Generalized Norm-compatible Systems on Unitary Shimura Varieties

We define and study in terms of integral Iwahori–Hecke algebras a new class of geometric operators acting on the Bruhat-Tits building of connected reductive groups over p-adic fields. These operators, which we call U-operators, generalize the geometric notion of "successors" for trees with a marked end. The first main contributions of the thesis are: (i) the integrality of the U-operator over the spherical Hecke algebra using the compatibility between Bernstein and Satake homomorphisms, (ii) in the unramified case, the U-operator attached to a cocharacter is a right root of the corresponding Hecke polynomial. In the second part of the thesis, we study some arithmetic aspects of special cycles on (products of) unitary Shimura varieties, these cycles are expected to yield new results towards the Bloch–Beilinson conjectures. As a global application of (ii), we obtain: (iii) the horizontal norm relations for these GGP cycles for arbitrary n, at primes where the unitary group splits. The general local theory developed in the first part of the thesis, has the potential to result in a number of global applications along the lines of (iii) (involving other Shimura varieties and also vertical norm relations) and offers new insights into topics such as the Blasius–Rogawski conjecture as well.