arXiv subjects
Publications and source records attributed to Bernhard Krötz.
By the collective name of {\it lattice counting} we refer to a setup introduced in Duke-Rudnick-Sarnak that aim to establish a relationship between arithmetic and randomness in the context of affine symmetric spaces. In this paper we extend the geometric setup from symmetric to real spherical spaces and continue to develop the approach with harmonic analysis which was initiated in Duke-Rudnick-Sarnak.
We apply the local structure theorem and the polar decomposition to a real spherical space Z=G/H and control the volume growth on Z. We define the Harish-Chandra Schwartz space on Z. We give a geometric criterion to ensure $L^p$-integrability of matrix coefficients on Z.
Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$-variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$. We prove a local structure theorem for $Z$. In the simplest case where $Z$ is homogeneous, the theorem provides an isomorphism of the open $P$-orbit with a bundle $Q \times_L S$. Here $Q$ is a parabolic subgroup with Levi decomposition $LU$, and $S$ is a homogeneous space for a quotient $D=L/L_n$ of $L$, where $L_n$ is normal in $L$, such that $D$ is compact modulo center.
Let G be a real reductive group and Z=G/H a unimodular homogeneous G-space. The space Z is said to satisfy VAI if all smooth vectors in the Banach representations L^{p}(Z) vanish at infinity, 1 <=p. For H connected we show that Z satisfies VAI if and only if it is of reductive type.
Let G be a connected real semi-simple Lie group and H a closed connected subgroup. Let P be a minimal parabolic subgroup of G. It is shown that H has an open orbit on the flag manifold G/P if and only if it has finitely many orbits on G/P. This confirms a conjecture by T. Matsuki.
Let Z be an algebraic homogeneous space Z=G/H attached to real reductive Lie group G. We assume that Z is real spherical, i.e., minimal parabolic subgroups have open orbits on Z. For such spaces we investigate their large scale geometry and provide a polar decomposition. This is obtained from the existence of simple compactifications of Z which is established in this paper.
Let G be a real semi-simple Lie group and H a closed subgroup which admits an open orbit on the flag manifold of a minimal parabolic subgroup. Let V be a Harish-Chandra module. A sharp finite bound is given for the dimension of the space of H-fixed distribution vectors for V and a related subrepresentation theorem is derived. Extended final version. To appear in Trans. Amer. Math. Soc.
Let Z be a homogeneous space Z=G/H of a real reductive Lie group G with a reductive subgroup H. The investigation concerns the quantitative decay of matrix coefficients on $Z$ under the assumption that Z is of spherical type, that is, minimal parabolic subgroups have open orbits on Z.
An elementary and self-contained approach to smooth Frechet globalizations of Harish-Chandra modules is presented. We provide applications to the theory of Eisenstein series and Automatic Continuity.
A triple space is a homogeneous space $G/H$ where $G=G_0\times G_0\times G_0$ is a threefold product group and $H\simeq G_0$ the diagonal subgroup of $G$. This paper concerns the geometry of the triple spaces with $G_0=\SL(2,\R)$, $\SL(2,\C)$ or $\SO_e(n,1)$ for $n\ge 2$. We determine the abelian subgroups $A\subset G$ for which there is a polar decomposition $G=KAH$, and we determine for which minimal parabolic subgroups $P\subset G$, the orbit $PH$ is open in $G/H$.
Let G be a real reductive group and G/H a unimodular homogeneous G space with a closed connected subgroup H. We establish estimates for the invariant measure on G/H. Using these, we prove that all smooth vectors in the Banach representation L^p(G/H) of G are functions that vanish at infinity if and only if G/H is of reductive type. An application to lattice counting on G/H is presented.
A theory of holomorphic extension of eigenfunctions on homogeneous harmonic spaces is developed.
For every moderate growth representation of a real Lie group G on a Frechet space E, we prove a factorization theorem of Dixmier--Malliavin type for the space of analytic vectors E^ω. There exists a natural algebra of superexponentially decreasing analytic functions A(G), such that E^ω = A(G) * E^ω. As a corollary we obtain that E^ωcoincides with the space of analytic vectors for the Laplace--Beltrami operator on G.
Let Y=G/H be a semisimple symmetric space. It is shown that the smooth vectors for the regular representation of G on L^p(Y) vanish at infinity.