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In mathematics, the '''Moy–Prasad  filtration''' is a family of [[filtration (mathematics)|filtrations]] of [[p-adic number|''p''-adic]] [[reductive group]]s and their [[Lie algebra]]s, named after Allen Moy and [[Gopal Prasad]]. The family is parameterized by the [[building (mathematics)|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 ''r''th 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.{{sfn | Bruhat | Tits | 1972 | loc=Section 6.4}} These subgroups are a special case of the Moy–Prasad  subgroups, defined when the group is [[Reductive group#Definitions|split]]. The main innovations of Moy and Prasad{{sfn | Moy | Prasad | 1994}} were to generalize Bruhat–Tits's construction to [[quasi-split group]]s, in particular [[algebraic torus|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 <math>\mathbb{Q}_p</math> and the ''p''-adic integers <math>\mathbb{Z}_p</math>. A reader unfamiliar with these rings may instead replace <math>\mathbb{Q}_p</math> by the [[Rational number|rational numbers]] <math>\mathbb{Q}</math> and <math>\mathbb{Z}_p</math> by the [[Integer|integers]] <math>\mathbb{Z}</math> without losing the main idea.

=== Multiplicative group ===
The simplest example of a ''p''-adic reductive group is <math>\mathbb{Q}_p^\times</math>, the [[multiplicative group]] of ''p''-adic [[Unit (ring theory)|units]]. Since <math>\mathbb{Q}_p^\times</math> is [[Abelian group|abelian]], it has a unique parahoric subgroup,  <math>\mathbb{Z}_p^\times</math>. The Moy–Prasad  subgroups of <math>\mathbb{Z}_p^\times</math> are the [[Higher unit group|higher unit groups]] <math>U^{(r)}</math>, where for simplicity <math>r</math> is a positive integer: <math display="block">(\mathbb{Z}_p^\times)_r=1+(p\,\mathbb{Z}_p)^r = \{u\in\mathbb{Z}_p^\times:u\equiv1\bmod{p^r}\}.</math>The Lie algebra of <math>\mathbb{Q}_p^\times</math> is <math>\mathbb{Q}_p</math>, and its Moy–Prasad  subalgebras are the nonzero [[Ideal (ring theory)|ideals]] of <math>\mathbb{Z}_p</math>:<math display="block">(\mathbb{Z}_p)_r=(p\,\mathbb{Z}_p)^r = \{p^ra : a\in\mathbb{Z}_p\}.</math>More generally, if <math>r</math> is a positive real number then we use the [[Floor and ceiling functions|floor function]] to define the <math>r</math>th Moy–Prasad  subgroup and subalgebra: <math display="block">(\mathbb{Z}_p^\times)_r := (\mathbb{Z}_p^\times)_{\lfloor r\rfloor},\qquad(\mathbb{Z}_p)_r := (\mathbb{Z}_p)_{\lfloor r\rfloor}</math>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 <math>r</math>th and <math>s</math>th Moy–Prasad  subgroups are equal if <math>s</math> is only slightly larger than <math>r</math>.

