Publication Type:Other

Publication Type:Journal Paper
Link/URL:http://www.math.ucalgary.ca/files/publications/cunning/20070329.pdf

Publication Type:Journal Paper
Link/URL:http://www.math.ucalgary.ca/files/publications/cunning/GOI3.pdf
Abstract:This paper concerns a class of orbital integrals in Lie algebras over $p$-adic fields. The values of these orbital integrals at the unit element in the Hecke algebra count points on varieties over finite fields. The construction, which is based on motivic integration, works both in characteristic zero and in positive characteristic. As an application, the Fundamental Lemma for this class of integrals is lifted from positive characteristic to characteristic zero. The results are based on a formula for orbital integrals as distributions inflated from orbits in the quotient spaces of the Moy-Prasad filtrations of the Lie algebra. This formula is established by Fourier analysis on these quotient spaces.

Publication Type:Journal Paper

Publication Type:Book
volume:26
Abstract:Ottawa Lectures on Admissible Representations of p-adic Groups offers researchers and graduate students a rare introduction to some of the major modern themes in the representation theory of p-adic groups: the classification and construction of their (complex) admissible representations; the calculation of their characters; and the realization of the celebrated local Langlands correspondence. Recent years have seen significant and rapid progress made toward each of these goals; the purpose of this book is to help bridge the gap from the classical literature to the forefront of research. The first part of this volume is devoted to the tools and techniques used to classify and construct smooth representations of p-adic groups: the Bernstein decomposition; Bruhat-Tits theory and filtrations of subgroups; and an overview of J.K. Yu''s construction of supercuspidal representations, together with J.-.L Kim''s proof of that it is exhaustive. The second part begins with an historical overview of character computations and continues with an introduction to motivic integration. The volume concludes, in the third part, with an introduction to the local Langlands programme and a proof of the local Langlands correspondence for algebraic tori. The chapters, written by leaders in this field, arose from lecture notes of mini-courses delivered at workshops held at the University of Ottawa in 2004 and 2007.

Publication Type:Preprint
Link/URL:http://www.math.ucalgary.ca/files/publications/cunning/Documenta1.pdf
Abstract:Abstract. In this paper we provide a geometric framework for the study of characters of depth-zero representations of unramified groups over local fields with finite residue fields which is built directly on Lusztig’s theory of character sheaves for groups over finite fields and uses ideas due to Schneider-Stuhler. Specifically, we introduce a class of coefficient systems on Bruhat-Tits buildings of perverse sheaves sheaves on affine algebraic groups over an algebraic closure of a finite field, and to each supercuspidal depth-zero representation of an unramified p-adic group we associate a formal sum of these coefficient systems, called a model for the representation. Then, using a character formula due to Schneider-Stuhler and a fixed-point formula in etale cohomology we show that each model defines a distribution which coincides with the Harish-Chandra character of the corresponding representation on the set of regular elliptic elements. The paper includes a detailed treatment of SL(2), Sp(4) and GL(n) as examples of the theory.

Publication Type:Conference Proceedings