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: $$(\mathbb{Z}_p^\times)_r=1+(p\,\mathbb{Z}_p)^r = \{u\in\mathbb{Z}_p^\times:u\equiv1\bmod{p^r}\}.$$The Lie algebra of $$\mathbb{Q}_p^\times$$ is $$\mathbb{Q}_p$$, and its Moy–Prasad  subalgebras are the nonzero ideals of $$\mathbb{Z}_p$$:$$(\mathbb{Z}_p)_r=(p\,\mathbb{Z}_p)^r = \{p^ra : a\in\mathbb{Z}_p\}.$$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 $$n\geq2$$), 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$$\text{GL}_n(\mathbb{Z}_p)_r=1+(p\,\text{M}_n(\mathbb{Z}_p))^r = \{u\in\text{M}_n(\mathbb{Z}_p):u\equiv1\bmod{p^r}\}.$$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$$\text{M}_n(\mathbb{Z}_p)_r=(p\,\text{M}_n(\mathbb{Z}_p))^r = \{u\in\text{M}_n(\mathbb{Z}_p):u\equiv0\bmod{p^r}\}.$$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 $$r\geq0$$, 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 $$r\geq0$$, 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 $$G(k)_{x,r+}=G(k)_{x,r+\varepsilon}$$.