##
**Locally analytic vectors in representations of locally \(p\)-adic analytic groups.**
*(English)*
Zbl 1430.22020

Memoirs of the American Mathematical Society 1175. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-7562-9/pbk; 978-1-4704-4052-7/ebook). iv, 158 p. (2017).

The aim of this book is to provide foundations for the locally analytic representation theory. Some of the material presented in this memoir is new, and other parts can be found in the cited papers by the author, and by Schneider and Teitelbaum (among others). It turns out that functional analysis currently provides the most important technical tool in the theory of locally analytical representations.

Classical examples of such representations are provided by finite-dimensional algebraic representations of \(p\)-adic reductive groups, and by smooth representations of such groups on \(p\)-adic vector spaces.

Locally analytic representation theory plays an important role in the analysis of \(p\)-adic symmetric spaces. The first non-classical examples of such representations were found by Morita in his analysis of the \(p\)-adic upper half-plane (the \(p\)-adic symmetric space attached to \(\text{GL}_2(\mathbb{Q}_p)\)). Schneider and Teitelbaum found further nontrivial examples in their analytic investigations of the \(p\)-adic symmetric spaces of \(\text{GL}_n(\mathbb{Q}_p)\), and initiated a systematic study of locally analytic representation theory.

The category of essentially admissible locally analytic representations provides the setting for the Jacquet module construction for locally analytic representations. These functors are in turn applied by Emerton to construct ‘eigenvarieties’ (generalizing the eigencurve of Coleman and Mazur) that \(p\)-adically interpolate systems of eigenvalues attached to automorphic Hecke eigenforms on reductive groups over number fields.

The book consists of a detailed introduction (including a long section concerning terminology, notation and conventions) and seven chapters.

In Chapter 1, the author develops the non-Archimedean functional analysis that will be required in the rest of this book. In this preliminary section, some functional analytic terminology is recalled, including basic aspects of the theory of Banach and Fréchet \(K\)-algebras, and the notion of Fréchet-Stein structure.

In Chapter 2, the author recalls the basic notions of non-Archimedean function theory: spaces of continuous, rigid analytic and locally analytic functions with values in locally convex \(K\)-vector spaces, spaces of distributions, and the restriction of scalar functor in both the rigid analytic and locally analytic settings.

In Chapter 3, the construction of the space of locally analytic vectors attached to a representation of a non-Archimedean locally \(L\)-analytic group \(G\) is presented. The author constructs first such a space when \(G\) is the group of points of an affinoid rigid analytic group defined over \(L\) in Section 3.3, and then extends the construction to certain non-affinoid rigid analytic groups in Section 3.4. Section 3.5 contains the construction of such a space attached to any \(G\)-representation of a locally \(L\)-analytic group \(G\). In Section 3.6, the author recalls the definition of a locally analytic representation of a locally \(L\)-analytic group on a convex \(K\)-space, and establishes some basic properties of this notion.

The aim of Section 4.1 is to develop some connections between the notions of smooth and locally finite representations, and the notion of locally analytic representations. In Section 4.2, it is assumed that \(G\) is the group of \(L\)-valued points of a connected reductive linear algebraic group \(G\) over \(L\). The main result of this section is that any irreducible locally algebraic representation is isomorphic to the tensor product of a smooth representation of \(G\) and a finite-dimensional algebraic representation of \(\mathbb{G}\).

One approach to analysing representations \(V\) of the group \(G\) is to pass to the dual space \(V'\). The aim of Chapter 5 is to recall this approach, and to relate it to the view-point of Chapter 3. In Section 5.2, the author recalls the description of algebras of analytic distributions via appropriate completions of universal enveloping algebras. This description is used in Section 5.3 to present a new construction of the Fréchet-Stein structure on the ring of locally analytic distributions on any compact open subgroup of \(G\).

All the results presented in Chapter 6 are essentially due to P. Schneider and J. Teitelbaum. Section 6.1 introduces the notion of an admissible locally analytic \(G\)-representation of a locally \(L\)-analytic group \(G\). It turns out that it is equivalent to the definition presented by Schneider and Teitelbaum. In Section 6.2, the author gives a proof (originally due to Schneider and Teitelbaum) of the result that any strongly admissible locally analytic \(G\)-representation is an admissible \(G\)-representation. In Section 6.3, admissible smooth and admissible locally algebraic representations are studied. In the final Section 6.5, strongly admissible locally analytic representations are characterized.

In Section 7, the author discusses the representations of groups of the form \(G\times\Gamma\), where \(G\) is a locally \(L\)-analytic group and \(\Gamma\) is a Hausdorff locally compact topological group that admits a countable basis of neighbourhoods of the identity consisting of open subgroups. The motivating example is a group of the form \(\mathbb{G}(\mathbb{A}_f)\), where \(G\) is a reductive group defined over some number field \(F\), \(\mathbb{A}_f\) denotes the ring of finite adèles of \(F\), and \(L\) is the completion of \(F\) at some finite prime. Section 7.2 extends various notions of admissibility introduced in Chapter 6 to the context of \(G\times\Gamma\) representations.

