782 research outputs found
Canonical extension of Whittaker distributions for GL(n,R)
No full textThe “multiplicity one theorem” asserts that the space of Whittaker functionals on irreducible representations of GL(r, R) is at most one-dimensional. This was originally proven by Piatetski-Shapiro [10] and Shalika [11]. In [4], Kostant showed that the dimension of the space of Whittaker functionals for any principal series representation of a quasisplit linear Lie group is exactly one. We give a new proof of the existence of Whittaker functionals on the principal series representations of GL(n, R) by an explicit construction using the integration pairing of Whittaker distributions against smooth functions in the principal series representations. This pairing gives the Jacquet integral. We derive formulas for a change of variables in the integral, that enable us to compute the Jacquet integral directly by means of integration by parts and thereby prove its analytic continuation. This legitimizes the pairing of Whittaker distributions and smooth functions, hence proving the existence of Whittaker functionals.Ph.D.Includes bibliographical reference
Théorème de Paley-Wiener pour les fonctions de Whittaker sur un groupe réductif p-adique
No full text57 pagesInternational audienceLet G be a p-adic reductive group and let U0 be the unipotent radical of a minimal parabolic subgroup of G. We introduce a Fourier transform defined on the space of smooth Whittaker functions on G which are compactly supported modulo U0. We determine its image. The proof follows the proof of Heiermann for the functions on the group.During the proof, we establish an inversion formula. This formula allows us to prove that an irreducible smooth representation of G, which has a Whittaker model in the space of smooth Whittaker functions on G which are compactly supported modulo U0, is cuspidal.This work gave us the opportunity to prepare a framework for the study of harmonic analysis on p-adic reductive symmetric spaces: B-matrices and constant term; a study of wave packets.Soit G un groupe réductif p-adique et U0 le radical unipotent d'un sous-groupe parabolique minimal de G. Nous introduisons une transformation de Fourier pour l'espace des fonctions de Whittaker lisses sur G et à support compact modulo U0. Nous en déterminons l'image. La preuve suit celle d'Heiermann pour les fonctions sur le groupe.Au cours de la preuve, une formule d'inversion est prouvée. Celle-ci permet de montrer qu'une représentation lisse irréductible de G, qui possède modèle de Whittaker dans les fonctions de Whittaker à support compact modulo U0, est cuspidale.Ce travail nous a donné l'opportunité de préparer un cadre pour l'analyse harmonique sur les espaces symétriques réductifs p-adiques: B-matrices et terme constant, propriétés des paquets d'ondes
The Miller F. Whittaker Library\u27s Renovation Project
No full textThe author describes the renovation project undertaken by the Miller F. Whittaker Library at South Carolina State University
Whittaker Limits of Difference Spherical Functions
No full textThe q-Whittaker function is introduced as a limit at t = 0 of the global q, t-spherical function, which extends the symmetric Macdonald polynomials to arbitrary eigenvalues. The limiting procedure generalizes that due to Etingof. The construction heavily depends on the technique of the q-Gaussians developed by the author (and Stokman in the non-reduced case). In this approach, the q-Whittaker function is given by a series convergent everywhere. One of the applications is a q-version of the Shintani–Casselman–Shalika formula, which is directly connected with the q, t-Mehta–Macdonald identities in terms of the Jackson integral. In type A, this formula generalizes that due to Gerasimov et al. The Harish-Chandra-type asymptotic formula is established for the global q, t-spherical functions, including the Whittaker limit
Yangians, Mirabolic Subalgebras, and Whittaker Vectors
No full textWe construct an element in a completion of the universal enveloping algebra of N, which we call the Kirillov projector, that connects the topics of the title: on the one hand, it is defined using the evaluation homomorphism from the Yangian of N, on the other hand, it gives a canonical projection onto the space of Whittaker vectors for any Whittaker module over the mirabolic subalgebra. Using the Kirillov projector, we deduce some categorical properties of Whittaker modules; for instance, we prove a mirabolic analog of Kostant's theorem. We also show that it quantizes a rational version of the Cremmer-Gervais -matrix. As an application, we construct a universal vertex-IRF transformation from the standard dynamical -matrix to this constant one in categorical terms.The author would like to thank Roman Bezrukavnikov, Pavel Etingof, Boris Feigin, Michael Finkelberg, Joel Kamnitzer, Vasily Krylov, and Leonid Rybnikov for helpful discussions and explanations, as well as the anonymous referees for their comments. The author would also like to thank the contributors of [41]; this project would not be possible without their libraries. This work was initially supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075–15–2022–287); the majority of it was carried out at the Massachusetts Institute of Technology. The author is very grateful to the Department of Mathematics of MIT for its hospitality and for the opportunity to avoid (a form of) politically motivated persecution in Russia
