomology & $L$-functions

Abstract: I will report on ongoing joint work with Jonathan Wang, relating the intersection complex of the arc space of a spherical variety to an unramified local $L$-function. This is a broad generalization of Tate's thesis ($G=\mathbb G_m$, $X=\mathbb A^1$), where the local unramified $L$-factors are represented by the characteristic function of the integers $\mathfrak o$ of a non-Archimedean field. For more general groups $G$ and possibly singular spherical $G$-varieties $X$, the characteristic function of $X(\mathfrak o)$ is not the correct object to consider, and has to be replaced by a function obtained as the Frobenius trace of the intersection complex of the arc space of $X$. In special cases of horospherical, toric, affine homogeneous spherical varieties, or certain reductive monoids, the relation of this function to $L$-functions was previously described in works of Braverman--Finkelberg--Gaitsgory--Mirković, Bouthier--Ngô and myself. Our current work describes these IC functions in a very general setting, relating the IC function of the arc space to an $L$-value determined by the geometry of the spherical variety.

number theoryrepresentation theory

Audience: researchers in the topic

Cross Atlantic Representation Theory and Other topics ONline (CARTOON) conference