Moy–Prasad filtration
# Moy–Prasad filtration

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 $\mathbb {Q} _{p}$ and the p-adic integers $\mathbb {Z} _{p}$. A reader unfamiliar with these rings may instead replace $\mathbb {Q} _{p}$ by the rational numbers $\mathbb {Q}$ and $\mathbb {Z} _{p}$ by the integers $\mathbb {Z}$ without losing the main idea.

### Multiplicative group

The simplest example of a p-adic reductive group is $\mathbb {Q} _{p}^{\times }$, the multiplicative group of p-adic units. Since $\mathbb {Q} _{p}^{\times }$ is abelian, it has a unique parahoric subgroup, $\mathbb {Z} _{p}^{\times }$. The Moy–Prasad subgroups of $\mathbb {Z} _{p}^{\times }$ are the higher unit groups $U^{(r)}$, where for simplicity $r$ is a positive integer:

$(\mathbb {Z} _{p}^{\times })_{r}=1+(p\,\mathbb {Z} _{p})^{r}=\{u\in \mathbb {Z} _{p}^{\times }:u\equiv 1{\bmod {p^{r}}}\}.$

The Lie algebra of $\mathbb {Q} _{p}^{\times }$ is $\mathbb {Q} _{p}$, and its Moy–Prasad subalgebras are the nonzero ideals of $\mathbb {Z} _{p}$:

$(\mathbb {Z} _{p})_{r}=(p\,\mathbb {Z} _{p})^{r}=\{p^{r}a:a\in \mathbb {Z} _{p}\}.$

More generally, if $r$ is a positive real number then we use the floor function to define the $r$th Moy–Prasad subgroup and subalgebra:

$(\mathbb {Z} _{p}^{\times })_{r}:=(\mathbb {Z} _{p}^{\times })_{\lfloor r\rfloor },\qquad (\mathbb {Z} _{p})_{r}:=(\mathbb {Z} _{p})_{\lfloor r\rfloor }$

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. In particular, it is usually the case that the $r$th and $s$th Moy–Prasad subgroups are equal if $s$ is only slightly larger than $r$.

### General linear group

Another important example of a p-adic reductive group is the general linear group ${\text{GL}}_{n}(\mathbb {Q} _{p})$; this example generalizes the previous one because ${\text{GL}}_{1}(\mathbb {Q} _{p})=\mathbb {Q} _{p}^{\times }$. Since ${\text{GL}}_{n}(\mathbb {Q} _{p})$ is nonabelian (when $n\geq 2$), it has infinitely many parahoric subgroups. One particular parahoric subgroup is ${\text{GL}}_{n}(\mathbb {Z} _{p})$. The Moy–Prasad subgroups of ${\text{GL}}_{n}(\mathbb {Z} _{p})$ are the subgroups of elements equal to the identity matrix $1$ modulo high powers of $p$. Specifically, when $r$ is a positive integer we define

${\text{GL}}_{n}(\mathbb {Z} _{p})_{r}=1+(p\,{\text{M}}_{n}(\mathbb {Z} _{p}))^{r}=\{u\in {\text{M}}_{n}(\mathbb {Z} _{p}):u\equiv 1{\bmod {p^{r}}}\}.$

where ${\text{M}}_{n}(\mathbb {Z} _{p})$ is the algebra of n × n matrices with coefficients in $\mathbb {Z} _{p}$. The Lie algebra of ${\text{GL}}_{n}(\mathbb {Q} _{p})$ is ${\text{M}}_{n}(\mathbb {Q} _{p})$, and its Moy–Prasad subalgebras are the spaces of matrices equal to the zero matrix modulo high powers of $p$; when $r$ is a positive integer we define

${\text{M}}_{n}(\mathbb {Z} _{p})_{r}=(p\,{\text{M}}_{n}(\mathbb {Z} _{p}))^{r}=\{u\in {\text{M}}_{n}(\mathbb {Z} _{p}):u\equiv 0{\bmod {p^{r}}}\}.$

Finally, as before, if $r$ is a positive real number then we use the floor function to define the $r$th Moy–Prasad subgroup and subalgebra:

