arXiv subjects

Jan Möllers

Publications and source records attributed to Jan Möllers.

At least 19 recordsLinked to original sources

arXiv2017

Upper bounds for geodesic periods over rank one locally symmetric spaces

We prove upper bounds for geodesic periods of automorphic forms over general rank one locally symmetric spaces. Such periods are integrals of automorphic forms restricted to special totally geodesic cycles of the ambient manifold and twisted with automorphic forms on the cycles. The upper bounds are in terms of the Laplace eigenvalues of the two automorphic forms, and they generalize previous results for real hyperbolic manifolds to the context of all rank one locally symmetric spaces.

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arXiv2017

The Berezin form on symmetric $R$-spaces and reflection positivity

For a symmetric $R$-space $K/L=G/P$ the standard intertwining operators provide a canonical $G$-invariant pairing between sections of line bundles over $G/P$ and its opposite $G/\overline{P}$. Twisting this pairing with an involution of $G$ which defines a non-compactly causal symmetric space $G/H$ we obtain an $H$-invariant form on sections of line bundles over $G/P$. Restricting to the open $H$-orbits in $G/P$ constructs the Berezin forms studied previously by G. van Dijk, S. C. Hille and V. F. Molchanov. We determine for which $H$-orbits in $G/P$ and for which line bundles the Berezin form is positive semidefinite, and in this case identify the corresponding representations of the dual group $G^c$ as unitary highest weight representations. We further relate this procedure of passing from representations of $G$ to representations of $G^c$ to reflection positivity.

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arXiv2016

Bessel operators on Jordan pairs and small representations of semisimple Lie groups

We provide a uniform construction of $L^2$-models for all small unitary representations in degenerate principal series of semisimple Lie groups which are induced from maximal parabolic subgroups with abelian nilradical. This generalizes previous constructions to the case of a maximal parabolic subgroup which is not necessarily conjugate to its opposite, and hence the previously used Jordan algebra methods have to be generalized to Jordan pairs. The crucial ingredients for the construction of the $L^2$-models are the Lie algebra action and the spherical vector. Working in the so-called Fourier transformed picture of the degenerate principal series, the Lie algebra action is given in terms of Bessel operators on Jordan pairs. We prove that precisely for those parameters of the principal series where small quotients occur, the Bessel operators are tangential to certain submanifolds. Further, we show that the small quotients are unitarizable if and only if these submanifolds carry equivariant measures. In this case we can express the spherical vectors in terms of multivariable K-Bessel functions, and some delicate estimates for these Bessel functions imply the existence of the $L^2$-models.

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arXiv2016

A Godement-Jacquet type integral and the metaplectic Shalika model

We present a novel integral representation for a quotient of global automorphic L-functions, the symmetric square over the exterior square. The pole of this integral characterizes a period of a residual representation of an Eisenstein series. As such, the integral itself constitutes a period, of an arithmetic nature. The construction involves the study of local and global aspects of a new model for double covers of general linear groups, the metaplectic Shalika model. In particular, we prove uniqueness results over p-adic and Archimedean fields, and a new Casselman-Shalika type formula.

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arXiv2015

On boundary value problems for some conformally invariant differential operators

We study boundary value problems for some differential operators on Euclidean space and the Heisenberg group which are invariant under the conformal group of a Euclidean subspace resp. Heisenberg subgroup. These operators are shown to be self-adjoint in certain Sobolev type spaces and the related boundary value problems are proven to have unique solutions in these spaces. We further find the corresponding Poisson transforms explicitly in terms of their integral kernels and show that they are isometric between Sobolev spaces and extend to bounded operators between certain $L^p$-spaces. The conformal invariance of the differential operators allows us to apply unitary representation theory of reductive Lie groups, in particular recently developed methods for restriction problems.

