Derived Langlands (eBook)
by victor snaith (Author)

 74,655 Words
 356 Pages
The Langlands Programme is one of the most important areas in modern pure mathematics. The importance of this volume lies in its potential to recast many aspects of the programme in an entirely new context. For example, the morphisms in the monomial category of a locally padic Lie group have a distributional description, due to Bruhat in his thesis. Admissible representations in the programme are often treated via convolution algebras of distributions and representations of Hecke algebras. The monomial embedding, introduced in this book, elegantly fits together these two uses of distribution theory. The author follows up this application by giving the monomial category treatment of the Bernstein Centre, classified by Deligne–Bernstein–Zelevinsky.
This book gives a new categorical setting in which to approach wellknown topics. Therefore, the context used to explain examples is often the more generally accessible case of representations of finite general linear groups. For example, Galois basechange and epsilon factors for locally padic Lie groups are illustrated by the analogous Shintani descent and Kondo–Gauss sums, respectively. General linear groups of local fields are emphasized. However, since the philosophy of this book is essentially that of homotopy theory and algebraic topology, it includes a short appendix showing how the buildings of Bruhat–Tits, sufficient for the general linear group, may be generalised to the tom Dieck spaces (now known as the Baum–Connes spaces) when G is a locally padic Lie group.
The purpose of this monograph is to describe a functorial embedding of the category of admissible krepresentations of a locally profinite topological group G into the derived category of the additive category of the admissible kmonomial module category. Experts in the Langlands Programme may be interested to learn that when G is a locally padic Lie group, the monomial category is closely related to the category of topological modules over a sort of enlarged Hecke algebra with generators corresponding to characters on compact open modulo the centre subgroups of G. Having set up this functorial embedding, how the ingredients of the celebrated Langlands Programme adapt to the context of the derived monomial module category is examined. These include automorphic representations, epsilon factors and Lfunctions, modular forms, Weil–Deligne representations, Galois base change and Hecke operators.
Contents: Finite Modulo the Centre Groups
 GL_{2} of a Local Field
 Automorphic Representations
 GL_{n}K in General
 Monomial Resolutions and Deligne Representations
 Kondo Style Invariants
 Hecke Operators and Monomial Resolutions
 Could Galois Descent be Functorial?
 PSHalgebras and the Shintani Correspondence
 Appendices:
 Galois Descent of Representations
 Remarks on a Paper of Guy Henniart
 Finite General Linear and Symmetric Groups
 Locally padic Lie Groups
Readership: Graduate students and researchers in automorphic forms reviewing the concepts from a localalgebraic point of view. Category Theory;Homological Algebra Number Theory;Representation Theory0Key Features:
 The book describes an embedding of the Langlands Programme into an entirely newcontext of a new derived category. Therefore, there are no competing titles
 As often as possible, phenomena from the Langlands Programme (e.g. Galois descent properties of admissible representations of reductive padic Lie groups) as viewed in the monomial category are illustrated by the analogous but simpler case of finite groups (e.g. Shintani descent)
 The book contains an excellent bibliography of 146 sources detailing the category theory, homological algebra, representation theory and algebraic topology background as well as many survey articles concerning the Langlands Programme
 Released: December 5, 2018
 Categories: Education
 Language: English
 Publisher: World Scientific Publishing Company
 ISBN10: 9813275766
 ISBN13: 9789813275768