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 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.
In their foundational work on the theory of buildings, Bruhat and Tits defined subgroups associated to concave functions of the root system.
The main innovations of Moy and Prasad[2] 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.
is a positive real number then we use the floor function to define the
This example illustrates the general phenomenon that although the Moy–Prasad filtration is indexed by the nonnegative real numbers, the filtration jumps only on a discrete, periodic subset, in this case, the natural numbers.
, and its Moy–Prasad subalgebras are the spaces of matrices equal to the zero matrix modulo high powers of
is a positive real number then we use the floor function to define the
Although the Moy–Prasad filtration is commonly used to study the representation theory of p-adic groups, one can construct Moy–Prasad subgroups over any Henselian, discretely valued field
In this and subsequent sections, we will therefore assume that the base field
is Henselian and discretely valued, and with ring of integers
The technical assumption needed for the Moy–Prasad isomorphism to exist is that
splits over a tamely ramified extension of the base field
[4] The Moy–Prasad can be used to define an important numerical invariant of a smooth representation
In a sequel to the paper defining their filtration, Moy and Prasad proved a structure theorem for depth-zero supercuspidal representations.
of a representation of this quotient that is cuspidal in the sense of Harish-Chandra (see also Deligne–Lusztig theory).
Moreover, every depth-zero supercuspidal representation is isomorphic to one of this form.
In the tame case, the local Langlands correspondence is expected to preserve depth, where the depth of an L-parameter is defined using the upper numbering filtration on the Weil group.
Our description of the construction follows Yu's article on smooth models.
[7] Since algebraic tori are a particular class of reductive groups, the theory of the Moy–Prasad filtration applies to them as well.
Since the reduced building of a torus is a point there is only one choice for
A torus is said to be induced if it is the direct product of finitely many tori of the form considered in the previous paragraph.
is defined as the integral points of the Néron lft-model of
This construction is independent of the choice of induced torus and embedding.
is defined by unramified descent from the quasi-split case, a standard trick in Bruhat–Tits theory.
of rational points: the former is an algebraic variety whereas the second is only an abstract group.
For this reason, there are many technical advantages to working not only with the abstract group