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arXiv2015

Knapp-Stein type intertwining operators for symmetric pairs

For a symmetric pair $(G,H)$ of reductive groups we construct a family of intertwining operators between spherical principal series representations of $G$ and $H$ that are induced from parabolic subgroups satisfying certain compatibility conditions. The operators are given explicitly in terms of their integral kernels and we prove convergence of the integrals for an open set of parameters and meromorphic continuation. We further discuss uniqueness of intertwining operators, and for the rank one cases $$ (G,H)=(SU(1,n;\mathbb{F}),S(U(1,m;\mathbb{F})\times U(n-m;\mathbb{F}))), \qquad \mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}, $$ and for the pair $$ (G,H)=(GL(4n,\mathbb{R}),GL(2n,\mathbb{C})) $$ we show that for a certain choice of maximal parabolic subgroups our operators generically span the space of intertwiners.

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arXiv2014

Restriction of some unitary representations of O(1,N) to symmetric subgroups

We find the complete branching law for the restriction of certain unitary representations of $O(1,n+1)$ to the subgroups $O(1,m+1)\times O(n-m)$, $0\leq m\leq n$. The unitary representations we consider belong either to the unitary spherical principal series, the spherical complementary series or are unitarizable subquotients of the spherical principal series. In the crucial case $0<m<n$ the decomposition consists of a continuous part and a discrete part. The continuous part is given by a direct integral of unitary principal series representations whereas the discrete part consists of finitely many representations which either belong to the complementary series or are unitarizable subquotients of the principal series. The explicit Plancherel formula is computed on the Fourier transformed side of the non-compact realization of the representations by using the spectral decomposition of a certain hypergeometric type ordinary differential operator. The main tool connecting this differential operator with the representations are second order Bessel operators which describe the Lie algebra action in this realization. To derive the spectral decomposition of the ordinary differential operator we use Kodaira's formula for the spectral decomposition of Schrödinger type operators.

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arXiv2014

Invariant differential operators on H-type groups and discrete components in restrictions of complementary series of rank one semisimple groups

We explicitly construct a finite number of discrete components in the restriction of complementary series representations of rank one semisimple groups $G$ to rank one subgroups $G_1$. For this we use the realizations of complementary series representations of $G$ and $G_1$ on Sobolev spaces on the nilpotent radicals $N$ and $N_1$ of the minimal parabolics in $G$ and $G_1$, respectively. The groups $N$ and $N_1$ are of H-type and we construct explicitly invariant differential operators between $N$ and $N_1$. These operators induce the projections onto the discrete components. Our construction of the invariant differential operators is carried out uniformly in the framework of H-type groups and also works for those H-type groups which do not occur as nilpotent radical of a parabolic subgroup in a semisimple group.

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arXiv2014

Discrete branching laws for minimal holomorphic representations

We find the explicit branching laws for the restriction of minimal holomorphic representations to symmetric subgroups in the case where the restriction is discretely decomposable. For holomorphic pairs the minimal holomorphic representation decomposes into a direct sum of lowest weight representations which is made explicit. For non-holomorphic pairs the restriction is shown to be irreducible and identified with a known representation. We further study a conjecture by Kobayashi on the behaviour of associated varieties under restriction and confirm this conjecture in the setting of this paper.

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arXiv2014

Estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles

We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles. The geodesic cycles we study are themselves hyperbolic manifolds of lower dimension. The restriction of an automorphic form to such a geodesic cycle can be expanded into eigenfunctions of the Laplacian on the geodesic cycle. We prove exponential decay for the coefficients in this expansion.

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arXiv2014

Structure of the degenerate principal series on symmetric R-spaces and small representations

Let $G$ be a simple real Lie group with maximal parabolic subgroup $P$ whose nilradical is abelian. Then $X=G/P$ is called a symmetric $R$-space. We study the degenerate principal series representations of $G$ on $C^\infty(X)$ in the case where $P$ is not conjugate to its opposite parabolic. We find the points of reducibility, the composition series and all unitarizable constituents. Among the unitarizable constituents we identify some small representations having as associated variety the minimal nilpotent $K_{\mathbb{C}}$-orbit in $\mathfrak{p}_{\mathbb{C}}^*$, where $K_{\mathbb{C}}$ is the complexification of a maximal compact subgroup $K\subseteq G$ and $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ the corresponding Cartan decomposition.

