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Taking up the works of Harish-Chandra, Langlands, Borel, Casselman,
Bernstein and Zelevinsky, among others, on the complex
representation theory of a p -adic reductive group G, the author
explores the representations of G over an algebraic closure Fl of a
finite field Fl with l1 p elements, which are called 'modular
representations'. The main feature of the book is to develop the
theory of types over Fl, and to use this theory to prove
fundamental results in the theory of modular representations.
"The present book is of evident importance to everyone interested
in the representation theory of p-adic groups....The monograph
starts on an elementary level laying proper foundations for the
things to come and then proceeds directly to results of recent
research."
--Zentralblatt
This book grew out of seminar held at the University of Paris 7
during the academic year 1985-86. The aim of the seminar was to
give an exposition of the theory of the Metaplectic Representation
(or Weil Representation) over a p-adic field. The book begins with
the algebraic theory of symplectic and unitary spaces and a general
presentation of metaplectic representations. It continues with
expos?'s on the recent work of Kudla (Howe Conjecture and
induction) and of Howe (proof of the conjecture in the unramified
case, representations of low rank). These lecture notes contain
several original results. The book assumes some background in
geometry and arithmetic (symplectic forms, quadratic forms,
reductive groups, etc.), and with the theory of reductive groups
over a p-adic field. It is written for researchers in p-adic
reductive groups, including number theorists with an interest in
the role played by the Weil Representation and -series in the
theory of automorphic forms.
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