Prop lie rings play a role in lie theory similar to that played by prop groups in group theory. We introduce here the notion of lie algebras and the example of main interest for us, the tangent space t eg of a lie group gat the identity e. Harmonic analysis on reductive padic groups and lie algebras 393 robert e. We then expand our class to contain those admissible padic lie extensions which lie in the. A totally disconnected, locally compact group g is said to be uniscalar if its scale function sg x g 3 n, as dened in g. Hundley automorphic representations and lfunctions for the general linear group, i.
The original source for much of the theory of padic analytic groups is lazards. The second part of the book contains, for the first time in a textbook, a detailed exposition of lazards algebraic approach to compact p adic lie groups, via his notion of a pvaluation. The book thoroughly discusses the analytic aspects of p adic manifolds and p adic lie groups. A compact topological group g admits the structure of a padic lie group if and only if it contains an open prop subgroup of finite rank. An adic representation of gq is a nite dimensional qvector space v equipped with a continuous homomorphism. Isomorphisms of padic group rings 595 correction by exhibiting the close connection between the lie palgebra struc ture of a grading of g and a nondegenerate bilinear form associated naturally to critical sections of rad arada. We call it the ladic fundamental lie algebra of xn. A p adic lie group g is called type r if l g is type r adg where l g is the lie algebra of g. Let g be a padic lie group of dimension d and v be a finite. If n is a representation of an group g on a complex vector space e, we briefly write n n, g, e. Abelian varieties, ladic representations, and lindependence. Hochschild cohomology and padic lie groups ucsd mathematics. Galois representations institute for advanced study. In mathematics, the etale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by grothendieck in order to prove the weil conjectures.
Thus this theorem is illustrative that our techniques allow one to study the l function of varieties for which the image of galois acting on the l adic cohomology is large. Sl2, c, and similarly, using an alternate metric completion of q, one can define a p adic lie group over the p adic numbers, a topological group in which each point has a p adic neighborhood. For example, padic lie groups play an important role in the theory of. Abelian ladic representations and elliptic curves mcgill university. Let gbe a compact padic analytic group whose lie algebra is split semisimple. Continuous representation theory of padic lie groups 3 principle asserts that the set. The author then sets up the analytic foundations of the theory of p adic lie groups and develops the relation between p adic lie groups and their lie algebras. Definable groups and compact padic lie groups article pdf available in journal of the london mathematical society 781 april 2008 with 81 reads how we measure reads. Discussion with a view towards chernweil theory is in chapter iv in.
Part i foundations 1 ultrametric spaces we begin by. Lazard proved that a small enough open subgroup uof gis a poincar e group, i. If a connected real lie group g has the additive exponential property then it is nilpotent. There is a particularly simple special case of this functoriality. It should be noted that autlg is an algebraic subgroup of gllg, the general linear group on lg. On classes of p adic lie groups c r e ra ja abstra ct w e consider non con tracting p adic lie groups and w e establish. On padic lseries, padic cohomology and class field theory. The universal prolouter monodromy representation of the moduli stack of hyperbolic curves 14 references 25 introduction. Eckhard meinrenken, lie groups and lie algebas, lecture notes 2010. The representation v is said to be rational if there exists a nite set s of primes such that v is unrami ed outside of s fgthat is, for all primes p 2 s fg, the action of. Featured on meta community and moderator guidelines for escalating issues via new response. An isomorphism of lie groups is a homomorphism fwhich admits an inverse also c 1 f as maps and such that f 1 is also a homomorphism of lie groups.
Continuous representation theory of adic lie groups. Let gbe a compact p adic analytic group whose lie algebra is split semisimple. 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. The universal pro l outer monodromy representation of the moduli stack of hyperbolic curves 14 references 25 introduction. To be more precise, let p be a prime number, let l be a. In general, padic uniformization replaces complex symmetric spaces xwith padic analytic spaces xone in each dimension.
