In mathematics, the Moy–Prasad filtration is a family of filtrations of p-adic reductive groups and their Lie algebras, named after Allen Moy and Gopal Prasad. The family is parameterized by the Bruhat–Tits building; that is, each point of the building gives a different filtration. Alternatively, since the initial term in each filtration at a point of the building is the parahoric subgroup for that point, the Moy–Prasad filtration can be viewed as a filtration of a parahoric subgroup of a reductive group. The chief application of the Moy–Prasad filtration is to the representation theory of p-adic groups, where it can be used to define a certain rational number called the depth of a representation. The representations of depth r can be better understood by studying the rth Moy–Prasad subgroups. This information then leads to a better understanding of the overall structure of the representations, and that understanding in turn has applications to other areas of mathematics, such as number theory via the Langlands program. For a detailed exposition of Moy-Prasad filtrations and the associated semi-stable points, see Chapter 13 of the book Bruhat-Tits theory: a new approach by Tasho Kaletha and Gopal Prasad.
History In their foundational work on the theory of buildings, Bruhat and Tits defined subgroups associated to concave functions of the root system. These subgroups are a special case of the Moy–Prasad subgroups, defined when the group is split. The main innovations of Moy and Prasad were to generalize Bruhat–Tits's construction to quasi-split groups, in particular tori, and to use the subgroups to study the representation theory of the ambient group.
Examples The following examples use the p-adic rational numbers Q p {\displaystyle \mathbb {Q} _{p}} and the p-adic integers Z p {\displaystyle \mathbb {Z} _{p}} . A reader unfamiliar with these rings may instead replace Q p {\displaystyle \mathbb {Q} _{p}} by the rational numbers Q {\displaystyle \mathbb {Q} } and Z p {\displaystyle \mathbb {Z} _{p}} by the integers Z {\displaystyle \mathbb {Z} } without losing the main idea.
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