68 research outputs found
Totally Ramified Maximal Tori and Bruhat-Tits theory
No full textSuppose is a nonarchimedean local field, is a maximally unramified extension of , and is a connected reductive -group. If is a -minisotropic maximal -torus in , then we use Bruhat-Tits theory to describe the stable classes in the -orbit of , the rational classes in the -orbit of , and the -embeddings, up to rational conjugacy, into of . We also provide, via Bruhat-Tits theory, a complete and explicit description of: the rational conjugacy classes of -minisotropic maximal tame -tori in ; the stable classes of -minisotropic maximal tame -tori in ; and the -embeddings, up to rational conjugacy, into of a -minisotropic maximal tame -torus of .Comments welcome! Appendices by Ram Ekstrom and Mitya Boyarchenko, Stephen DeBacker, Anna Spice, Loren Spice, and Cheng-Chiang Tsai. Version two includes some new results and correction
Arithmetic of the Yoshida Lift.
Get PDFCo-Chairs: Stephen M. DeBacker and Christopher M. Skinner
This thesis concerns the arithmetic properties of the Yoshida lift, Y, which is a scalar- valued holomorphic Siegel modular form of degree 2 obtained as the theta lift of a pair of automorphic forms f1,f2 on D×, where D is a definite quaternion algebra over Q.
Specifically, we define a refined version of the Yoshida lift, Y, which has the special property that it preserves p-integral structures and is not identically zero under mild conditions. For p-integrality, we compute a formula for the Fourier coefficients aT of Y by exploiting an inherent freedom in the definition of Y. The formula for aT in turn allows us to compute the Bessel model of the Yoshida lift, and apply an argument of Cornut–Vatsal to conclude that Y is non-zero. Furthermore, if we assume Artin’s conjecture on primitive roots, then we show that Y is in fact not zero modulo p.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/77876/1/jxj_1.pd
Interview with Mark Reeder on Depth-zero supercuspidal L-packets and their stability, by Mark Reeder and Stephen DeBacker
No full textIn this article Professors DeBacker and Reeder verify the local Langlands correspondence for pure inner forms of unramified p-adic groups and tame Langlands parameters in "general position". For each such parameter, the authors explicitly construct, in a natural way, a finite set ("L-packet") of depth-zero supercuspidal representations of the appropriate p-adic group, and they verify some expected properties of this L-packet. In particular, the authors prove, with some conditions on the base field, that the appropriate sum of characters of the representations in the author's L-packet is stable.Title supplied by cataloger
Admissible invariant distributions on reductive
No full textHarish-Chandra presented these lectures on admissible invariant distributions for p-adic groups at the Institute for Advanced Study in the early 1970s. He published a short sketch of this material as his famous "Queen's Notes". This book, which was prepared and edited by DeBacker and Sally, presents a faithful rendering of Harish-Chandra's original lecture notes. The main purpose of Harish-Chandra's lectures was to show that the character of an irreducible admissible representation of a connected reductive p-adic group G is represented by a locally summable function on G. A key ingredient in this proof is the study of the Fourier transforms of distributions on \mathfrak g, the Lie algebra of G. In particular, Harish-Chandra shows that if the support of a G-invariant distribution on \mathfrak g is compactly generated, then its Fourier transform has an asymptotic expansion about any semisimple point of \mathfrak g. Harish-Chandra's remarkable theorem on the local summability of characters for p-adic groups was a major result in representation theory that spawned many other significant results. This book presents, for the first time in print, a complete account of Harish-Chandra's original lectures on this subject, including his extension and proof of Howe's Theorem. In addition to the original Harish-Chandra notes, DeBacker and Sally provide a nice summary of developments in this area of mathematics since the lectures were originally delivered. In particular, they discuss quantitative results related to the local character expansion
Some Results on Tori in p-adic Groups
Get PDFThe rational conjugacy classes of tori play an important role in the representation theory of reductive p-adic groups defined over a finite extension k of the p-adic numbers. In an upcoming paper, Adler and DeBacker give a parameterization of the rational conjugacy classes of embeddings of maximal k-tori using Bruhat-Tits theory, and our primary goal will be to move towards an analogous parameterization of maximal theta-split k-tori, which play a prominent role in the theory of p-adic symmetric spaces. In particular, we will parameterize the rational conjugacy classes of maximal theta-split tori in finite groups of Lie type and in groups defined over the maximal unramified extension K of k. We will then use Bruhat-Tits theory to parameterize the theta-split k-tori which split over K and then determine which of these tori can emerge as the maximal K-split subtorus of a maximal theta-split k-torus. We will also provide a similar parameterization of a class of unramified tori which we will call unramified theta-perfect tori. These tori will play an important role in future work where we will use them to determine how the conjugacy classes of theta-split tori over K split into rational conjugacy classes. Finally, in the case of symplectic groups, we will compare DeBacker’s parameterization of maximal unramified tori to another parameterization due to Waldspurger.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169650/1/jachaley_1.pd
