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 GLn(K), where n = 2 and K is a finite field extension of the field of p-adic numbers. The group GLn(K) acts on the Drinfeld space of dimension n - 1, hence on its complex of differential forms, yielding representations of GLn(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 GLn(K), which can be seen as a p-adic analogue of Harish-Chandra’s (gln,K)-modules (in the latter, K is a maximal compact subgroup of GLn(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 GLn(K), and an explicit partial splitting of this complex is constructed in the derived category of (duals of) locally analytic representations of GLn(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.
