Similarly we can form the adelic ring a f of a general number eld f. Continuous representation theory of p adic lie groups 1263 where lg is the group of krational points of the connected reductive group over k whose root datum is dual to the root datum of g over l. Fix a maximal torus t of g which splits in an unramified extension of k, and which has the same split rank as the. This text contains an elementary introduction, with exercises, to representations of padic reductive groups. Foreach p, the gfrepresentation ontpe encodes all euler factors at primes.
These notes are an introduction to the pmodular or \modp representation theory of padic reductive groups. These are notes i am taking for shrenik shahs ongoing course on padic reductive groups o ered at columbia university in fall 2017 math gr8674. The category ofall smooth representations ofg and gintertwinings among them, is denoted by algg. Conference and workshop on the representation theory of padic groups, july 10th19th. Representation theory of reductive p adic groups, besides its importance for harmonic analyses, is very important for langlands program. Introduction to the theory of admissible representations of p. It is easy to see that if g is an algebraic group over f then gf is an l group. The theory of representation of padic reductive groups has nowadays attained a mature stage of. Vigneras and others published induced rrepresentations of padic reductive groups find, read and cite all the research you need on researchgate. Introduction to the theory of admissible representations of padic reductive groups w. Generalized semiinvariant distributions on padic spaces. Since qun p is not p adically complete any more, we need to work with qdun p instead. Let c cq p be the vector space of cvalued compactly supported.
This volume is an outgrowth of the program modular representation theory of finite and padic groups held at the institute for mathematical sciences at national university of singapore during the period of 126 april 20. Note above that the additive group ga is the union of its compact open subgroups. Modular representation theory of finite and padic groups. C is viewed as a topological group via the isomorphism from hom cq p. This plays a very important role for harmonic analysis on unipotent padic groups. Representations of padic groups lectures by joseph. Structure and representation theory of p adic reductive groups notes taken by pakhin lee. Basic representation theory of reductive padic groups. Baum and others published representation theory of adic groups. Representations of real and padic groups lecture notes. A locally pro nite group is a group gsatisfying any one of the. Integration, functions, and distributions see weil wei95 for integration and fourier analysis on p adic elds.
Let k be a local field with residue field k of characteristic p. Representations of padic groups for the modularity seminar. These examples are all special cases of the true source of adic representations, which is adic etale cohomology. In mathematics, the p adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. One seeks linear algebra data in the other cohomology groups attached to x. Generalized semiinvariant distributions on p adic spaces 1729 we suggest in this paper that, to describe all semiinvariant distributions on the spaces. Conference and workshop on the representation theory of p adic groups, july 10th19th. For the representation theory of p adic groups, the correct analogy is going to be representations of nite groups, rather than the theory for reductive. Introduction to the theory of admissible representations of p adic reductive groups w. Harmonic analysis and representation theory of p adic reductive groups updated on 19th may 2016 these are the notes of my spring 2016 class at the university of chicago on the representation theory of p adic groups. Harmonic analysis and representation theory of padic.
Existence of invariant norms in padic representations of. We will focus on the group gl 2q p, but try to provide statements that generalize to an arbitrary padic reductive group gfor example, gl nq p. In this subsection, we always assume that a is a category with in. Structure and representation theory of padic reductive groups. In sii, we build up the machinery which enables us, after jacquet, to reduce the classification of the irreducible admissible representations for a padic group g to. For this reason, the theory of padic representations of galois groups turns out to be a very convenient framework for studying the arithmetic of lfunctions. On irreducible representations of compact padic analytic groups. The extension is achieved by an alternative interpretation of the concept of closeness or absolute value.
Introduction over the past several years, operator algebraists have become increasingly interested in the problem of calculating the ktheory of group calgebras. Such a correspondence has been shown for g gl2 when the. The padic hodge theory of semistable galois representations. This site is like a library, use search box in the widget to get ebook that you want. As complex representations of nite groups are discussed in earlier courses, some results concerning representations of gin the case when f is a nite eld will be assumed. The overflow blog coming together as a community to connect. An informal characterization of cug is that it is the smallest full subcategory of smooth g representations such that. Representations of reductive padic groups under revision. The irreducible representations of the heisenberg group with nontrivial central char acter. In particular, one obtains a natural notion of parabolic subgroups of a p adic group g gf and their levi factors, which allows one to consider the process of parabolic induction.
We make a few remarks about the connection between representations of padic groups and automorphic forms. Introduction to p adic analytic number theory download. The representation theory of p adic groups is closely related to the structure of the underlying algebraic group. Structure and representation theory of padic reductive groups notes taken by pakhin lee abstract. In the following article, the authors want to give their contribution to the subject with focus on an p adic analogue of the classical hilbert space featuring phenomena such as selfduality etc.
We exemplify this relation in the case of the padic group of type g2. This is an adic gkrepresentation of dimension 2g, where g is the dimension of a. On irreducible representations of compact p adic analytic groups by konstantin ardakov and simon wadsley abstract we prove that the canonical dimension of a coadmissible representation of a semisimple p adic lie group in a p adic banach space is either zero or at least half the dimension of a nonzero coadjoint orbit. These notes will keep updating as the lectures progress. Representations mod p of the p adic group for the hecke f p algebra of a general reductive p adic group g with respect to a pro p iwahori sub group is probably accessible. Continuous representation theory of padic lie groups. This latter approach will allow to show that padic integers form a ring, denoted by zp. Representations of padic symplectic groups markotadi. Harmonic analysis and representation theory of padic reductive groups updated on 19th may 2016 these are the notes of my spring 2016 class at the university of chicago on the representation theory of padic groups. Introduction to admissible representations of padic groups. Moreover, very interesting examples of padic galois representations arise from geometry. In this section we shall recall some well known facts from the representation theory of reductive padic groups. It also gives us often better understanding of representation theory of reductive lie groups. Harmonic analysis and representation theory of padic reductive.
We let eur be the maximal unrami ed extension of ewith ring of integers o eur, and we can take the padic completion ob eur, and by inverting pwe get ebur. Click download or read online button to get introduction to p adic analytic number theory book now. A question on representation theory of padic groups. Many interesting properties of p adic representations are encoded in the. N, z, q, rand care the sets of respectively the natural numbers i. On the unicity of types for representations of reductive p. In this case a continuous representation may have in. The purpose of the padic langlands programme is to create a similar correspondence between some of the padic representations of the reductive group, and some of the padic representations of the galois group. But it does hold for groups g gk, whenever g is a unipotent group over a padic. Pdf induced rrepresentations of padic reductive groups. Let k be a padic field, and g a reductive connected algebraic group over k. Madore revised 7th december 2000 in all that follows, pwill stand for a prime number. G k glv, where v is a finitedimensional vector space over q p.