=== General linear group ===
Another important example of a ''p''-adic reductive group is the [[general linear group]] <math>\text{GL}_n(\mathbb{Q}_p)</math>; this example generalizes the previous one because <math>\text{GL}_1(\mathbb{Q}_p) = \mathbb{Q}_p^\times</math>. Since <math>\text{GL}_n(\mathbb{Q}_p)</math> is nonabelian (when <math>n\geq2</math>), it has infinitely many parahoric subgroups. One particular parahoric subgroup is <math>\text{GL}_n(\mathbb{Z}_p)</math>. The Moy–Prasad  subgroups of <math>\text{GL}_n(\mathbb{Z}_p)</math> are the subgroups of elements equal to the [[identity matrix]] <math>1</math> modulo high powers of <math>p</math>. Specifically, when <math>r</math> is a positive integer we define<math display="block">\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}\}.</math>where <math>\text{M}_n(\mathbb{Z}_p)</math> is the [[matrix algebra|algebra of ''n × n'' matrices]] with coefficients in <math>\mathbb{Z}_p</math>. The Lie algebra of <math>\text{GL}_n(\mathbb{Q}_p)</math> is <math>\text{M}_n(\mathbb{Q}_p)</math>, and its Moy–Prasad  subalgebras are the spaces of matrices equal to the [[zero matrix]] modulo high powers of <math>p</math>; when <math>r</math> is a positive integer we define<math display="block">\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}\}.</math>Finally, as before, if <math>r</math> is a positive real number then we use the [[Floor and ceiling functions|floor function]] to define the <math>r</math>th Moy–Prasad  subgroup and subalgebra:<math display="block">\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}</math>In this example, the Moy–Prasad  groups would more commonly be denoted by <math>\text{GL}_n(\mathbb{Q}_p)_{x,r}</math> instead of <math>\text{GL}_n(\mathbb{Z}_p)_r</math>, where <math>x</math> is a point of the building of <math>\text{GL}_n(\mathbb{Q}_p)</math> whose corresponding parahoric subgroup is <math>\text{GL}_n(\mathbb{Z}_p).</math>

==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 ring|Henselian]], [[Discrete valuation|discretely valued]] field <math>k</math>, not just over a nonarchimedean local field. In this and subsequent sections, we will therefore assume that the base field <math>k</math> is Henselian and discretely valued, and with ring of integers <math>\mathcal{O}_k</math>. Nonetheless, the reader is welcome to assume for simplicity that <math>k=\mathbb{Q}_p</math>, so that <math>\mathcal{O}_k=\mathbb{Z}_p</math>.

Let <math>G</math> be a reductive <math>k</math>-group, let <math>r\geq0</math>, and let <math>x</math> be a point of the extended Bruhat-Tits building of <math>G</math>. The <math>r</math>th Moy–Prasad  subgroup of <math>G(k)</math> at <math>x</math> is denoted by <math>G(k)_{x,r}</math>. Similarly, the <math>r</math>th Moy–Prasad  Lie subalgebra of <math>\mathfrak{g}</math> at <math>x</math> is denoted by <math>\mathfrak{g}_{x,r}
</math>; it is a [[Free module|free]] <math>\mathcal{O}_k</math>-module spanning <math>\mathfrak{g}_{x,r}
</math>, or in other words, a [[Lattice (module)|lattice]]. (In fact, the Lie algebra <math>\mathfrak{g}_{x,r}
</math> can also be defined when <math>r<0</math>, though the group <math>G(k)_{x,r}</math> cannot.)

Perhaps the most basic property of the Moy–Prasad  filtration is that it is decreasing: if <math>r\leq s
</math> then <math>\mathfrak{g}_{x,r}\supseteq\mathfrak{g}_{x,s}
</math> and <math>G(k)_{x,r}\supseteq G(k)_{x,s}</math>. It is standard to then define the subgroup and subalgebra<math display="block">G(k)_{x,r+}:=\bigcup_{s>r} G(k)_{x,s},
\qquad
\mathfrak{g}_{x,r+}:=\bigcup_{s>r} \mathfrak{g}_{x,s}.</math>This convention is just a notational shortcut because for any <math>r
</math>, there is an <math>\varepsilon>0
</math> such that <math>\mathfrak{g}_{x,r+}=\mathfrak{g}_{x,r+\varepsilon}</math> and <math>G(k)_{x,r+}=G(k)_{x,r+\varepsilon}</math>.

