451 research outputs found
Branching integrals and Casselman phenomenon
Let be a real semisimple Lie group, its maximal complex subgroup, and
its complexification.
It is known that all the -finite matrix elements on admit holomorphic
continuation to branching functions on having singularities at the a
prescribed divisor. We propose a geometric explanation of this phenomenon. The
note also contsins a general survey of holomorphic continuations of
infinite-dimensional representations.Comment: 13pp, an addendum is adde
Reporting and Sharing Financial Information with XBRL
In less than a decade, XBRL (eXtensible Business Reporting Language) has become a standard for reporting financial data in an XML format. This paper is an introduction to XBRL, the technical documents needed to accomplish the reporting, potential problems found in current reporting mechanisms, and future directions for use of XBRL
Uniqueness of Bessel models: the archimedean case
In the archimedean case, we prove uniqueness of Bessel models for general
linear groups, unitary groups and orthogonal groups.Comment: 22 page
On q-deformed gl(l+1)-Whittaker function II
A representation of a specialization of a q-deformed class one lattice
gl(\ell+1}-Whittaker function in terms of cohomology groups of line bundles on
the space QM_d(P^{\ell}) of quasi-maps P^1 to P^{\ell} of degree d is proposed.
For \ell=1, this provides an interpretation of non-specialized q-deformed
gl(2)-Whittaker function in terms of QM_d(\IP^1). In particular the (q-version
of) Mellin-Barnes representation of gl(2)-Whittaker function is realized as a
semi-infinite period map. The explicit form of the period map manifests an
important role of q-version of Gamma-function as a substitute of topological
genus in semi-infinite geometry. A relation with Givental-Lee universal
solution (J-function) of q-deformed gl(2)-Toda chain is also discussed.Comment: Extended version submitted in Comm. Math. Phys., 24 page
A general form of Gelfand-Kazhdan criterion
We formalize the notion of matrix coefficients for distributional vectors in
a representation of a real reductive group, which consist of generalized
functions on the group. As an application, we state and prove a Gelfand-Kazhdan
criterion for a real reductive group in very general settings.Comment: 16 pages, to appear in Manuscripta Mathematic
Denominators of Eisenstein cohomology classes for GL_2 over imaginary quadratic fields
We study the arithmetic of Eisenstein cohomology classes (in the sense of G.
Harder) for symmetric spaces associated to GL_2 over imaginary quadratic
fields. We prove in many cases a lower bound on their denominator in terms of a
special L-value of a Hecke character providing evidence for a conjecture of
Harder that the denominator is given by this L-value. We also prove under some
additional assumptions that the restriction of the classes to the boundary of
the Borel-Serre compactification of the spaces is integral. Such classes are
interesting for their use in congruences with cuspidal classes to prove
connections between the special L-value and the size of the Selmer group of the
Hecke character.Comment: 37 pages; strengthened integrality result (Proposition 16), corrected
statement of Theorem 3, and revised introductio
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
The strong thirteen spheres problem
The thirteen spheres problem is asking if 13 equal size nonoverlapping
spheres in three dimensions can touch another sphere of the same size. This
problem was the subject of the famous discussion between Isaac Newton and David
Gregory in 1694. The problem was solved by Schutte and van der Waerden only in
1953.
A natural extension of this problem is the strong thirteen spheres problem
(or the Tammes problem for 13 points) which asks to find an arrangement and the
maximum radius of 13 equal size nonoverlapping spheres touching the unit
sphere. In the paper we give a solution of this long-standing open problem in
geometry. Our computer-assisted proof is based on a enumeration of the
so-called irreducible graphs.Comment: Modified lemma 2, 16 pages, 12 figures. Uploaded program packag
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