160 research outputs found
Vertex operators, solvable lattice models and metaplectic Whittaker functions
We show that spherical Whittaker functions on an -fold cover of the
general linear group arise naturally from the quantum Fock space representation
of introduced by Kashiwara, Miwa and Stern
(KMS). We arrive at this connection by reconsidering solvable lattice models
known as `metaplectic ice' whose partition functions are metaplectic Whittaker
functions. First, we show that a certain Hecke action on metaplectic Whittaker
coinvariants agrees (up to twisting) with a Hecke action of Ginzburg,
Reshetikhin, and Vasserot. This allows us to expand the framework of KMS by
Drinfeld twisting to introduce Gauss sums into the quantum wedge, which are
necessary for connections to metaplectic forms. Our main theorem interprets the
row transfer matrices of this ice model as `half' vertex operators on quantum
Fock space that intertwine with the action of
.
In the process, we introduce new symmetric functions termed
\textit{metaplectic symmetric functions} and explain how they relate to
Whittaker functions on an -fold metaplectic cover of GL. These resemble
\textit{LLT polynomials} introduced by Lascoux, Leclerc and Thibon; in fact the
metaplectic symmetric functions are (up to twisting) specializations of
\textit{supersymmetric LLT polynomials} defined by Lam. Indeed Lam constructed
families of symmetric functions from Heisenberg algebra actions on the Fock
space commuting with the -action. We explain
that half vertex operators agree with Lam's construction and this
interpretation allows for many new identities for metaplectic symmetric and
Whittaker functions, including Cauchy identities. While both metaplectic
symmetric functions and LLT polynomials can be related to vertex operators on
the -Fock space, only metaplectic symmetric functions are connected to
solvable lattice models.Comment: v3 changes: minor edit
Eisenstein series and automorphic representations
We provide an introduction to the theory of Eisenstein series and automorphic
forms on real simple Lie groups G, emphasising the role of representation
theory. It is useful to take a slightly wider view and define all objects over
the (rational) adeles A, thereby also paving the way for connections to number
theory, representation theory and the Langlands program. Most of the results we
present are already scattered throughout the mathematics literature but our
exposition collects them together and is driven by examples. Many interesting
aspects of these functions are hidden in their Fourier coefficients with
respect to unipotent subgroups and a large part of our focus is to explain and
derive general theorems on these Fourier expansions. Specifically, we give
complete proofs of the Langlands constant term formula for Eisenstein series on
adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic
spherical Whittaker function associated to unramified automorphic
representations of G(Q_p). In addition, we explain how the classical theory of
Hecke operators fits into the modern theory of automorphic representations of
adelic groups, thereby providing a connection with some key elements in the
Langlands program, such as the Langlands dual group LG and automorphic
L-functions. Somewhat surprisingly, all these results have natural
interpretations as encoding physical effects in string theory. We therefore
also introduce some basic concepts of string theory, aimed toward
mathematicians, emphasising the role of automorphic forms. In particular, we
provide a detailed treatment of supersymmetry constraints on string amplitudes
which enforce differential equations of the same type that are satisfied by
automorphic forms. Our treatise concludes with a detailed list of interesting
open questions and pointers to additional topics which go beyond the scope of
this book.Comment: 326 pages. Detailed and example-driven exposition of the subject with
highlighted applications to string theory. v2: 375 pages. Substantially
extended and small correction
A reduction principle for Fourier coefficients of automorphic forms
In this paper we analyze a general class of Fourier coefficients of
automorphic forms on reductive adelic groups
and their covers. We prove that any such
Fourier coefficient is expressible through integrals and sums involving
'Levi-distinguished' Fourier coefficients. By the latter we mean the class of
Fourier coefficients obtained by first taking the constant term along the
nilradical of a parabolic subgroup, and then further taking a Fourier
coefficient corresponding to a -distinguished nilpotent orbit in
the Levi quotient. In a follow-up paper we use this result to establish
explicit formulas for Fourier expansions of automorphic forms attached to
minimal and next-to-minimal representations of simply-laced reductive groups.Comment: 35 pages. v2: Extended results and paper split into two parts with
second part appearing soon. New title to reflect new focus of this part. v3:
Minor corrections and updated reference to the second part that has appeared
as arXiv:1908.08296. v4: Minor corrections and reformulation
The Cohomological Sarnak-Xue Density Hypothesis for
We prove the cohomological version of the Sarnak-Xue Density Hypothesis for
over a totally real field and for inner forms split at all finite
places. The proof relies on recent lines of work in the Langlands program: (i)
Arthur's Endoscopic Classification of Representations of classical groups, and
(ii) the Generalized Ramanujan-Petersson Theorem, proved for cohomological
self-dual cuspidal representations of general linear groups. We give
applications for the growth of cohomology of arithmetic manifolds,
density-Ramanujan complexes, cutoff phenomena and optimal strong approximation.Comment: 71 page
Colored five-vertex models and Demazure atoms
Type A Demazure atoms are pieces of Schur functions, or sets of tableaux
whose weights sum to such functions. Inspired by colored vertex models of
Borodin and Wheeler, we will construct solvable lattice models whose partition
functions are Demazure atoms; the proof of this makes use of a Yang-Baxter
equation for a colored five-vertex model. As a biproduct, we construct Demazure
atoms on Kashiwara's crystal and give new algorithms for
computing Lascoux-Sch\"utzenberger keys
Eulerianity of Fourier coefficients of automorphic forms
We study the question of Eulerianity (factorizability) for Fourier
coefficients of automorphic forms, and we prove a general transfer theorem that
allows one to deduce the Eulerianity of certain coefficients from that of
another coefficient. We also establish a `hidden' invariance property of
Fourier coefficients. We apply these results to minimal and next-to-minimal
automorphic representations, and deduce Eulerianity for a large class of
Fourier and Fourier-Jacobi coefficients. In particular, we prove Eulerianity
for parabolic Fourier coefficients with characters of maximal rank for a class
of Eisenstein series in minimal and next-to-minimal representations of groups
of ADE-type that are of interest in string theory.Comment: 28 pages. v2: Clarified connection to Fourier-Jacobi coefficients and
references added. v3: Minor correction
Re-thinking the “ecological envelope” of Eastern Baltic cod (Gadus morhua): conditions for productivity, reproduction, and feeding over time
Hypoxia is presently seen as the principal driver behind the decline of the former dominating Eastern Baltic cod stock (EBC; Gadus morhua). It has been proposed that both worsening conditions for reproduction and lower individual growth, condition, and survival are linked to hypoxia. Here, we elucidate the ecological envelope of EBC in terms of salinity stratification, oxygen content, and benthic animal biomasses, and how it has affected EBC productivity over time. The spawning conditions started deteriorating in the Gotland Deep in the 1950s due to oxygen depletion. In contrast, in the Bornholm Basin, hydrographic conditions have remained unchanged over the last 60 years. Indeed, the current extent of both well-oxygenated areas and the frequency of hypoxia events do not differ substantially from periods with high EBC productivity in the 1970s–1980s. Furthermore, oxygenated and therefore potentially suitable feeding areas are abundant in all parts of the Baltic Sea, and our novel analysis provides no evidence of a reduction in benthic food sources for EBC over the last 30 years. We find that while reproduction failure is intricately linked to hydrographic dynamics, a relationship between the spread of hypoxia and the decline in EBC productivity during the last decades cannot be substantiated.Peer reviewe
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