This beautiful book is a very welcome addition to the mathematical literature. The author explains the difficult subjects and provides the reader with a detailed insight into this difficult but fascinating new branch of representation theory. It will be a very useful reference for anybody interested in representation theory, automorphic forms and number theory.

Classical examples of such representations are provided by finite-dimensional algebraic representations of \(p\)-adic reductive groups, and by smooth representations of such groups on \(p\)-adic vector spaces.

Locally analytic representation theory plays an important role in the analysis of \(p\)-adic symmetric spaces. The first non-classical examples of such representations were found by Morita in his analysis of the \(p\)-adic upper half-plane (the \(p\)-adic symmetric space attached to \(\text{GL}_2(\mathbb{Q}_p)\)). Schneider and Teitelbaum found further nontrivial examples in their analytic investigations of the \(p\)-adic symmetric spaces of \(\text{GL}_n(\mathbb{Q}_p)\), and initiated a systematic study of locally analytic representation theory.

The category of essentially admissible locally analytic representations provides the setting for the Jacquet module construction for locally analytic representations. These functors are in turn applied by Emerton to construct ‘eigenvarieties’ (generalizing the eigencurve of Coleman and Mazur) that \(p\)-adically interpolate systems of eigenvalues attached to automorphic Hecke eigenforms on reductive groups over number fields.

The book consists of a detailed introduction (including a long section concerning terminology, notation and conventions) and seven chapters.

In Chapter 1, the author develops the non-Archimedean functional analysis that will be required in the rest of this book. In this preliminary section, some functional analytic terminology is recalled, including basic aspects of the theory of Banach and Fréchet \(K\)-algebras, and the notion of Fréchet-Stein structure.

In Chapter 2, the author recalls the basic notions of non-Archimedean function theory: spaces of continuous, rigid analytic and locally analytic functions with values in locally convex \(K\)-vector spaces, spaces of distributions, and the restriction of scalar functor in both the rigid analytic and locally analytic settings.

In Chapter 3, the construction of the space of locally analytic vectors attached to a representation of a non-Archimedean locally \(L\)-analytic group \(G\) is presented. The author constructs first such a space when \(G\) is the group of points of an affinoid rigid analytic group defined over \(L\) in Section 3.3, and then extends the construction to certain non-affinoid rigid analytic groups in Section 3.4. Section 3.5 contains the construction of such a space attached to any \(G\)-representation of a locally \(L\)-analytic group \(G\). In Section 3.6, the author recalls the definition of a locally analytic representation of a locally \(L\)-analytic group on a convex \(K\)-space, and establishes some basic properties of this notion.

The aim of Section 4.1 is to develop some connections between the notions of smooth and locally finite representations, and the notion of locally analytic representations. In Section 4.2, it is assumed that \(G\) is the group of \(L\)-valued points of a connected reductive linear algebraic group \(G\) over \(L\). The main result of this section is that any irreducible locally algebraic representation is isomorphic to the tensor product of a smooth representation of \(G\) and a finite-dimensional algebraic representation of \(\mathbb{G}\).

One approach to analysing representations \(V\) of the group \(G\) is to pass to the dual space \(V'\). The aim of Chapter 5 is to recall this approach, and to relate it to the view-point of Chapter 3. In Section 5.2, the author recalls the description of algebras of analytic distributions via appropriate completions of universal enveloping algebras. This description is used in Section 5.3 to present a new construction of the Fréchet-Stein structure on the ring of locally analytic distributions on any compact open subgroup of \(G\).

All the results presented in Chapter 6 are essentially due to P. Schneider and J. Teitelbaum. Section 6.1 introduces the notion of an admissible locally analytic \(G\)-representation of a locally \(L\)-analytic group \(G\). It turns out that it is equivalent to the definition presented by Schneider and Teitelbaum. In Section 6.2, the author gives a proof (originally due to Schneider and Teitelbaum) of the result that any strongly admissible locally analytic \(G\)-representation is an admissible \(G\)-representation. In Section 6.3, admissible smooth and admissible locally algebraic representations are studied. In the final Section 6.5, strongly admissible locally analytic representations are characterized.

In Section 7, the author discusses the representations of groups of the form \(G\times\Gamma\), where \(G\) is a locally \(L\)-analytic group and \(\Gamma\) is a Hausdorff locally compact topological group that admits a countable basis of neighbourhoods of the identity consisting of open subgroups. The motivating example is a group of the form \(\mathbb{G}(\mathbb{A}_f)\), where \(G\) is a reductive group defined over some number field \(F\), \(\mathbb{A}_f\) denotes the ring of finite adèles of \(F\), and \(L\) is the completion of \(F\) at some finite prime. Section 7.2 extends various notions of admissibility introduced in Chapter 6 to the context of \(G\times\Gamma\) representations.

This beautiful book is a very welcome addition to the mathematical literature. The author explains the difficult subjects and provides the reader with a detailed insight into this difficult but fascinating new branch of representation theory. It will be a very useful reference for anybody interested in representation theory, automorphic forms and number theory.

Reviewer: Andrzej Dąbrowski (Szczecin)

### MSC:

22E50 | Representations of Lie and linear algebraic groups over local fields |