The Moy–Prasad  filtration satisfies the following additional properties.{{sfn | Hakim | Murnaghan | 2010 | loc=Section 2.5}}

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

Under certain technical assumptions on <math>G</math>, an additional important property is satisfied. By the commutator subgroup property, the quotient <math>G(k)_{x,r:s}</math> is abelian if <math>r\leq s\leq 2r</math>. In this case there is a canonical isomorphism <math>\mathfrak{g}_{x,r:s}\cong G(k)_{x,r:s}</math>, called the '''Moy–Prasad  isomorphism'''. The technical assumption needed for the Moy–Prasad  isomorphism to exist is that <math>G</math> be '''tame''', meaning that <math>G</math> splits over a [[Tame ramification|tamely ramified]] extension of the base field <math>k</math>. If this assumption is violated then <math>\mathfrak{g}_{x,r:s}</math> and <math>G(k)_{x,r:s}</math> are not necessarily isomorphic.{{sfn | Yu | 2015 |p=|loc=section 5}}

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

In a sequel to the paper defining their filtration, Moy and Prasad proved a structure theorem for depth-zero supercuspidal representations.{{sfn | Moy | Prasad | 1996 | loc=Proposition 6.6}} Let <math>x</math> be a point in a minimal facet of the building of <math>G</math>; that is, the parahoric subgroup <math>G(k)_{x,0}</math> is a maximal parahoric subgroup. The quotient <math>G(k)_{x,0:0+}</math> is a finite group of Lie type. Let <math>\tau</math> be the inflation to <math>G(k)_{x,0}</math> of a representation of this quotient that is cuspidal in the sense of Harish-Chandra (see also [[Deligne–Lusztig theory]]). The stabilizer <math>G(k)_x</math> of <math>x</math> in <math>G(k)</math> contains the parahoric group <math>G(k)_{x,0}</math> as a finite-index normal subgroup. Let <math>\rho</math> be an irreducible representation of <math>G(k)_x</math> whose restriction to <math>G(k)_{x,0}</math> contains <math>\tau</math> as a subrepresentation. Then the [[compact induction]] of <math>\rho</math> to <math>G(k)</math> 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.{{sfn | Chen | Kamgarpour | 2014 | loc=Section 1}}

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

Our description of the construction follows Yu's article on smooth models.{{sfn|Yu|2015}}

===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 <math>G=T</math> is a torus. Since the reduced building of a torus is a point there is only one choice for <math>x</math>, and so we will suppress <math>x</math> from the notation and write <math>T(k)_r:=T(k)_{x,r}</math>.

First, consider the special case where <math>T</math> is the [[Weil restriction]] of <math>\mathbb{G}_\text{m}</math> along a finite separable extension <math>\ell</math> of <math>k</math>, so that <math>T(k)=\ell^\times</math>. In this case, we define <math>T(k)_r</math> as the set of <math>a\in\ell^\times</math> such that <math>\text{val}_k(x-1)\geq r</math>, where <math>\text{val}_k:\ell\to\mathbb{R}</math> is the unique [[Valuation (algebra)#Extension of valuations|extension]] of the valuation of <math>k</math> to <math>\ell</math>.

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 <math>r</math>th Moy–Prasad  subgroup of an induced torus is defined as the product of the <math>r</math>th Moy–Prasad  subgroup of these factors.

Second, consider the case where <math>r=0</math> but <math>T</math> is an arbitrary torus. Here the Moy–Prasad  subgroup <math>T(k)_0</math> is defined as the integral points of the [[Néron model|Néron lft-model]] of <math>T</math>.{{sfn|Bosch|Lütkebohmert|Raynaud|1990|loc=Chapter 10}} This definition agrees with the previously given one when <math>T</math> 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 <math>T</math>, then, we choose an embedding of <math>T</math> in an induced torus <math>S</math> and define <math>T(k)_r:=T(k)_0\cap S(k)_r</math>. 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 <math>G(k)_{x,r}</math> in the case where <math>G</math> is [[Reductive group#Definitions|split]]. After, we will comment on the general definition.

Let <math>T</math> be a maximal split torus of <math>G</math> whose apartment contains <math>x</math>, and let <math>\Phi</math> be the [[root system]] of <math>G</math> with respect to <math>T</math>.

