Moy–Prasad filtration

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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

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In their foundational work on the theory of buildings, Bruhat and Tits defined subgroups associated to concave functions of the root system.[1] These subgroups are a special case of the Moy–Prasad subgroups, defined when the group is split. 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.

Examples

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The following examples use the p-adic rational numbers   and the p-adic integers  . A reader unfamiliar with these rings may instead replace   by the rational numbers   and   by the integers   without losing the main idea.

Multiplicative group

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The simplest example of a p-adic reductive group is  , the multiplicative group of p-adic units. Since   is abelian, it has a unique parahoric subgroup,  . The Moy–Prasad subgroups of   are the higher unit groups  , where for simplicity   is a positive integer:  The Lie algebra of   is  , and its Moy–Prasad subalgebras are the nonzero ideals of  : More generally, if   is a positive real number then we use the floor function to define the  th Moy–Prasad subgroup and subalgebra:  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  th and  th Moy–Prasad subgroups are equal if   is only slightly larger than  .

General linear group

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Another important example of a p-adic reductive group is the general linear group  ; this example generalizes the previous one because  . Since   is nonabelian (when  ), it has infinitely many parahoric subgroups. One particular parahoric subgroup is  . The Moy–Prasad subgroups of   are the subgroups of elements equal to the identity matrix   modulo high powers of  . Specifically, when   is a positive integer we define where   is the algebra of n × n matrices with coefficients in  . The Lie algebra of   is  , and its Moy–Prasad subalgebras are the spaces of matrices equal to the zero matrix modulo high powers of  ; when   is a positive integer we define Finally, as before, if   is a positive real number then we use the floor function to define the  th Moy–Prasad subgroup and subalgebra: In this example, the Moy–Prasad groups would more commonly be denoted by   instead of  , where   is a point of the building of   whose corresponding parahoric subgroup is  

Properties

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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  , not just over a nonarchimedean local field. In this and subsequent sections, we will therefore assume that the base field   is Henselian and discretely valued, and with ring of integers  . Nonetheless, the reader is welcome to assume for simplicity that  , so that  .

Let   be a reductive  -group, let  , and let   be a point of the extended Bruhat-Tits building of  . The  th Moy–Prasad subgroup of   at   is denoted by  . Similarly, the  th Moy–Prasad Lie subalgebra of   at   is denoted by  ; it is a free  -module spanning  , or in other words, a lattice. (In fact, the Lie algebra   can also be defined when  , though the group   cannot.)

Perhaps the most basic property of the Moy–Prasad filtration is that it is decreasing: if   then   and  . It is standard to then define the subgroup and subalgebra This convention is just a notational shortcut because for any  , there is an   such that   and  .

The Moy–Prasad filtration satisfies the following additional properties.[3]

  • A jump in the Moy–Prasad filtration is defined as an index (that is, nonnegative real number)   such that  . The set of jumps is discrete and countably infinite.
  • If   then   is a normal subgroup of   and   is an ideal of  . It is a notational convention in the subject to write   and   for the associated quotients.
  • The quotient   is a reductive group over the residue field of  , namely, the maximal reductive quotient of the special fiber of the  -group underlying the parahoric  . In particular, if   is a nonarchimedean local field (such as  ) then this quotient is a finite group of Lie type.
  •   and  ; here the first bracket is the commutator and the second is the Lie bracket.
  • For any automorphism   of   we have   and  , where   is the derivative of  .
  • For any uniformizer   of   we have  .

Under certain technical assumptions on  , an additional important property is satisfied. By the commutator subgroup property, the quotient   is abelian if  . In this case there is a canonical isomorphism  , called the Moy–Prasad isomorphism. The technical assumption needed for the Moy–Prasad isomorphism to exist is that   be tame, meaning that   splits over a tamely ramified extension of the base field  . If this assumption is violated then   and   are not necessarily isomorphic.[4]

Depth of a representation

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The Moy–Prasad can be used to define an important numerical invariant of a smooth representation   of  , the depth of the representation: this is the smallest number   such that for some point   in the building of  , there is a nonzero vector of   fixed by  .

In a sequel to the paper defining their filtration, Moy and Prasad proved a structure theorem for depth-zero supercuspidal representations.[5] Let   be a point in a minimal facet of the building of  ; that is, the parahoric subgroup   is a maximal parahoric subgroup. The quotient   is a finite group of Lie type. Let   be the inflation to   of a representation of this quotient that is cuspidal in the sense of Harish-Chandra (see also Deligne–Lusztig theory). The stabilizer   of   in   contains the parahoric group   as a finite-index normal subgroup. Let   be an irreducible representation of   whose restriction to   contains   as a subrepresentation. Then the compact induction of   to   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.[6]

Construction

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Although we defined   to lie in the extended building of  , it turns out that the Moy–Prasad subgroup   depends only on the image of   in the reduced building, so that nothing is lost by thinking of   as a point in the reduced building.

