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Sin resume

Whittaker supports for representations of reductive groups

Let $F$ be either $\mathbb{R}$ or a finite extension of $\mathbb{Q}_p$, and let $G$ be a finite central extension of the group of $F$-points of a reductive group defined over $F$. Also let $\pi$ be a smooth representation of $G$ (Frechet of moderate growth if $F=\mathbb{R}$). For each nilpotent orbit $\mathcal{O}$ we consider a certain Whittaker quotient $\pi_{\mathcal{O}}$ of $\pi$. We define the Whittaker support WS$(\pi)$ to be the set of maximal $\mathcal{O}$ among those for which $\pi_{\mathcal{O}}\neq 0$. In this paper we prove that all $\mathcal{O}\in\mathrm{WS}(\pi)$ are quasi-admissible nilpotent orbits, generalizing some of the results in [Moe96,JLS16]. If $F$ is $p$-adic and $\pi$ is quasi-cuspidal then we show that all $\mathcal{O}\in\mathrm{WS}(\pi)$ are $F$-distinguished, i.e. do not intersect the Lie algebra of any proper Levi subgroup of $G$ defined over $F$. We also give an adaptation of our argument to automorphic representations, generalizing some results from [GRS03,Shen16,JLS16,Cai] and confirming some conjectures from [Ginz06]. Our methods are a synergy of the methods of the above-mentioned papers, and of our preceding paper [GGS17]

Analytic continuation of equivariant distributions

We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein’s theory of analytic continuation of holonomic distributions. We use this to construct H-equivariant functionals on principal series representations of G, where G is a real reductive group and H is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the p-adic case using a recent result of Hong and Sun

Acromegaly, Mr Punch and caricature.

The origin of Mr Punch from the Italian Pulcinella of the Commedia dell'arte is well known but his feature, large hooked nose, protruding chin, kyphosis and sternal protrusion all in an exaggerated form also suggest the caricature of an acromegalic. This paper looks at the physical characteristics of acromegaly, the origin of Mr Punch and the development of caricature linking them together in the acromegalic caricature that now has a life of its own

Derivatives for smooth representations of GL(n,R) and GL(n,C)

The notion of derivatives for smooth representations of GL(n) in the p-adic case was defined by J. Bernstein and A. Zelevinsky. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations by S. Sahi and called the "adduced" representation. In this paper we define derivatives of all order for smooth admissible Frechet representations (of moderate growth). The archimedean case is more problematic than the p-adic case; for example arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation. In [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS12].Comment: First version of this preprint was split into 2. The proofs of two theorems which are technically involved in analytic difficulties were separated into "Twisted homology for the mirabolic nilradical" preprint. All the rest stayed in v2 of this preprint. v3: version to appear in the Israel Journal of Mathematic

Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

In this paper we analyze Fourier coefficients of automorphic forms on adelic split simply-laced reductive groups $G(\mathbb{A})$. Let $\pi$ be a minimal or next-to-minimal automorphic representation of $G(\mathbb{A})$. We prove that any $\eta\in \pi$ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro--Shalika formula for cusp forms on $GL_n$. We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient in terms of these Whittaker coefficients. A consequence of our results is the non-existence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for $G$ of type $D_5$ and $E_8$ with a view towards applications to scattering amplitudes in string theory

Optimizing wetland restoration to improve water quality at a regional scale

Published by IOP Publishing Ltd. Excessive phosphorus (P) export to aquatic ecosystems can lead to impaired water quality. There is a growing interest among watershed managers in using restored wetlands to retain P from agricultural landscapes and improve water quality. We develop a novel framework for prioritizing wetland restoration at a regional scale. The framework uses an ecosystem service model and an optimization algorithm that maximizes P reduction for given levels of restoration cost. Applying our framework in the Lake Champlain Basin, we find that wetland restoration can reduce P export by 2.6% for a budget of $50 M and 5.1$200 M. Sensitivity analysis shows that using finer spatial resolution data for P sources results in twice the P reduction benefits at a similar cost by capturing hot-spots on the landscape. We identify 890 wetlands that occur in more than 75% of all optimal scenarios and represent priorities for restoration. Most of these wetlands are smaller than 7 ha with contributing area less than 100 ha and are located within 200 m of streams. Our approach provides a simple yet robust tool for targeting restoration efforts at regional scales and is readily adaptable to other restoration strategies

Optimizing investments in national-scale forest landscape restoration in Uganda to maximize multiple benefits

Forest loss and degradation globally has resulted in declines in multiple ecosystem services and reduced habitat for biodiversity. Forest landscape restoration offers an opportunity to mitigate these losses, conserve biodiversity, and improve human well-being. As part of the Bonn Challenge, a global effort to restore 350 million hectares of deforested and degraded land by 2030, over 30 countries have recently made commitments to national forest landscape restoration. In order to achieve these goals, decision-makers require information on the potential benefits and costs of forest landscape restoration to efficiently target investments. In response to this need, we developed an approach using a suite of ecosystem service mapping tools and a multi-objective spatial optimization technique that enables decision-makers to estimate the potential benefits and opportunity costs of restoration, visualize tradeoffs associated with meeting multiple objectives, and prioritize where restoration could deliver the greatest benefits.Wedemonstrate the potential of this approach in Uganda, one of the nations committed to the Bonn Challenge. Using maps of the potential benefits and costs of restoration and efficiency frontiers for optimal restoration scenarios, we were able to communicate how ecosystem services benefits vary spatially across the country and how different weights on ecosystem services objectives can affect the allocation of restoration across Uganda. This work provides a generalizable approach to improve investments in forest landscape restoration and illuminates the tradeoffs associated with alternative restoration strategies.UKAid from the UK government through the International Union for Conservation of Nature’s KnowFor program as well as by the Natural Capital Project, a partnership between the University of Minnesota, Stanford University, the World Wildlife Fund, and the Nature Conservancy. MG was supported by the National Research Foundation of South Africa (Grant Number 98889).http://http://iopscience.iop.org1748-9326am2017Plant Scienc