The aim of the last lecture is to explain how to adapt the results of the previous lectures to. Let g be a padic lie group and lg be the lie algebra of g. Nilpotent conjugacy classes of reductive padic lie. Contents i foundations 1 1 ultrametric spaces 1 2 nonarchimedean elds 6 3 convergent series 12 4 di erentiability 15 5 power series 22 6 locally analytic functions 36 ii manifolds 41 7 charts and atlases 41 8 manifolds 43 9 the tangent space 52 10 the topological vector space canm. Exponential generation and largeness for compact padic lie groups. Continuous representation theory of padic lie groups. We also prove a similar decomposition for lattices in a solvable locally compact group.
Reference for ladic lie algebras mathematics stack exchange. Every prop lie ring admits the structure of a lie algebra over the p adic integers. Some knowledge of algebra and calculus is required of the reader, but the text is easily accessible to graduate students, and to mathematicians at large. For example, padic lie groups play an important role in the theory of prop groups i. We also characterize, for linear lie groups or padic lie groups, the pairs with the relative howemoore property with respect to a closed. Lie algebras, algebraic groups, and lie groups james milne. The name lie algebra was given by hermann weyl in the 1930s. The links to gt, ca, ag, and ags in the pdf file will work if the files are placed in the. For example, padic lie groups play an important role in the theory of propgroups i.
Given a fixed integer n, we consider closed subgroups of glnp, where p. We call it the l adic fundamental lie algebra of xn. Lie groups and lie algebras, addisonwesley 1975 translated from french 2 j. Let g be a compact padic lie group of dimension at least 2, and let m be a finitely generated zpmodule with gaction. The links to gt, ca, ag, and ags in the pdf file will work if the files are placed in. These notes are an introduction to lie algebras, algebraic groups, and. The definition of finite groups of lie type due to chevalley involves restricting from a lie algebra over the complex numbers to a lie. A finite nilpotent lie ring of characteristic a power of p is called finitep. On classes of padic lie groups new york journal of mathematics. Hochschild cohomology and padic lie groups ucsd mathematics. It is a free pronilpotent lie algebra with generators corresponding to the loops around 0 and all nth roots of unity. To every lie group we can associate a lie algebra whose underlying vector space is the tangent space of the.
This paper is a continuation of 3 in which the rst two authors have introduced the spherical hecke algebra and the satake isomorphism for an untwisted a ne kacmoody group over a nonarchimedian local eld. In this paper we develop the theory of the iwahorihecke algebra associated to. The concept of the associated lie algebra over k will be used at once to obtain the. Prop lie rings play a role in lie theory similar to that played by prop groups in group theory. We give an explicit formula for the subalgebra zeta function of a general 3dimensional lie algebra over the p adic integers z p. This is the same as saying that the number of v adic worlds in which the algebra remains a division algebra should be even, and that these can be speci. Are nilpotent conjugacy classes of reductive lie algebras over p adic. We make a detailed study of pro p lie rings in terms of various properties, including their topology, prufer rank, subring growth, and p adic module structure. In general, only topological groups having similar local properties to r n for some positive integer n can be lie groups of course they must also have a differentiable structure.