Our description of the construction follows Yu's article on smooth models.[7]

Tori

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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   is a torus. Since the reduced building of a torus is a point there is only one choice for  , and so we will suppress   from the notation and write  .

First, consider the special case where   is the Weil restriction of   along a finite separable extension   of  , so that  . In this case, we define   as the set of   such that  , where   is the unique extension of the valuation of   to  .

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

Second, consider the case where   but   is an arbitrary torus. Here the Moy–Prasad subgroup   is defined as the integral points of the Néron lft-model of  .[8] This definition agrees with the previously given one when   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  , then, we choose an embedding of   in an induced torus   and define  . This construction is independent of the choice of induced torus and embedding.

Reductive groups

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For simplicity, we will first outline the construction of the Moy–Prasad subgroup   in the case where   is split. After, we will comment on the general definition.

Let   be a maximal split torus of   whose apartment contains  , and let   be the root system of   with respect to  .

For each  , let   be the root subgroup of   with respect to  . As an abstract group   is isomorphic to  , though there is no canonical isomorphism. The point   determines, for each root  , an additive valuation  . We define  .

Finally, the Moy–Prasad subgroup   is defined as the subgroup of   generated by the subgroups   for   and the subgroup  .

If   is not split, then the Moy–Prasad subgroup   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   is split, to the case where   is only quasi-split, using the relative root system. From here, the Moy–Prasad subgroup can be defined for an arbitrary   by passing to the maximal unramified extension   of  , a field over which every reductive group, and in particular  , is quasi-split, and then taking the fixed points of this Moy–Prasad group under the Galois group of   over  .

Group schemes

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The  -group   carries much more structure than the group   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  , but also the variety  . Similarly, although we described   as an abstract group, a certain subgroup of  , it is desirable for   to be the group of integral points of a group scheme   defined over the ring of integers, so that  . In fact, it is possible to construct such a group scheme  .

Lie algebras

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Let   be the Lie algebra of  . 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   of  ; they are free  -modules, that is,  -lattices in the  -vector space  . When  , it turns out that   is just the Lie algebra of the  -group scheme  .

Indexing set

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We have defined the Moy–Prasad filtration at the point   to be indexed by the set   of real numbers. It is common in the subject to extend the indexing set slightly, to the set   consisting of   and formal symbols   with  . The element   is thought of as being infinitesimally larger than  , and the filtration is extended to this case by defining  . Since the valuation on   is discrete, there is   such that  .

See also

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Citations

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  1. ^ Bruhat & Tits 1972, Section 6.4.
  2. ^ Moy & Prasad 1994.
  3. ^ Hakim & Murnaghan 2010, Section 2.5.
  4. ^ Yu 2015, section 5.
  5. ^ Moy & Prasad 1996, Proposition 6.6.
  6. ^ Chen & Kamgarpour 2014, Section 1.
  7. ^ Yu 2015.
  8. ^ Bosch, Lütkebohmert & Raynaud 1990, Chapter 10.

References

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  • Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel (1990). Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Vol. 21. Berlin, New York: Springer-Verlag. doi:10.1007/978-3-642-51438-8. ISBN 978-3-540-50587-7. MR 1045822.
  • Bruhat, F.; Tits, J. (1972). "Groupes Réductifs Sur Un Corps Local". Publications mathématiques de l'IHÉS. 41 (1): 5–251. doi:10.1007/BF02715544. ISSN 0073-8301. MR 0327923.
  • Chen, Tsao-Hsien; Kamgarpour, Masoud (2014). "Preservation of depth in local geometric Langlands correspondence". arXiv:1404.0598 [math.RT].
  • Hakim, J.; Murnaghan, F. (8 July 2010). "Distinguished Tame Supercuspidal Representations". International Mathematics Research Papers. Oxford University Press (OUP). arXiv:0709.3506. doi:10.1093/imrp/rpn005. ISSN 1687-3017. MR 2431732.
  • Moy, Allen; Prasad, Gopal (1994). "Unrefined minimal K-types for p-adic groups". Inventiones Mathematicae. 116 (1): 393–408. doi:10.1007/BF01231566. hdl:2027.42/46580. ISSN 0020-9910. MR 1253198.
  • Moy, Allen; Prasad, Gopal (1996). "Jacquet functors and unrefined minimal K-types". Commentarii Mathematici Helvetici. 71 (1). European Mathematical Society Publishing House: 98–121. doi:10.1007/bf02566411. ISSN 0010-2571. MR 1371680.
  • Yu, Jiu-Kang (2015). "Smooth models associated to concave functions in Bruhat-Tits theory". Autour des schémas en groupes, Vol. III. Panor. Synthèses. Vol. 47. pp. 227–258. MR 3525846.