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arXiv2013

Branching laws for small unitary representations of GL(n,C)

The unitary principal series representations of $G=GL(n,\mathbb{C})$ induced from a character of the maximal parabolic subgroup $P=(GL(1,\mathbb{C})\times GL(n-1,\mathbb{C}))\ltimes\mathbb{C}^{n-1}$ attain the minimal Gelfand--Kirillov dimension among all infinite-dimensional unitary representations of $G$. We find the explicit branching laws for the restriction of these representations to symmetric subgroups of $G$.

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arXiv2013

A geometric quantization of the Kostant-Sekiguchi correpondence for scalar type unitary highest weight representations

For any Hermitian Lie group $G$ of tube type we give a geometric quantization procedure of certain $K_{\mathbb{C}}$-orbits in $\mathfrak{p}_{\mathbb{C}}^*$ to obtain all scalar type highest weight representations. Here $K_{\mathbb{C}}$ is the complexification of a maximal compact subgroup $K\subseteq G$ with corresponding Cartan decomposition $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ of the Lie algebra of $G$. We explicitly realize every such representation $π$ on a Fock space consisting of square integrable holomorphic functions on its associated variety $Ass(π)\subseteq\mathfrak{p}_{\mathbb{C}}^*$. The associated variety $Ass(π)$ is the closure of a single nilpotent $K_{\mathbb{C}}$-orbit $\mathcal{O}^{K_{\mathbb{C}}}\subseteq\mathfrak{p}_{\mathbb{C}}^*$ which corresponds by the Kostant-Sekiguchi correspondence to a nilpotent coadjoint $G$-orbit $\mathcal{O}^G\subseteq\mathfrak{g}^*$. The known Schrödinger model of $π$ is a realization on $L^2(\mathcal{O})$, where $\mathcal{O}\subseteq\mathcal{O}^G$ is a Lagrangian submanifold. We construct an intertwining operator from the Schrödinger model to the new Fock model, the generalized Segal-Bargmann transform, which gives a geometric quantization of the Kostant-Sekiguchi correspondence (a notion invented by Hilgert, Kobayashi, Ørsted and the author). The main tool in our construction are multivariable $I$- and $K$-Bessel functions on Jordan algebras which appear in the measure of $\mathcal{O}^{K_{\mathbb{C}}}$, as reproducing kernel of the Fock space and as integral kernel of the Segal-Bargmann transform. As a corollary to our construction we also obtain the integral kernel of the unitary inversion operator in the Schrödinger model in terms of a multivariable $J$-Bessel function as well as explicit Whittaker vectors.

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arXiv2012

Heat kernel analysis for Bessel operators on symmetric cones

We investigate the heat equation corresponding to the Bessel operators on a symmetric cone $Ω=G/K$. These operators form a one-parameter family of elliptic self-adjoint second order differential operators and occur in the Lie algebra action of certain unitary highest weight representations. The heat kernel is explicitly given in terms of a multivariable $I$-Bessel function on $Ω$. Its corresponding heat kernel transform defines a continuous linear operator between $L^p$-spaces. The unitary image of the $L^2$-space under the heat kernel transform is characterized as a weighted Bergmann space on the complexification $G_{\mathbb C}/K_{\mathbb C}$ of $Ω$, the weight being expressed explicitly in terms of a multivariable $K$-Bessel function on $Ω$. Even in the special case of the symmetric cone $Ω=\mathbb{R}_+$ these results seem to be new.

