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Abstract a Tannakian Description for Parahoric ABSTRACT Title of dissertation: A TANNAKIAN DESCRIPTION FOR PARAHORIC BRUHAT-TITS GROUP SCHEMES Kevin Michael Wilson Jr., Doctor of Philosophy, 2010 Dissertation directed by: Professor Thomas Haines, Department of Mathematics Let K be a field which is complete with respect to a discrete valuation and let O be the ring of integers in K. We study the Bruhat-Tits building Be(G) and the parahoric Bruhat-Tits group schemes associated to a connected reductive split GFe linear algebraic group G defined over O. In order to study these objects we use the theory of Tannakian duality, devel- oped by Saavedra Rivano, which shows how to recover G from its category of finite rank projective representations over O. We also use Moy-Prasad filtrations in order to define lattice chains in any such representation. Using these two tools, we give a Tannakian description to Be(G). We also define a functor Aut associated to a facet F ⊂ (G) in terms of Fe e Be lattice chains in a Tannakian way. We show that Aut is representable by an affine Fe group scheme of finite type, has generic fiber G , and satisfies Aut (O ) = (O ) K Fe E GFe E for every unramified Galois extension E of K. Moreover, we show that there is a canonical morphism from to Aut , which GFe Fe we conjecture to be an isomorphism. We prove that it is an isomorphism when the residue characteristic of K is zero and G is arbitrary, when G = GLn and K is arbitrary, and when Fe is the minimal facet containing the origin and G and K are arbitrary. A TANNAKIAN DESCRIPTION FOR PARAHORIC BRUHAT-TITS GROUP SCHEMES by Kevin Michael Wilson Jr. Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2010 Advisory Committee: Professor Thomas Haines, Chair/Advisor Professor Jeffrey Adams Professor Jonathan Rosenberg Professor Niranjan Ramachandran Professor M. Coleman Miller c Copyright by Kevin Michael Wilson Jr. 2010 Acknowledgments I would like to take this opportunity to thank all of the people that have made this thesis possible. First, and foremost, I would like to thank my advisor, Professor Thomas Haines, for all of the time and energy he has put into developing me as a mathematician. He has truly been a fantastic advisor, and I could not have asked for a better thesis problem to work on. Moreover, I greatly appreciate all of the financial support he has provided me with, which has allowed me to focus on my research and to attend several conferences. I would also like to thank the Maryland Mathematics Department as a whole for all of the support, both financial and otherwise, that they have given me through- out my time here. I am especially grateful for the NSF VIGRE fellowship that was awarded to me for my first four years here. I owe a special thank you to Profes- sor Jeffrey Adams for all of the knowledge he has imparted to me over the years through the wonderful classes he taught and the various discussions we have had. I also owe a debt of gratitude to Professor Niranjan Ramachandran for pointing out an important reference which is used in the proof of Proposition 4.2.13. Graduate school would have been much less enjoyable without the help and friendship of the many people I have met here. In particular, I would like to thank Ted Clifford, Walter Ray-Dulany, Sean Rostami, Chris Zorn, Nick Long, Moshe Adrian, and Ben Lauser for all of the fun times and interesting discussions we have had over the years. Finally, I would like to thank my parents, Kevin and Betty Wilson, and my fianc´ee,Beth McLaughlin, for always being there, and for providing all of the love and support I could ever want or need. ii Table of Contents 1 Introduction and Summary of Results 1 1.1 Outline of Thesis . .4 2 Preliminaries 7 2.1 Notation . .7 2.1.1 Lattice Chains . .7 2.1.2 Buildings Notation . .8 2.1.3 Categorical Notation . 11 2.2 Representations of G ........................... 12 2.2.1 Existence of Faithful Representations . 12 2.2.2 Tensor Generator . 16 2.2.3 Weyl Modules . 20 2.3 Tannakian Duality . 25 3 A Tannakian Description for Bruhat-Tits Buildings 27 3.1 Moy-Prasad Filtrations Associated to Any x 2 Ae ........... 27 3.2 Moy-Prasad Filtrations of Any Finitely Generated Representation . 28 3.3 Dual Filtration Associated to a Moy-Prasad Filtration . 29 3.4 Parahoric Subgroups Stabilize Moy-Prasad Filtrations . 30 3.5 Moy-Prasad Filtrations Associated to Any Point in Be(G)....... 33 3.6 Recovering the Facet Containing y 2 Be(G) From fVy;rg ........ 35 3.7 Tannakian Description for Be(G)..................... 