Ober­se­mi­nar Zah­len­the­o­rie: Ta­ku­ma Ha­ya­shi (Osa­ka Me­tro­po­li­tan Uni­ver­si­ty): Ra­ti­o­na­li­ty of equi­va­ri­a­nt twis­ted D-mo­du­les on flag va­ri­e­ties

Abstract: To study rationality of special $L$-values, it is a basic problem to find and analyze rational structures of automorphic representations and their cohomology groups. As Raghuram--Shahidi pointed out, it is morally necessary to work out the representations at the infinite places as well as at the finite places. 
To construct rational forms of complex irreducible Harish-Chandra modules, Michael Harris proposed to use twisted D-modules on flag varieties. In fact, complex irreducible Harish-Chandra modules with regular infinitesimal character are constructed and classified by twisted D-modules on complex flag varieties (the Beilinson--Bernstein correspondence). He proposed to consider its rational analog. 
Fabian Januszewski then pointed out that some rationality problems arise when applying Harris' strategy. In particular, some of complex twisted D-modules cannot have $F$-forms but $F'$-forms for some finite extension $F'/F$ of a base field $F\subset \mathbb{C}$. 
In this talk, I will explain how to classify simple equivariant twisted D-modules on flag varieties over fields $F$ of characteristic zero. I will show that they are absolutely simple in many examples, including certain symmetric pairs arising from exceptional groups. For instance,  when $F=\mathbb{Q}$, it implies that the complex irreducible equivariant twisted D-modules admit $\mathbb{Q}$-forms. We also discuss how to obtain minimal fields of definition from the classification result in general. 
This talk is partially based on a joint work with Fabian Januszewski.