math.AP↗

arXiv2012

Minimal representations of conformal groups and generalized Laguerre functions

We give a unified construction of the minimal representation of a finite cover $G$ of the conformal group of a (non necessarily euclidean) Jordan algebra $V$. This representation is realized on the $L^2$-space of the minimal orbit $\mathcal{O}$ of the structure group $L$ of $V$. We construct its corresponding $(\mathfrak{g},\mathfrak{k})$-module and show that it can be integrated to a unitary irreducible representation of $G$ on $L^2(\mathcal{O})$. In particular, we obtain a unified approach to the two most prominent minimal representations, namely the Segal--Shale--Weil representation of the metaplectic group $\textup{Mp}(n,\mathbb{R})$ and the minimal representation of $\textup{O}(p+1,q+1)$ which was recently studied by T. Kobayashi, G. Mano and B. Ørsted. In the second part we investigate special functions which give rise to $\mathfrak{k}$-finite vectors in the representation. Various properties of these special functions such as differential equations, recurrence relations and integral formulas connect to the representation theory involved. Finally, we define the conformal inversion operator $\mathcal{F}_{\mathcal{O}}$ by the action of the longest Weyl group element. $\mathcal{F}_{\mathcal{O}}$ is a unitary operator on $L^2(\mathcal{O})$ of order 2. We show that the action of $\mathcal{F}_{\mathcal{O}}$ on radial functions is given by a special case of Meijer's $G$-transform.

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arXiv2012

Minimal representations via Bessel operators

We construct an L^2-model of "very small" irreducible unitary representations of simple Lie groups G which, up to finite covering, occur as conformal groups Co(V) of simple Jordan algebras V. If $V$ is split and G is not of type A_n, then the representations are minimal in the sense that the annihilators are the Joseph ideals. Our construction allows the case where G does not admit minimal representations. In particular, applying to Jordan algebras of split rank one we obtain the entire complementary series representations of SO(n,1)_0. A distinguished feature of these representations in all cases is that they attain the minimum of the Gelfand--Kirillov dimensions among irreducible unitary representations. Our construction provides a unified way to realize the irreducible unitary representations of the Lie groups in question as Schroedinger models in L^2-spaces on Lagrangian submanifolds of the minimal real nilpotent coadjoint orbits. In this realization the Lie algebra representations are given explicitly by differential operators of order at most two, and the key new ingredient is a systematic use of specific second-order differential operators (Bessel operators) which are naturally defined in terms of the Jordan structure.

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arXiv2012

Fock model and Segal-Bargmann transform for minimal representations of Hermitian Lie groups

For any Hermitian Lie group G of tube type we construct a Fock model of its minimal representation. The Fock space is defined on the minimal nilpotent K_C-orbit X in p_C and the L^2-inner product involves a K-Bessel function as density. Here K is a maximal compact subgroup of G, and g_C=k_C+p_C is a complexified Cartan decomposition. In this realization the space of k-finite vectors consists of holomorphic polynomials on X. The reproducing kernel of the Fock space is calculated explicitly in terms of an I-Bessel function. We further find an explicit formula of a generalized Segal-Bargmann transform which intertwines the Schroedinger and Fock model. Its kernel involves the same I-Bessel function. Using the Segal--Bargmann transform we also determine the integral kernel of the unitary inversion operator in the Schroedinger model which is given by a J-Bessel function.

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arXiv2011

Special functions associated to a certain fourth order differential equation

We develop a theory of "special functions" associated to a certain fourth order differential operator $\mathcal{D}_{μ,ν}$ on $\mathbb{R}$ depending on two parameters $μ,ν$. For integers $μ,ν\geq-1$ with $μ+ν\in2\mathbb{N}_0$ this operator extends to a self-adjoint operator on $L^2(\mathbb{R}_+,x^{μ+ν+1}dx)$ with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, $L^2$-norms, integral representations and various recurrence relations. This fourth order differential operator $\mathcal{D}_{μ,ν}$ arises as the radial part of the Casimir action in the Schrödinger model of the minimal representation of the group $O(p,q)$, and our "special functions" give $K$-finite vectors.

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