\(l\)-modular representations of a \(p\)-adic reductive group with \(l\neq p\).
(Représentations \(l\)-modulaires d’un groupe réductif \(p\)-adique avec \(l\neq p\).)

*(English)*Zbl 0859.22001
Progress in Mathematics (Boston, Mass.). 137. Boston, MA: Birkhäuser. xviii, 233 p. (1996).

Let \(F\) be a \(p\)-adic field and \(G\) the group of \(F\)-rational points of a reductive connected group over \(F\). Then \(G\) is a totally disconnected locally compact, or, what amounts to the same, a locally profinite group. For a complex representation \(V\) one usually considers the “skeleton” of smooth vectors \(V^\infty\), consisting of all vectors with open stabilizer in \(G\). The algebraic representation \(V^\infty\) then essentially determines \(V\) and it is well known that in this way the representation theory of \(G\) can be made an essentially algebraic theory. Since the topology of the complex numbers does not show up anymore in the notion of smooth representations, the latter can be considered over any ground field. By obvious reasons it seems natural to restrict to an algebraically closed ground field so that only the characteristic really plays a role.

The present monograph features an introduction to the theory of \(\overline{\mathbb{F}}_l\)-valued smooth representations, where \(l\neq p\) is a prime number, \(\mathbb{F}_l\) is the prime field with \(l\) elements and \(\overline{F}_l\) is an algebraic closure of \(\mathbb{F}_l\). As a by-product purely algebraic proofs are given for a number of results in the theory of (complex) smooth representations which originally have been proven by means of analytic methods.

Chapter I gives the general theory of smooth representations of an arbitrary locally profinite group over any commutative ring. The main topics here are induction of representations and the general theory of Hecke algebras.

In chapter II the theory is specialized to reductive \(p\)-adic groups. One encounters parabolic induction, cuspidal and supercuspidal representations and the theory of minimal types for \(\overline{\mathbb{F}}_l\)-valued representations. This theory requires some knowledge about the Bruhat-Tits building, the latter only being cited in the present book.

In the last chapter we are given the theory of types by Howe-Moy and Bushnell-Kutzko in its \(\overline{\mathbb{F}}_l\)-valued form. This theory allows one to reduce problems concerning irreducible cuspidal representations to the easier “level 0 case”. To give an example, it is shown that the cuspidal representations which are supercuspidal are those occurring in the mod-\(l\) reduction of the generalized Steinberg representations.

The present book is of evident importance to everyone interested in the representation theory of \(p\)-adic groups. Unfortunately not very much is said about the Bruhat-Tits building. This is justified, however, by the desire to bound the number of pages and by the fact that in the case of major interest, \(\text{GL}_n\), the theory of types can be formulated directly without use of geometry. The monograph starts on an elementary level laying proper foundations for the things to come and then proceeds directly to results of recent research.

The present monograph features an introduction to the theory of \(\overline{\mathbb{F}}_l\)-valued smooth representations, where \(l\neq p\) is a prime number, \(\mathbb{F}_l\) is the prime field with \(l\) elements and \(\overline{F}_l\) is an algebraic closure of \(\mathbb{F}_l\). As a by-product purely algebraic proofs are given for a number of results in the theory of (complex) smooth representations which originally have been proven by means of analytic methods.

Chapter I gives the general theory of smooth representations of an arbitrary locally profinite group over any commutative ring. The main topics here are induction of representations and the general theory of Hecke algebras.

In chapter II the theory is specialized to reductive \(p\)-adic groups. One encounters parabolic induction, cuspidal and supercuspidal representations and the theory of minimal types for \(\overline{\mathbb{F}}_l\)-valued representations. This theory requires some knowledge about the Bruhat-Tits building, the latter only being cited in the present book.

In the last chapter we are given the theory of types by Howe-Moy and Bushnell-Kutzko in its \(\overline{\mathbb{F}}_l\)-valued form. This theory allows one to reduce problems concerning irreducible cuspidal representations to the easier “level 0 case”. To give an example, it is shown that the cuspidal representations which are supercuspidal are those occurring in the mod-\(l\) reduction of the generalized Steinberg representations.

The present book is of evident importance to everyone interested in the representation theory of \(p\)-adic groups. Unfortunately not very much is said about the Bruhat-Tits building. This is justified, however, by the desire to bound the number of pages and by the fact that in the case of major interest, \(\text{GL}_n\), the theory of types can be formulated directly without use of geometry. The monograph starts on an elementary level laying proper foundations for the things to come and then proceeds directly to results of recent research.

Reviewer: A.Deitmar (Heidelberg)

##### MSC:

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

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