This book gives a complete description of the de Rham complex of the Drinfeld space of dimension n − 1 as a complex of representations of GL_n(K), where n ≥ 2 and K is a finite field extension of the field of p-adic numbers. The group GL_n(K) acts on the Drinfeld space of dimension n − 1, hence on its complex of differential forms, yielding representations of GL_n(K) that mathematicians began to study in the 1980s. Understanding these representations was one of the main motivations for the development of the theory of locally analytic representations of GL_n(K), which can be seen as a p-adic analogue of Harish-Chandra’s (gl_n,K)-modules (in the latter, K is a maximal compact subgroup of GL_n(R)). 

A transparent description is provided of the global sections of the de Rham complex of the Drinfeld space of dimension n-1 as a complex of (duals of) locally analytic representations of GL_n(K), and an explicit partial splitting of this complex is constructed in the derived category of (duals of) locally analytic representations of GL_n(K). Multiple intermediate results on Ext groups of locally analytic representations are established, which may be useful in other contexts. Requiring a light background in locally analytic representations, modules over enveloping algebras, and rigid spaces, the book is aimed at a general audience of number theorists and representation theorists.

Christophe Breuil is the directeur de recherche CNRS at Université Paris-Saclay. His research interests include p-adic number theory, representation theory and algebraic geometry, with a focus on applications to the locally analytic, p-adic and mod p Langlands programs. He has published around 50 articles and preprints on this topic, including 4 books.

Zicheng Qian is an assistant professor (tenure track) at the Morningside Center of Mathematics of Beijing. His research interests includemod p and p-adic aspects of the Langlands program. He has published 6 articles and preprints on this topic, including 3 books.