Lectures On Harmonic Analysis For Reductive -Adic Groups
No full textThis paper is based on lectures delivered at the Institute for Mathematical Sciences at the National University of Singapore in 2002. The author thanks the Institute for its support. is very natural and therefore a good place to begin the course. In order to form a group with respect to addition, the additive identity and additive inverses are tossed in to the mix to give us the integers 24380 35 This set does not form a group with respect to multiplication; it is therefore enlarged to , the field of rational numbers. Everything so far has been very natural. At this point, the incompleteness of the rationals is demonstrated by proving that the square root of two is not rational. To compensate for this, the fact that the rationals are ordered (that is, there is a notion of nonpositive and nonnegative) is invoked to define the absolute value, , of any rational number if As usual, the absolute value gives you a metric with respect to which you complete the rationals. We recall that two Cauchy sequences of rational numbers are said to be equivalent with respect to provided that is a Cauchy sequence which converges to zero. The set of real numbers is then defined to be the set of equivalence classes of Cauchy sequences with respect to , that is, the completion of with respect . It turns out, however, that the normal absolute value provides just one of an infinite number of incompatible ways to complete the rational numbers , and, from the proper perspective, this completion is a somewhat unnatural object. Let be any prime. If is a nonzero rational number, then there is a unique integer ! #"!$ with &% and '% . We can ..
Motivic integration and the regular Shalika germ.
No full textLet G denote an adjoint semisimple algebraic group defined over the integers. There is an analogy between Haar integration on the p-adic integer valued points of G and motivic integration on the complex power series valued points of G . A result of D. Shelstad from p-adic harmonic analysis states that the regular Shalika germ of such a group is an asymptotically constant function. In the present work it is demonstrated that the complex power series analogue of the regular Shalika germ is similarly an asymptotically constant function. There are three main steps involved in this demonstration. The first step is a description of certain affine schemes, together with morphisms from these schemes to the group. This allows the calculation of the complex power series analogue of the regular Shalika germ via motivic integration over such an affine scheme. The second step is the association of a less complicated shadow scheme to each such affine scheme. This association is achieved by analysis of the coordinate rings of the affine schemes. The third step is the use of the transformation rule from motivic integration. This is used to equate a motivic integral on each affine scheme with one on its associated shadow scheme. It is again used to equate the resulting motivic integrals on two different shadow schemes. This exactly means that the complex power series analogue of the regular Shalika germ is an asymptotically constant function.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/123887/2/3106104.pd
The computational complexity of convex bodies.
No full textFor a convex body B, the membership question is the following: given a point x, is x in B? In this dissertation, we study the computational complexity of convex bodies in terms of the membership question. Since this question can be quite difficult to answer, we also study the computational complexity of testing membership for sets approximating a convex body. We give two new approximation constructions, along with some metric analysis of these approximations.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/126843/2/3276318.pd
Parameterizing Conjugacy Classes of Unramified Tori via Bruhat-Tits Theory
Get PDFSuppose is a nonarchimedean local field, is a maximally unramified extension of , and is a connected reductive -group. In this paper we provide parameterizations via Bruhat-Tits theory of: the rational conjugacy classes of -tori in that split over ; the rational and stable conjugacy classes of the -split components of the centers of unramified twisted Levi subgroups of ; and the rational conjugacy classes of unramified twisted generalized Levi subgroups of . We also provide parameterizations of analogous objects for finite groups of Lie type.Comments welcome! Appendix by Jeffrey D. Adler. Version 3 includes minor correction
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