${\text{GL}}_{n}(\mathbb {Z} _{p})_{r}:={\text{GL}}_{n}(\mathbb {Z} _{p})_{\lfloor r\rfloor },\qquad {\text{M}}_{n}(\mathbb {Z} _{p})_{r}:={\text{M}}_{n}(\mathbb {Z} _{p})_{\lfloor r\rfloor }$

In this example, the Moy–Prasad groups would more commonly be denoted by ${\text{GL}}_{n}(\mathbb {Q} _{p})_{x,r}$ instead of ${\text{GL}}_{n}(\mathbb {Z} _{p})_{r}$, where $x$ is a point of the building of ${\text{GL}}_{n}(\mathbb {Q} _{p})$ whose corresponding parahoric subgroup is ${\text{GL}}_{n}(\mathbb {Z} _{p}).$

## Properties

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 $k$, not just over a nonarchimedean local field. In this and subsequent sections, we will therefore assume that the base field $k$ is Henselian and discretely valued, and with ring of integers ${\mathcal {O}}_{k}$. Nonetheless, the reader is welcome to assume for simplicity that $k=\mathbb {Q} _{p}$, so that ${\mathcal {O}}_{k}=\mathbb {Z} _{p}$.

Let $G$ be a reductive $k$-group, let $r\geq 0$, and let $x$ be a point of the extended Bruhat-Tits building of $G$. The $r$th Moy–Prasad subgroup of $G(k)$ at $x$ is denoted by $G(k)_{x,r}$. Similarly, the $r$th Moy–Prasad Lie subalgebra of ${\mathfrak {g}}$ at $x$ is denoted by ${\mathfrak {g}}_{x,r}$; it is a free ${\mathcal {O}}_{k}$-module spanning ${\mathfrak {g}}_{x,r}$, or in other words, a lattice. (In fact, the Lie algebra ${\mathfrak {g}}_{x,r}$ can also be defined when $r<0$, though the group $G(k)_{x,r}$ cannot.)

Perhaps the most basic property of the Moy–Prasad filtration is that it is decreasing: if $r\leq s$ then ${\mathfrak {g}}_{x,r}\supseteq {\mathfrak {g}}_{x,s}$ and $G(k)_{x,r}\supseteq G(k)_{x,s}$. It is standard to then define the subgroup and subalgebra

$G(k)_{x,r+}:=\bigcup _{s>r}G(k)_{x,s},\qquad {\mathfrak {g}}_{x,r+}:=\bigcup _{s>r}{\mathfrak {g}}_{x,s}.$

This convention is just a notational shortcut because for any $r$, there is an $\varepsilon >0$ such that ${\mathfrak {g}}_{x,r+}={\mathfrak {g}}_{x,r+\varepsilon }$ and $G(k)_{x,r+}=G(k)_{x,r+\varepsilon }$.

The Moy–Prasad filtration satisfies the following additional properties.

* A jump in the Moy–Prasad filtration is defined as an index (that is, nonnegative real number) $r$ such that $G(k)_{x,r+}\neq G(k)_{x,r}$. The set of jumps is discrete and countably infinite.
* If $r\leq s$ then $G(k)_{x,s}$ is a normal subgroup of $G(k)_{x,r}$ and ${\mathfrak {g}}_{x,s}$ is an ideal of ${\mathfrak {g}}_{x,r}$. It is a notational convention in the subject to write $G(k)_{x,r:s}:=G(k)_{x,r}/G(k)_{x,s}$ and ${\mathfrak {g}}_{x,r:s}:={\mathfrak {g}}_{x,r}/{\mathfrak {g}}_{x,s}$ for the associated quotients.
* The quotient $G(k)_{x,0:0+}$ is a reductive group over the residue field of ${\mathcal {O}}_{k}$, namely, the maximal reductive quotient of the special fiber of the ${\mathcal {O}}_{k}$-group underlying the parahoric $G(k)_{x,0}$. In particular, if $k$ is a nonarchimedean local field (such as $\mathbb {Q} _{p}$) then this quotient is a finite group of Lie type.
* $[G(k)_{x,r},G(k)_{x,s}]\subseteq G(k)_{x,r+s}$ and $[{\mathfrak {g}}_{x,r},{\mathfrak {g}}_{x,s}]\subseteq {\mathfrak {g}}_{x,r+s}$; here the first bracket is the commutator and the second is the Lie bracket.
* For any automorphism $\theta$ of $G$ we have $\theta (G(k)_{x,r})=G(k)_{\theta (x),r}$ and ${\text{d}}\theta ({\mathfrak {g}}_{x,r})={\mathfrak {g}}_{\theta (x),r}$, where ${\text{d}}\theta$ is the derivative of $\theta$.
* For any uniformizer $\varpi$ of $k$ we have $\varpi {\mathfrak {g}}_{x,r}={\mathfrak {g}}_{x,r+1}$.

