2,351 research outputs found

### Shintani functions, real spherical manifolds, and symmetry breaking operators

For a pair of reductive groups $G \supset G'$, we prove a geometric criterion
for the space $Sh(\lambda, \nu)$ of Shintani functions to be finite-dimensional
in the Archimedean case.
This criterion leads us to a complete classification of the symmetric pairs
$(G,G')$ having finite-dimensional Shintani spaces.
A geometric criterion for uniform boundedness of $dim Sh(\lambda, \nu)$ is
also obtained.
Furthermore, we prove that symmetry breaking operators of the restriction of
smooth admissible representations yield Shintani functions of moderate growth,
of which the dimension is determined for $(G, G') = (O(n+1,1), O(n,1))$.Comment: to appear in Progress in Mathematics, Birkhause

### Can (Electric-Magnetic) Duality Be Gauged?

There exists a formulation of the Maxwell theory in terms of two vector
potentials, one electric and one magnetic. The action is then manifestly
invariant under electric-magnetic duality transformations, which are rotations
in the two-dimensional internal space of the two potentials, and local. We ask
the question: can duality be gauged? The only known and battled-tested method
of accomplishing the gauging is the Noether procedure. In its decanted form, it
amounts to turn on the coupling by deforming the abelian gauge group of the
free theory, out of whose curvatures the action is built, into a non-abelian
group which becomes the gauge group of the resulting theory. In this article,
we show that the method cannot be successfully implemented for
electric-magnetic duality. We thus conclude that, unless a radically new idea
is introduced, electric-magnetic duality cannot be gauged. The implication of
this result for supergravity is briefly discussed.Comment: Some minor typos correcte

### Non-Linear Realisation of the Pure N=4, D=5 Supergravity

We perform the non-linear realisation or the coset formulation of the pure
N=4, D=5 supergravity. We derive the Lie superalgebra which parameterizes a
coset map whose induced Cartan-Maurer form produces the bosonic field equations
of the pure N=4, D=5 supergravity by canonically satisfying the Cartan-Maurer
equation. We also obtain the first-order field equations of the theory as a
twisted self-duality condition for the Cartan-Maurer form within the
geometrical framework of the coset construction.Comment: 12 page

### Some results on invariant theory

First published in the Bulletin of the American Mathematical Society in Vol.68 1962, published by the American Mathematical Societ

### Eigenfunctions of the Laplacian and associated Ruelle operator

Let $\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\'e
disk \DD and $\Delta$ the corresponding hyperbolic Laplace operator. Any
smooth eigenfunction $f$ of $\Delta$, equivariant by $\Gamma$ with real
eigenvalue $\lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral
representation by a distribution \dd_{f,s} (the Helgason distribution) which
is equivariant by $\Gamma$ and supported at infinity \partial\DD=\SS^1. The
geodesic flow on the compact surface \DD/\Gamma is conjugate to a suspension
over a natural extension of a piecewise analytic map T:\SS^1\to\SS^1, the
so-called Bowen-Series transformation. Let $\ll_s$ be the complex Ruelle
transfer operator associated to the jacobian $-s\ln |T'|$. M. Pollicott showed
that \dd_{f,s} is an eigenfunction of the dual operator $\ll_s^*$ for the
eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic
eigenfunction $\psi_{f,s}$ of $\ll_s$ for the eigenvalue 1, given by an
integral formula \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}}
\dd_{f,s} (d\eta), \noindent where $J(\xi,\eta)$ is a $\{0,1\}$-valued
piecewise constant function whose definition depends upon the geometry of the
Dirichlet fundamental domain representing the surface \DD/\Gamma

### Invariant differential equations on homogeneous manifolds

First published in the Bulletin of the American Mathematical Society in Vol.83, 1977, published by the American Mathematical Societ

### Radon-Fourier transforms on symmetric spaces and related group representations

First published in the Bulletin of the American Mathematical Society in Vol.71, 1965, published by the American Mathematical Societ

### Fundamental solutions of invariant differential operators on symmetric spaces

First published in the Bulletin of the American Mathematical Society in 1963, published by the American Mathematical Societ

### Wigner transform and pseudodifferential operators on symmetric spaces of non-compact type

We obtain a general expression for a Wigner transform (Wigner function) on
symmetric spaces of non-compact type and study the Weyl calculus of
pseudodifferential operators on them

### Duality and Radon transform for symmetric spaces

First published in the Bulletin of the American Mathematical Society in Vol.69, 1963, published by the American Mathematical Societ

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