On Jordan-Hölder series of some locally analytic representations
Journal of The American Mathematical Society
Let G be a split reductive p-adic group. This paper is about the Jordan-Hölder series of locally analytic G-representations which are induced from locally algebraic representations of a parabolic subgroup P ⊂ G. We construct for every representation M of Lie(G) in the BGG-category O, which is equipped with an algebraic P -action, and for every smooth P -representation V , a locally analytic representation F G P (M, V ) of G. This gives rise to a bi-functor to the category of locally analytic
... locally analytic representations. We prove that it is exact and give a criterion for the topological irreducibility of F G P (M, V ) in terms of M and V . Independently of our approach, O. Jones considers in  the relation between the Bernstein-Gelfand-Gelfand resolution and locally analytic principal series representations of G (and, more generally, of subgroups of G possessing an Iwahori decomposition). In [16, Thm. 26] he proves the existence of an exact sequence which coincides with the exact sequence 4.11.3 in paragraph 4.11. The results about irreducibility in section 5 can then be used to decide if this sequence is actually a composition series, and, if not, how it can be refined to give such a series. Notation and conventions: We denote by p a prime number and consider fields L ⊂ K which are both finite extensions of Q p . Let O L and O K be the rings of integers of L, resp. K, and let | · | K be the absolute value on K such that |p| K = p −1 . The field L is our "base field", whereas we consider K as our "coefficient field". For a locally convex K-vector space V we denote by V b its strong dual, i.e., the K-vector space of continuous linear forms equipped with the strong topology of bounded convergence. Sometimes, in particular when V is finite-dimensional, we simplify notation and write V instead of V b . All finite-dimensional K-vector spaces are equipped with the unique Hausdorff locally convex topology. We let G 0 be a split reductive group scheme over O L and T 0 ⊂ B 0 ⊂ G 0 a maximal split torus and a Borel subgroup scheme, respectively. We denote by G, B, T the base change of G 0 , B 0 and T 0 to L. By G 0 = G 0 (O L ), B 0 = B 0 (O L ), etc., and G = G(L), B = B(L), etc., we denote the corresponding groups of O L -valued points and L-valued points, respectively. Standard parabolic subgroups of G (resp. G) are ON JORDAN-HÖLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 5 those which contain B (resp. B). For each standard parabolic subgroup P (or P ) we let L P (or L P ) be the unique Levi subgroup which contains T (resp. T ). Finally, Gothic letters g, p, etc., will denote the Lie algebras of G, P, etc.: g = Lie(G), t = Lie(T), b = Lie(B), p = Lie(P), l P = Lie(L P ), etc.. Base change to K is usually denoted by the subscript K , for instance, g K = g ⊗ L K. We make the general convention that we denote by U (g), U (p), etc., the corresponding enveloping algebras, after base change to K, i.e., what would be usually denoted by U (g) ⊗ L K, U (p) ⊗ L K etc. Similarly, we use the abbreviations D(G) = D(G, K), D(P ) = D(P, K) etc. for the locally L-analytic distributions with values in K. Preliminaries on locally analytic representations and g-modules 2.1. Locally analytic representations. We start with recalling some basic facts on locally analytic representations as introduced by Schneider and Teitelbaum  . Here all contributions of this complex are objects in O. Hence the cohomology of this complex which computesH i Recall that we denote for a character µ ∈ X * (T) by L(µ) the irreducible highest weight U (g)-module of weight µ. 4 We note that for formulating Thm. 7.1 we have used in loc.cit. the identification of P d−i K with V (X d−i+1 , . . . , X d ). Therefore the standard parabolic subgroups used there are lower (block) triangular. Afterwards we used the conjugacy of V (X 0 , . . . , X i−1 ) and V (X d−i+1 , . . . , X d ) within P d K via the action of G on P d K .