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

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

If <math>G</math> is not split, then the Moy–Prasad  subgroup <math>G(k)_{x,r}</math> 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 <math>G</math> is split, to the case where <math>G</math> is only quasi-split, using the [[relative root system]]. From here, the Moy–Prasad  subgroup can be defined for an arbitrary <math>G</math> by passing to the maximal unramified extension <math>k^\text{nr}</math> of <math>k</math>, a field over which every reductive group, and in particular <math>G</math>, is quasi-split, and then taking the fixed points of this Moy–Prasad  group under the Galois group of <math>k^\text{nr}</math> over <math>k</math>.

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

===Lie algebras===
Let <math>\mathfrak{g}</math> be the Lie algebra of <math>G</math>. 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 <math>\mathfrak{g}_{x,r}</math> of <math>\mathfrak{g}</math>; they are free <math>\mathcal{O}_k</math>-modules, that is, <math>\mathcal{O}_k</math>-lattices in the <math>k</math>-vector space <math>\mathfrak{g}</math>. When <math>r\geq0</math>, it turns out that <math>\mathfrak{g}_{x,r}</math> is just the Lie algebra of the <math>\mathcal{O}_k</math>-group scheme <math>G_{x,r}</math>.

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

==See also==
* [[Congruence subgroup]]

==Citations==
{{reflist}}

==References==
{{refbegin}}
* {{cite book | last1=Bosch | first1=Siegfried | last2=Lütkebohmert | first2=Werner | last3=Raynaud | first3=Michel | author3-link= Michel Raynaud | title=Néron models | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] (3)| isbn=978-3-540-50587-7 |mr=1045822 | year=1990 | volume=21 | doi=10.1007/978-3-642-51438-8}}
*{{cite journal | last1=Bruhat | first1=F. | last2=Tits | first2=J. | title=Groupes Réductifs Sur Un Corps Local | journal=Publications mathématiques de l'IHÉS | volume=41 | issue=1 | year=1972 | pages=5–251 | issn=0073-8301 | doi=10.1007/BF02715544 | url=http://www.numdam.org/item/PMIHES_1972__41__5_0/ | mr=0327923}}
*{{cite arXiv | last1=Chen | first1=Tsao-Hsien | first2=Masoud | last2=Kamgarpour | eprint=1404.0598 | title=Preservation of depth in local geometric Langlands correspondence | class=math.RT | date=2014 }}
* {{cite journal | last=Hakim | first=J. | last2=Murnaghan | first2=F. | title=Distinguished Tame Supercuspidal Representations | journal=International Mathematics Research Papers | publisher=Oxford University Press (OUP) | date=8 July 2010 | arxiv=0709.3506 | issn=1687-3017 | doi=10.1093/imrp/rpn005 | mr=2431732 }}
*{{cite journal | last1=Moy | first1=Allen | last2=Prasad | first2=Gopal | title=Unrefined minimal K-types for ''p''-adic groups | journal=Inventiones Mathematicae | volume=116 | issue=1 | year=1994 | pages=393–408 | issn=0020-9910 | doi=10.1007/BF01231566 | mr=1253198 | hdl=2027.42/46580 | hdl-access=free }}
*{{cite journal | last=Moy | first=Allen | last2=Prasad | first2=Gopal | title=Jacquet functors and unrefined minimal K-types | journal=[[Commentarii Mathematici Helvetici]] | publisher=European Mathematical Society Publishing House | volume=71 | issue=1 | year=1996 | issn=0010-2571 | doi=10.1007/bf02566411 | pages=98–121 | mr=1371680}}
*{{cite encyclopedia | last=Yu | first=Jiu-Kang | encyclopedia=Autour des schémas en groupes, Vol. III | title=Smooth models associated to concave functions in Bruhat-Tits theory | series=Panor. Synthèses | volume=47 | date=2015 | pages=227–258 | mr=3525846}}
{{refend}}


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[[Category:Representation theory of algebraic groups]]