A prop lie ring is an inverse limit of finitep lie rings. It should be noted that aut l g is an algebraic subgroup of gl l g, the general linear group on l g. Continuous representation theory of padic 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. A complex lie group is defined in the same way using complex manifolds rather than real ones example. G of isomorphism classes of irreducible smoothrepresentations of g has a distinguished partitioning into. Introduction let pbe a prime and ga padic lie group of dimension d. Review of abelian l g n c university of california, berkeley. A formal argument using the exactness of both l and r shows that the. A padic lie group g is called type r if lg is type r adg where lg is the lie algebra of g. Endomorphism algebras of admissible adic representations. The second part of the book contains, for the first time in a textbook, a detailed exposition of lazards algebraic approach to compact padic lie groups, via his notion of a pvaluation. They are automatically 0 for i sufficiently large 27. For a lie group, the subscript n is called the rank of the group, which is a measure of how large the group is. Apr 05, 2001 uniscalar p adic lie groups uniscalar p adic lie groups glockner, helge.
Section 4 is devoted to a more careful analysis of l,g. On irreducible representations of compact padic analytic. If a locally compact group can be approximated by padic lie groups for variable primes p, then it is a propadic lie group for some prime p. The author then sets up the analytic foundations of the theory of padic lie groups and develops the relation between padic lie groups and their lie algebras. Contents introduction 1 notations and conventions 3 1. A compact topological group g admits the structure of a p adic lie group if and only if it contains an open prop subgroup of finite rank. The euler characteristics considered in this paper have the following form. Introduction let pbe a prime and ga p adic lie group of dimension d. Let pbe an odd very good prime for g, and let g c be a complex semisimple algebraic group with the same root system as g. Onishchik, lie transformation groups encyclopaedia of mathematical sciences, volume 20, springer 1993.
The first, based on a graduate course schneider teaches at munster, gives a padicanalytic account of the theory which is much like the classical theory of lie groups. Willis, the structure of totally disconnected, locally compact groups, math. Let aut l g be the group of all lie algebra automorphisms of l g. Lie groups and lie algebras, addisonwesley 1975 translated from french 2 j. Let g be a padic lie group and lg be the lie algebra of g. We defined tangent plane, tangent bundle and lie group. On the rank of compact p adic lie groups springerlink. On classes of padic lie groups new york journal of mathematics. On classes of padic lie groups university at albany, suny. Let m be a finitely generated zmodule on which g acts, and suppose that the homology groups hg, m are finite for all i. A variant of the etale site, well adapted to the needs of. Every prop lie ring admits the structure of a lie algebra over the padic integers. V on the inertia group ip is trivial, and furthermore. Automorphy for some ladic lifts of automorphic mod lgalois representations.
Big galois representations and padic lfunctions 2 twodimensional odd semisimple odd representation. Serre, lie algebras and lie groups, benjamin 1965 translated from french. This implies that there is a canonical isomorphism. If is a rational prime distinct from the characteristic of k, the galois group galkk.
Uniscalar p adic lie groups, forum mathematicum 10. It is a free pronilpotent lie algebra with generators corresponding to the loops around 0 and all nth roots of unity. Let g be a p adic lie group and lg be the lie algebra of g. If xq is a smooth projective variety and we choose an embedding q. In particular we prove that b,g is equal to the number of generators of the lie algebra of g. James milne, section 19 of lectures on etale cohomology. While considering a prop group g or profinite groups in general we shall be interested only in closed subgroups and continuous. Part i foundations 1 ultrametric spaces we begin by establishing some very basic and elementary notions. A note on the exponential map of a real or padic lie group. C then the natural action of gq on the cohomology hixc,q l. We use this regulator map to study padic representations of global galois groups over certain abelian extensions of number. Why is every ladic galois representation conjugate to one.
The functoriality principle asserts that the set g of isomorphism classes of irreducible smooth. Akhil mathew, l l adic cohomology and exponential sums. The total quotient ring qh is the ring of fractions by the multiplicative set of all nonzero divisors. Continuous representation theory of padic lie groups. Although a , for example, is certainly larger than e 8, mathematicians know how to describe the representations of a n for every n, so there is nothing special about a. Lie groups richard borcherds, mark haiman, nicolai reshetikhin, vera serganova, and theo johnsonfreyd october 5, 2016. The prolouter galois representations associated to hyperbolic curves 4 2.
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