Fix a maximal torus t of g which splits in an unramified extension of k, and which has the same split rank as the. 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. On the unicity of types for representations of reductive p. Conference and workshop on the representation theory of padic groups, july 10th19th. 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.
The category ofall smooth representations ofg and gintertwinings among them, is denoted by algg. This site is like a library, use search box in the widget to get ebook that you want. We make a few remarks about the connection between representations of padic groups and automorphic forms. C is viewed as a topological group via the isomorphism from hom cq p. Representations of real and padic groups lecture notes. An informal characterization of cug is that it is the smallest full subcategory of smooth g representations such that. A question on representation theory of padic groups. Introduction to the theory of admissible representations of padic reductive groups w. Introduction to the theory of admissible representations of p. 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. Harmonic analysis and representation theory of padic. 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.
Modular representation theory of finite and padic groups. Representations of padic groups for the modularity seminar. Introduction over the past several years, operator algebraists have become increasingly interested in the problem of calculating the ktheory of group calgebras. We now prove the padic version of the stonevon neumann theorem. Introduction to admissible representations of padic groups. We exemplify this relation in the case of the padic group of type g2. 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. 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. Generalized semiinvariant distributions on p adic spaces 1729 we suggest in this paper that, to describe all semiinvariant distributions on the spaces. The adelic ring a of q is a certain subring of the direct product r q p q pof all of these completions. Such a correspondence has been shown for g gl2 when the. The irreducible representations of the heisenberg group with nontrivial central char acter. Structure and representation theory of padic reductive groups notes taken by pakhin lee abstract.
In this subsection, we always assume that a is a category with in. But it does hold for groups g gk, whenever g is a unipotent group over a padic. These notes are an introduction to the pmodular or \modp representation theory of padic reductive groups. This text contains an elementary introduction, with exercises, to representations of padic reductive groups. Harmonic analysis and representation theory of padic reductive. This latter approach will allow to show that padic integers form a ring, denoted by zp. 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. Pdf induced rrepresentations of padic reductive groups. Representations of padic groups lectures by joseph. Representations of reductive padic groups under revision. Introduction to p adic analytic number theory download. 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.
These examples are all special cases of the true source of adic representations, which is adic etale cohomology. 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. Similarly we can form the adelic ring a f of a general number eld f. Introduction to the theory of admissible representations of p adic reductive groups w. G k glv, where v is a finitedimensional vector space over q p. On irreducible representations of compact padic analytic groups. 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. It also gives us often better understanding of representation theory of reductive lie groups. 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. In this section we shall recall some well known facts from the representation theory of reductive padic groups. This is an adic gkrepresentation of dimension 2g, where g is the dimension of a. Structure and representation theory of padic reductive groups.
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. The extension is achieved by an alternative interpretation of the concept of closeness or absolute value. The theory of representation of padic reductive groups has nowadays attained a mature stage of. It is easy to see that if g is an algebraic group over f then gf is an l group. In this course, we will be particularly interested in the case where f is a p adic eld p adic elds will be described in section 2.
Basic representation theory of reductive padic groups. In this article, a padic representation of k or of g k, the absolute galois group of k will be a continuous representation. Madore revised 7th december 2000 in all that follows, pwill stand for a prime number. 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. Many interesting properties of p adic representations are encoded in the. In this case a continuous representation may have in. In this section, g is any hausdorff topological group that has a fundamental system of neighbour hoods of 1.
Let k be a padic field, and g a reductive connected algebraic group over k. Since qun p is not p adically complete any more, we need to work with qdun p instead. Integration, functions, and distributions see weil wei95 for integration and fourier analysis on p adic elds. Existence of invariant norms in padic representations of. Generalized semiinvariant distributions on padic spaces. Conference and workshop on the representation theory of p adic groups, july 10th19th.
Moreover, very interesting examples of padic galois representations arise from geometry. These notes will keep updating as the lectures progress. Click download or read online button to get introduction to p adic analytic number theory book now. The overflow blog coming together as a community to connect. Baum and others published representation theory of adic groups. 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. Structure and representation theory of p adic reductive groups notes taken by pakhin lee. 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. Let c cq p be the vector space of cvalued compactly supported. A locally pro nite group is a group gsatisfying any one of the. N, z, q, rand care the sets of respectively the natural numbers i. Representations of padic symplectic groups markotadi. The representation theory of p adic groups is closely related to the structure of the underlying algebraic group. Foreach p, the gfrepresentation ontpe encodes all euler factors at primes.
Note above that the additive group ga is the union of its compact open subgroups. Vigneras and others published induced rrepresentations of padic reductive groups find, read and cite all the research you need on researchgate. 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.
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. One seeks linear algebra data in the other cohomology groups attached to x. It is easy to see that if g is an algebraic group over f then gf is an lgroup. Let k be a local field with residue field k of characteristic p. Representation theory of reductive p adic groups, besides its importance for harmonic analyses, is very important for langlands program. This plays a very important role for harmonic analysis on unipotent padic groups.
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