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The trace form gives a connection between the representation ring and the space of invariant bilinear forms of a Lie algebra $L$. This thesis reviews the definition of the trace of an endomorphism of a finitely generated projective module over a commutative ring $R$. We then use this to look at the trace form of a finitely generated projective representation of a Lie algebra $L$ over $R$ and its representation ring. While doing so, we prove a few trace formulas which are useful in the theory of the Dynkin index, an invariant introduced by Dynkin in 1952 to study homomorphisms between simple Lie algebras.

Around 1990, Arthur proved a local (ordinary) trace formula for real or p-adic connected reductive groups. The local trace formula is a powerful tool in the local harmonic analysis of reductive groups. One of the aims of this thesis is to establish a local twisted trace formula for certain non-connected reductive groups, which is a twisted version of Arthur’s local trace formula. As an application of the local twisted trace formula, we will prove some twisted orthogonality relations, which are generalizations of Arthur’s results about orthogonality relations for tempered elliptic characters. To establish these relations, we will also give a classification of twisted elliptic representations.

Around 1990, Arthur proved a local (ordinary) trace formula for real or p-adic connected reductive groups. The local trace formula is a powerful tool in the local harmonic analysis of reductive groups. One of the aims of this thesis is to establish a local twisted trace formula for certain non-connected reductive groups, which is a twisted version of Arthur’s local trace formula. As an application of the local twisted trace formula, we will prove some twisted orthogonality relations, which are generalizations of Arthur’s results about orthogonality relations for tempered elliptic characters. To establish these relations, we will also give a classification of twisted elliptic representations.

Thesis (Ph. D.)--Carleton University, 2001. / Includes bibliographical references (p. 141-144). Also available in electronic format on the Internet.

Thesis (Doctoral)--Universität Bonn, 1990. / Includes bibliographical references.

Nous étudions la stratification de Newton des variétés de Shimura de type PEL aux places de bonne réduction. Nous considérons la strate basique de certaines variétés de Shimura simples de type PEL modulo une place de bonne réduction. Sous des hypothèses simplificatrices nous prouvons une relation entre la cohomologie l-adique de ce strate basique et la cohomologie de la variété de Shimura complexe. En particulier, nous obtenons des formules explicites pour le nombre de points dans la strate basique sur des corps finis, en termes de représentations automorphes. Nous obtenons les résultats à l'aide de la formule des traces et de la troncature de la formule de Kottwitz pour le nombre de points sur une variété de Shimura sur un corps fini. Nous montrons, en utilisant la formule des traces, que n'importe quelle strate de Newton d'une variété de Shimura de type PEL de type (A) est non vide en une place de bonne réduction. Ce résultat a déjà été établi par Viehmann-Wedhorn; nous donnons une nouvelle preuve de ce théorème. Considérons la strate basique des variétés de Shimura associées à certains groupes unitaires dans les cas où cette strate est une variété finie. Alors, nous démontrons un résultat d' équidistribution pour les opérateurs de Hecke agissant sur cette strate. Nous relions le taux de convergence avec celui de la conjecture de Ramanujan. Dans nos formules ne figurent que des représentations automorphes cuspidales sur Gl_n pour lesquelles cette conjecture est connue, et nous obtenons donc des estimations très bonnes sur la vitesse de convergence. En collaboration avec Erez Lapid nous calculons le module de Jacquet d'une représentation en échelle pour tout sous-groupe parabolique standard du groupe général linéaire sur un corps local non-archimédien. / We study the Newton stratification of Shimura varieties of PEL type, at the places of good reduction. We consider the basic stratum of certain simple Shimura varieties of PEL type at a place of good reduction. Under simplifying hypotheses we prove a relation between the l-adic cohomology of this basic stratum and the cohomology of the complex Shimura variety. In particular we obtain explicit formulas for the number of points in the basic stratum over finite fields, in terms of automorphic representations. We obtain our results using the trace formula and truncation of the formula of Kottwitz for the number of points on a Shimura variety over a finite field. We prove, using the trace formula that any Newton stratum of a Shimura variety of PEL-type of type (A) is non-empty at a prime of good reduction. This result is already established by Viehmann-Wedhorn; we give a new proof of this theorem. We consider the basic stratum of Shimura varieties associated to certain unitary groups in cases where this stratum is a finite variety. Then, we prove an equidistribution result for Hecke operators acting on the basic stratum. We relate the rate of convergence to the bounds from the Ramanujan conjecture of certain particular cuspidal automorphic representations on Gl_n. The Ramanujan conjecture turns out to be known for these automorphic representations, and therefore we obtain very sharp estimates on the rate of convergence. We prove that any connected reductive group G over a non-Archimedean local field has a cuspidal representation. Together with Erez Lapid we compute the Jacquet module of a Ladder representation at any standard parabolic subgroup of the general linear group over a non-Archimedean local field.

This thesis consists of a summary of four papers dealing with resonances of Dirac operators on Euclidean 3-space. In Paper I we show that the Complex Absorbing Potential (CAP) method is valid in the semiclassical limit for resonances sufficiently close to the real line if the potential is smooth and compactly supported. In Paper II  we continue the investigations initiated in Paper I but here we study clouds of resonances close to the real line and show that in some sense the CAP method remains valid also for multiple resonances. In Paper III we study perturbations of Dirac operators with smooth decaying scalar potentials  and show that these possess many resonances near certain points related to the maximum and the minimum of the potential. In Paper IV we show a trace formula of Poisson type for Dirac operators having compactly supported potentials which is related to resonances. The techniques mainly stem from complex function theory and scattering theory.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第22230号 / 理博第4544号 / 新制||理||1653(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 池田 保, 教授 雪江 明彦, 教授 並河 良典 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

In this licentiate thesis, we study the action of Hecke operators on Drinfeld cusp forms via the theory of crystals over function fields. The thesis contains one preliminary chapter, in which we recall some basic theory of Drinfeld modules and Drinfeld modular forms, as well as the Eichler-Shimura theory developed by Böckle. The core of the thesis consists of Chapter II, in which we prove a Lefschetz trace formula for crystals over stacks and deduce a Ramanujan bound for Drinfeld modular forms, and Chapter III, in which we compute traces and slopes of Hecke operators. We formulate several questions and conjectures based on our data. We also include an appendix in which we discuss the relationship between traces of an operator in positive characteristic and its eigenvalues.