37 4 A Tannakian Description for Bruhat-Tits Group Schemes 42 4.1 Definition of the Functor . 44 4.2 Comparison of Aut with ....................... 46 Fe GFe 4.2.1 Generic Fiber . 47 4.2.2 OE Points . 48 4.2.3 Representable by an Affine Scheme . 55 4.2.4 Finite Type . 59 5 Special Cases 67 5.1 x =0.................................... 67 5.2 Char(k)=0................................ 67 5.3 G = GLn ................................. 70 6 Dependence of Aut on the Facet F 75 Fe e 6.1 Comparing Aut and Aut ........................ 75 Fe Ge 6.2 The Iwahori Case . 76 iii A Lattice Chain Description for BT Group Schemes Associated to GLn 80 A.1 Background and Examples . 80 A.2 Definitions . 88 A.3 Results . 91 B Background Results 101 Bibliography 103 iv Chapter 1 Introduction and Summary of Results The notion of a building was introduced by Jacques Tits in the 1950's to provide a geometric tool for the study of semi-simple Lie groups over general fields in a systematic way. See [T2], [Br], and [Rou]. These buildings are simplicial complexes made up of apartments which are spheres, and hence they are called spherical buildings. In the 1960's, Fran¸coisBruhat and Jacques Tits ([BT1]) defined another build- ing associated to a semi-simple Lie group, or more generally a reductive group, over a field equipped with a non-archimedean discrete valuation. These buildings are (poly)-simplicial complexes whose apartments are affine spaces, and they are called affine buildings or Bruhat-Tits buildings. They are the analogue in the non- archimedean setting of the symmetric space associated to a real reductive Lie group. Let G be a connected reductive group defined over a non-archimedean field K with ring of integers O, and let Be(G(K)) denote the Bruhat-Tits building associated to G. In their seminal works [BT1] and [BT2], Bruhat and Tits proved many things about the structure of Be(G(K)) and also defined some important objects which relate to Be(G(K)). At least for our purposes, the most important related object they defined was the parahoric Bruhat-Tits group scheme associated to a (poly)-simplex F, called GFe e 1 a facet, in Be(G). It is a smooth connected affine group scheme over O with generic fiber G such that, at least when G is simply connected, (O) is the stabilizer in GFe G(K) of the facet F. More generally, the group (O) is a subgroup of the fixer of e GFe Fe in G(K) called a parahoric subgroup. Parahoric subgroups are important objects in the theory of Shimura varieties (see [RZ], [H2]) and also in representation theory. In particular, in [MP1] and [MP2] Allen Moy and Gopal Prasad defined an R-filtration on parahoric subgroups and on the Lie algebra associated to G. This allowed them to define the depth of an admissible representation, which is an important invariant of such representations. Moreover, in the depth zero case they related the representation theory of G to the well understood representation theory of an associated finite group. Thus it is important to understand the parahoric Bruhat-Tits group schemes, which are somewhat mysterious objects in general. Bruhat and Tits themselves did significant work on understanding these objects in [BT2], [BT3], and [BT4]. In the latter two papers, they described the Bruhat-Tits building of a classical group G in terms of graded lattice chains on the standard representation V of G. Under this identification, the group schemes can be realized as stabilizers of these lattice GFe chains in V . Moreover, using these concrete descriptions, the Moy-Prasad filtrations on a parahoric subgroup can be described in terms of lattice chains. See [Yu]. More recently, in [GY1] and [GY2] Wee Teck Gan and Jiu-Kang Yu described the Bruhat-Tits buildings and group schemes associated to the exceptional groups G2, F4, and E6 in terms of a set of norms on a representation V , or equivalently a set of graded lattice chains in V . In each of these cases they choose one representation 2 of the particular group which remains fixed throughout their descriptions. The goal of this thesis is to give a uniform lattice chain description to all of the Bruhat-Tits buildings and group schemes associated to the split classical groups and the exceptional groups. In all of the above cases, a particular representation is chosen through which these objects are described. However, the choice made depends greatly on the case you are in. In order to get a more uniform description it seemed natural to use the entire category of representations associated to G, and for this we need the theory of Tannakian duality.
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