Searcharxiv⌕ Search

arXiv subjects

Robert J. Stanton

Publications and source records attributed to Robert J. Stanton.

7 recordsLinked to original sources
arXiv2021

Holomorphic torsion and geometric zeta functions for certain Hermitian locally symmetric manifolds

We give a dynamical description, in terms of a Weil-type zeta function, to the holomorphic torsion with coefficients for certain compact Hermitian locally symmetric manifolds, whose connected group G of isometries of the universal cover has only one conjugacy class of cuspidal maximal parabolic subgroup and satisfies a technical Ansatz relative to the given coefficients. A distinguishing feature of our zeta function is that its construction involves in an essential way the geometry of a standard compactification of the universal cover. The two senior authors are indebted to their junior colleague, Jan Frahm, for his laborious work shedding light on the scope of the validity of the Ansatz, and for writing up the attached Appendix. The results therein show that for real rank one groups G the Ansatz is satisfied with respect to any coefficients, for some rank two groups G it is satisfied with respect to certain coefficients, and also that there are groups G which do not obey the Ansatz.

math.RT↗
arXiv2020

Extensions of real bounded symmetric domains

For a real bounded symmetric domain, G/K, we construct various natural enlargements to which several aspects of harmonic analysis on G/K and G have extensions. Our starting point is the realization of G/K as a totally real submanifold in a bounded domain G_h/K_h. We describe the boundary orbits and relate them to the boundary orbits of G_h/K_h. We relate the crown and the split-holomorphic crown of G/K to the crown Ξ_h of G_h/K_h. We identify an extension of a representation of K to a larger group L_c and use that to extend sections of vector bundles over the Borel compactification of G/K to its closure. Also, we show there is an analytic extension of K-finite matrix coefficients of G to a specific Matsuki cycle space.

math.RT↗
arXiv2012

The geometry of special symplectic representations

We show there is a class of symplectic Lie algebra representations over any field of characteristic not 2 or 3 that have many of the exceptional algebraic and geometric properties of both symmetric three forms in two dimensions and alternating three forms in six dimensions. All nonzero orbits are coisotropic and the covariants satisfy relations generalising classical identities of Eisenstein and Mathews. The main algebraic result is that suitably generic elements of these representation spaces can be uniquely written as the sum of two elements of a naturally defined Lagrangian subvariety. We give universal explicit formulae for the summands and show how they lead to the existence of geometric structure on appropriate subsets of the representation space. Over the reals this structure reduces to either a conic, special pseudo-K\" ahler metric or a conic, special para-K\" ahler metric.

math.RT↗
arXiv2009

The special symplectic structure of binary cubics

Let $k$ be a field of characteristic not 2 or 3. Let $V$ be the $k$-space of binary cubic polynomials. The natural symplectic structure on $k^2$ promotes to a symplectic structure $ω$ on $V$ and from the natural symplectic action of $\textrm{Sl}(2,k)$ one obtains the symplectic module $(V,ω)$. We give a complete analysis of this symplectic module from the point of view of the associated moment map, its norm square $Q$ (essentially the classical discriminant) and the symplectic gradient of $Q$. Among the results are a symplectic derivation of the Cardano-Tartaglia formulas for the roots of a cubic, detailed parameters for all $\textrm{Sl}(2,k)$ and $\textrm{Gl}(2,k)$-orbits, in particular identifying a group structure on the set of $\textrm{Sl}(2,k)$-orbits of fixed nonzero discriminant, and a purely symplectic generalization of the classical Eisenstein syzygy for the covariants of a binary cubic. Such fine symplectic analysis is due to the special symplectic nature inherited from the ambient exceptional Lie algebra $\mathfrak G_2$.

math.SG↗
arXiv2006

Holomorphic extension of representations: (I) automorphic functions

Let G be a connected, real, semisimple Lie group contained in its complexification G_C, and let K be a maximal compact subgroup of G. We construct a K_C-G double coset domain in G_C, and we show that the action of G on the K-finite vectors of any irreducible unitary representation of G has a holomorphic extension to this domain. For the resultant holomorphic extension of K-finite matrix coefficients we obtain estimates of the singularities at the boundary, as well as majorant/minorant estimates along the boundary. We obtain L^\infty bounds on holomorphically extended automorphic functions on G/K in terms of Sobolev norms, and we use these to estimate the Fourier coefficients of combinations of automorphic functions in a number of cases, e.g. of triple products of Maass forms.

math.RT↗