Under certain technical assumptions on $G$, an additional important property is satisfied. By the commutator subgroup property, the quotient $G(k)_{x,r:s}$ is abelian if $r\leq s\leq 2r$. In this case there is a canonical isomorphism ${\mathfrak {g}}_{x,r:s}\cong G(k)_{x,r:s}$, called the Moy–Prasad isomorphism. The technical assumption needed for the Moy–Prasad isomorphism to exist is that $G$ be tame, meaning that $G$ splits over a tamely ramified extension of the base field $k$. If this assumption is violated then ${\mathfrak {g}}_{x,r:s}$ and $G(k)_{x,r:s}$ are not necessarily isomorphic.

## Depth of a representation

The Moy–Prasad can be used to define an important numerical invariant of a smooth representation $(\pi ,V)$ of $G(k)$, the depth of the representation: this is the smallest number $r$ such that for some point $x$ in the building of $G$, there is a nonzero vector of $V$ fixed by $G(k)_{x,r+}$.

In a sequel to the paper defining their filtration, Moy and Prasad proved a structure theorem for depth-zero supercuspidal representations. Let $x$ be a point in a minimal facet of the building of $G$; that is, the parahoric subgroup $G(k)_{x,0}$ is a maximal parahoric subgroup. The quotient $G(k)_{x,0:0+}$ is a finite group of Lie type. Let $\tau$ be the inflation to $G(k)_{x,0}$ of a representation of this quotient that is cuspidal in the sense of Harish-Chandra (see also Deligne–Lusztig theory). The stabilizer $G(k)_{x}$ of $x$ in $G(k)$ contains the parahoric group $G(k)_{x,0}$ as a finite-index normal subgroup. Let $\rho$ be an irreducible representation of $G(k)_{x}$ whose restriction to $G(k)_{x,0}$ contains $\tau$ as a subrepresentation. Then the compact induction of $\rho$ to $G(k)$ is a depth-zero supercuspidal representation. 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.

## Construction

Although we defined $x$ to lie in the extended building of $G$, it turns out that the Moy–Prasad subgroup $G(k)_{x,r}$ depends only on the image of $x$ in the reduced building, so that nothing is lost by thinking of $x$ as a point in the reduced building.

Our description of the construction follows Yu's article on smooth models.

### Tori

Since algebraic tori are a particular class of reductive groups, the theory of the Moy–Prasad filtration applies to them as well. It turns out, however, that the construction of the Moy–Prasad subgroups for a general reductive group relies on the construction for tori, so we begin by discussing the case where $G=T$ is a torus. Since the reduced building of a torus is a point there is only one choice for $x$, and so we will suppress $x$ from the notation and write $T(k)_{r}:=T(k)_{x,r}$.

First, consider the special case where $T$ is the Weil restriction of $\mathbb {G} _{\text{m}}$ along a finite separable extension $\ell$ of $k$, so that $T(k)=\ell ^{\times }$. In this case, we define $T(k)_{r}$ as the set of $a\in \ell ^{\times }$ such that ${\text{val}}_{k}(x-1)\geq r$, where ${\text{val}}_{k}:\ell \to \mathbb {R}$ is the unique extension of the valuation of $k$ to $\ell$.

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. The $r$th Moy–Prasad subgroup of an induced torus is defined as the product of the $r$th Moy–Prasad subgroup of these factors.

Second, consider the case where $r=0$ but $T$ is an arbitrary torus. Here the Moy–Prasad subgroup $T(k)_{0}$ is defined as the integral points of the Néron lft-model of $T$. This definition agrees with the previously given one when $T$ is an induced torus.

