The purpose of this note is to show that the Betti realization of motives is compatible with Grothendieck's six operations and the nearby cycles functors, which in the motivic world, were previously studied by the author. We first review the construction of the Betti realization. Then, we establish some general criteria which, applied to the Betti realization, give the compatibilities we seek except for the one concerning the nearby cycles functors. The latter will be treated in a separate section.

This paper gives a complete description of the spherical unitary dual of split classical real and p-adic groups. The proof makes heavy use of the affine graded Hecke algebra.

We investigate the special fibres of Siegel modular varieties with Iwahori level structure. On these spaces, we have the Newton stratification, and the Kottwitz–Rapoport (KR) stratification; one would like to understand how these stratifications are related to each other. We give a simple description of all KR strata which are entirely contained in the supersingular locus as disjoint unions of Deligne–Lusztig varieties. We also give an explicit numerical description of the KR stratification in terms of abelian varieties.

The notion of a prolongation of an algebraic variety is developed in an abstract setting that generalizes the difference and (Hasse) differential contexts. An interpolating map that compares these prolongation spaces with algebraic jet spaces is introduced and studied.

We study behaviours of the ‘equianharmonic’ parameter of the Grothendieck–Teichmüller group introduced by Lochak and Schneps. Using geometric construction of a certain one-parameter family of quartics, we realize the Galois action on the fundamental group of a punctured Mordell elliptic curve in the standard Galois action on a specific subgroup of the braid group . A consequence is to represent a matrix specialization of the ‘equianharmonic’ parameter in terms of special values of the adelic beta function introduced and studied by Anderson and Ihara.