Estimates on non-decaying Whittaker functions
No full textSince the Fourier coefficients of an automorphic form along the nilpotent radical of parabolic subgroup are expressed in terms of Whittaker functions, a better understanding of their growth in every direction would be useful in the study of automorphic forms. Bump and Huntley (1995) used an integral formula which was found by Vinogradov, Takhtadzhyan (1978), and Stade (1988) to obtain precise information of the spherical Whittaker functions M(ν1,ν2) (y1, y2) as both y1 and y2 → ∞. To (1995) used a method similar to the characteristic method in the theory of differential equations to compute the leading exponents of asymptotic expansions of a basis of Whittaker functions in the positive Weyl chamber for a split semi-simple Lie group over R,, which, in particular, yields a solution to Zuckerman's conjecture for SL(3, R). Templier (2015) has recently used an integral representation by Givental to show To's result: the exponential growth of M(ν1,ν2) (y1, y2) for y1, y2 ≥ 1 as either or both y1, y2 → ∞. In this thesis I use a new formula which was derived by Ishii and Stade (2007) to obtain the asymptotic expansions of M(ν1,ν2) (t,1/tp) and M(ν1,ν2)(1/tp, t) as t → ∞ where 2 ≤ p ∈ 1/2Z then successfully prove an analog of the Multiplicity One Theorem in these directions, namely that in certain circumstances the moderate growth condition in the theory of automorphic forms is automatic.Ph.D.Includes bibliographical referencesby Tien Duy Trin
Formule de Plancherel pour les fonctions de Whittaker sur un groupe réductif p-adique
No full textInternational audienceWe prove the Plancherel formula for Whittaker functions on a reductive p-adic group. This a sequel to our work on Paley-Wiener theorem. Our proof is close to the proof written by Waldspurger of the Harish-Chandra Plancherel formula for smooth functions on the group and use many of his results. One simplification is the easy proof of the Fourier transfom, which follows from a result of Joseph Bernstein.Nous prouvons la formule de Plancherel pour les fonctions de Whittaker sur un groupe réductif p-adique. Les méthodes sont proches de celles de lapreuve de Waldspurger, d’après Harish-Chandra, pour les fonctions lisses sur le groupe.Au delà du résultat, ce travail met en place un cadre qui devrait s’avérer utile pour d’autres formules de Plancherel, notamment pour les espaces symétriques réductifs p-adiques. En particulier, il met en valeur le role des matrices B et de leur propriété d’adjonction
Whittaker Chambers: A Biography, by Sam Tanenhaus. (New York: Random House, 1997. Pp. 638. $35.00)
No full textFree-lance writer Sam Tanenhaus has made an extraordinary contribution to twentieth-century American history in his stunning new biography of Whittaker Chambers. Tanenhaus combines scholastic rigidity with his eminently readable style to create a work of the highest standard of historical biography. Whittaker Chambers emerges as a complex, enigmatic figure, misunderstood by most in his own time and by his liberal critics yet today. The author is successful in understanding and intimating the complex forces that drew Chambers to, and ultimately from, communism, and eventually toward his confrontation with Alger Hiss
Fostering Cultural-Ecological Reconnections at the Ursinus Food Forest
No full textProfessor of Environmental Studies Patrick Hurley works to better understand the often unrecognized ways that social-political dimensions shape forest conservation interactions. This interactive tour and talk about the Ursinus Food Forest will explore Dr. Patrick Hurley’s very real Ursinus Quest: the effort to apply concepts of multifunctionality and food forestry to approximately two acres of previously farmed land at the Whittaker Environmental Research Station. The onsite engagement with students and the many species on site will introduce the diverse learning experiences that characterize the intergenerational commitment to transforming this place through the creation of a living-learning laboratory.https://digitalcommons.ursinus.edu/baden/1009/thumbnail.jp
Whittaker S, Arguing for Adaptation, Routledge Green Finance Handbook, Supplementary Files
No full textThese Supplementary Files support the analysis described in the published paper: Arguing for climate change adaptation finance – A bibliometric study in the Routledge Green Finance Handbook. They contain items S1 to S7 which are referenced in the main paper. The analysis is the author’s own work and originates from the literature extracted for the Systematic Literature Review and bibliometric analysis undertaken by the author in 2021. The analysis supplements the findings presented in the paper. Original data can be made available upon request to the author
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