It turns out that every torus can be embedded in an induced torus. To define the Moy–Prasad subgroups of a general torus $T$, then, we choose an embedding of $T$ in an induced torus $S$ and define $T(k)_{r}:=T(k)_{0}\cap S(k)_{r}$. This construction is independent of the choice of induced torus and embedding.

### Reductive groups

For simplicity, we will first outline the construction of the Moy–Prasad subgroup $G(k)_{x,r}$ in the case where $G$ is split. After, we will comment on the general definition.

Let $T$ be a maximal split torus of $G$ whose apartment contains $x$, and let $\Phi$ be the root system of $G$ with respect to $T$.

For each $\alpha \in \Phi$, let $U_{\alpha }$ be the root subgroup of $G$ with respect to $\alpha$. As an abstract group $U_{\alpha }$ is isomorphic to $\mathbb {G} _{\text{a}}$, though there is no canonical isomorphism. The point $x$ determines, for each root $\alpha$, an additive valuation $v_{\alpha ,x}:U_{\alpha }(k)\to \mathbb {R}$. We define $U_{\alpha }(k)_{x,r}:=\{u\in U_{\alpha }(k):v_{\alpha ,x}(u)\geq r\}$.

Finally, the Moy–Prasad subgroup $G(k)_{x,r}$ is defined as the subgroup of $G(k)$ generated by the subgroups $U_{\alpha }(k)_{x,r}$ for $\alpha \in \Phi$ and the subgroup $T(k)_{r}$.

If $G$ is not split, then the Moy–Prasad subgroup $G(k)_{x,r}$ is defined by unramified descent from the quasi-split case, a standard trick in Bruhat–Tits theory. More specifically, one first generalizes the definition of the Moy–Prasad subgroups given above, which applies when $G$ is split, to the case where $G$ is only quasi-split, using the relative root system. From here, the Moy–Prasad subgroup can be defined for an arbitrary $G$ by passing to the maximal unramified extension $k^{\text{nr}}$ of $k$, a field over which every reductive group, and in particular $G$, is quasi-split, and then taking the fixed points of this Moy–Prasad group under the Galois group of $k^{\text{nr}}$ over $k$.

### Group schemes

The $k$-group $G$ carries much more structure than the group $G(k)$ 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 $G(k)$, but also the variety $G(k)$. Similarly, although we described $G(k)_{x,r}$ as an abstract group, a certain subgroup of $G(k)$, it is desirable for $G(k)_{x,r}$ to be the group of integral points of a group scheme $G_{x,r}$ defined over the ring of integers, so that $G(k)_{x,r}=G_{x,r}({\mathcal {O}}_{k})$. In fact, it is possible to construct such a group scheme $G_{x,r}$.

### Lie algebras

Let ${\mathfrak {g}}$ be the Lie algebra of $G$. In a similar procedure as for reductive groups, namely, by defining Moy–Prasad filtrations on the Lie algebra of a torus and the Lie algebra of a root group, one can define the Moy–Prasad Lie algebras ${\mathfrak {g}}_{x,r}$ of ${\mathfrak {g}}$; they are free ${\mathcal {O}}_{k}$-modules, that is, ${\mathcal {O}}_{k}$-lattices in the $k$-vector space ${\mathfrak {g}}$. When $r\geq 0$, it turns out that ${\mathfrak {g}}_{x,r}$ is just the Lie algebra of the ${\mathcal {O}}_{k}$-group scheme $G_{x,r}$.

### Indexing set

We have defined the Moy–Prasad filtration at the point $x$ to be indexed by the set $\mathbb {R}$ of real numbers. It is common in the subject to extend the indexing set slightly, to the set ${\widetilde {\mathbb {R} }}$ consisting of $\mathbb {R}$ and formal symbols $r+$ with $r\in \mathbb {R}$. The element $r+$ is thought of as being infinitesimally larger than $r$, and the filtration is extended to this case by defining $G(k)_{x,r+}:=\bigcup _{s>r}G(k)_{x,s}$. Since the valuation on $k$ is discrete, there is $\varepsilon >0$ such that $G(k)_{x,r+}=G(k)_{x,r+\varepsilon }$.

