Lorentz Center - Generalizations of Symmetric Spaces from 25 Nov 2013 through 29 Nov 2013
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    Generalizations of Symmetric Spaces
    from 25 Nov 2013 through 29 Nov 2013

 

Erik van den Ban

Convexity and holomorphy of Eisenstein integrals for semisimple symmetric spaces

 

Let X = G/H be a semisimple symmetric space.

Eisenstein integrals are functions on X which are essentially matrix coefficients of K-finite vectors and H-fixed distribution vectors of (generalized) principal series

appearing in the Plancherel decomposition of X. These principal series representations are constructed by means of induction from parabolic subgroups.

The use of different types of parabolic subgroups leads to different normalisations of Eisenstein integrals. In the case of the group G ~ G \times G / diagonal(G) the use of parabolic subgroups of the form P x P leads to (Harish-Chandra's) Eisenstein integrals depending holomorphically on the continuous spectral parameter, whereas the use of parabolic subgroups of the form P x \bar{P} leads to differently normalized Eisenstein integrals with singularities in that parameter (joint work with Job Kuit). We will relate this issue of holomorphy to a new convexity theorem (joint work with Dana Balibanu) for semisimple symmetric spaces .

 

 

Ralph Bremigan

Group actions and (hyper)k\"ahler structures

 

Since at least the late 1970s, moment-map techniques have been used to study actions of complex reductive groups (e.g., linear and projective representations), obtaining stratifications, and relating the collection of orbits in the original space to a ``critical'' sub-collection of orbits by a maximal compact subgroup.  Recent work of H. Heinzner, G. Schwarz, and H. St\"otzel has extended these approaches to actions of real reductive groups; interesting examples include real projective representations and actions of real groups on flag manifolds.  Further examples of interest can be obtained in certain hyperk\"ahler settings, where a 2-sphere of related actions can be studied.  These settings include complex (co)adjoint orbits and cotangent bundles to flag manifolds.  Issues involving computability of the hyperk\"ahler structures arise, however. 

 

 

Mahir Bilen Can

A new class of spherical varieties and their Borel orbits

 

A J-irreducible monoid is the Zariski closure in End(V) of the image of an irreducible

representation of an algebraic group. In this talk, using the theory of J-irreducible monoids, we

1. introduce a new class of spherical embeddings of (complex algebraic) symmetric spaces,

2. characterize the parametrizing sets of their Borel orbits,

3. study an example in detail to test our theory.

This talk is based on our joint work with Roger Howe and Lex Renner.

 

 

Patrick Delorme

Harmonic analysis on reductive p-adic reductive symmetric spaces.

 

We prove that certain neighborhoods at infinity of a symmetric space identify to ones of parabolic  degeneration up to the action of small compact open subgroups.

This is done mainly using asymptotic properties of Eisenstein integrals established some years ago by Nathalie Lagier.

These identifications allow  to use the techniques due to Yiannis Sakellaridis and Akshay Venkatesh for the Plancherel formula of spherical varieties when the group is split and the field is of characteristic zero.

This leads to the Plancherel formula for general symmetric spaces (the group is not necesseraly split and the characteristic of the field is simply different from 2)

 

 

Walter Freyn
S-representations and twin buildings for hyperbolic Kac-Moody algebras

 

In this talk, I will review the geometry of S-representations of hyperbolic Kac-Moody algebras and prove embedding results for the twin building. These results describe the local structure of (conjectured) hyperbolic Kac-Moody symmetric spaces.

 

 

Tobias Hartnick

An introduction to Hermitian higher Teichmüller theory

 

Given a symmetric space X and a group \Gamma one can study the geometry of the moduli space M(X, \Gamma) of all locally symmetric spaces with universal cover X and fundamental group \Gamma. For \Gamma a surface group and X the hyperbolic plane this is the content of classical Teichmüller theory. In recent years there have been many attempts to form "higher Teichmüller theories" by studying M(X, \Gamma) for various classes of higher rank symmetric spaces X and various classes of groups \Gamma. In these higher rank situations rigidity rules out interesting moduli spaces for lattices, but one can still study deformation spaces of smaller (typically Gromov-hyperbolic) groups of infinite covolume.

 

In this talk we focus on the case where \Gamma is a surface group and X is Hermitian, which is among the best understood cases. We will discuss various geometric and dynamical conditions which ensure that an action of \Gamma on X is free and properly discontinuous. Among these are maximality of certain characteristic numbers, Anosov properties of certain associated flows, regularity of boundary maps and positivity properties with respect to the causal geometry of the associated Shilov boundary. Some of these conditions admit a cohomological reformulation in terms of bounded cohomology. We will indicate how these cohomological conditions can be used to establish properties of the representations in question. This is based on joint work with Gabi Ben Simon, Marc Burger, Alessandra Iozzi and Anna Wienhard.

 

 

Joachim Hilgert

Radon transforms for limits of symmetric spaces

 

In this talk we explain some new results with Gestur Ólafsson on the possibility to define and invert horocyclic Radon transforms on inductive limits of Riemannian symmetric spaces of non-compact type.

 

 

Max Horn

Spin covers of "maximal compact" subgroups of real Kac-Moody groups

 

Let G be real Kac-Moody group, \theta a Cartan-Chevalley involution of G and K the subgroup of G consisting of the elements fixed by \theta.

For example, if G=SL(n) then K=SO(n). It is well-known that SO(n) admits a two-fold cover, the spin group Spin(n). We describe how to construct such spin covers for "maximal compact" subgroup K for arbitrary simply laced Kac-Moody groups. Finally, we briefly discuss how to extend this to more general diagrams.

 

 

Stefan Kolb

Radial part calculations for affine sl2 and the Heun-KZB heat equation.


In their seminal work in the 70s Olshanetsky and Perelomov used radial part calculations for symmetric spaces to prove integrability of the Calogero-Moser Hamiltonian for special parameters. In this talk we will extend their argument to affine sl2 with the Chevalley involution. The resulting operator is identified with a blend of the Inozemtsev Hamiltonian and the KZB-heat equation in dimension one. The corresponding zonal spherical functions give rise to symmetric theta functions.

 

 

Linus Kramer

The classification of compact buildings and related geometries

 

In my talk I will explain the various classification results on compact  spherical buildings, and also the recent classification of compact building-like geometries. If time permits, I will also mention some applications.

 

 

Bernhard Kroetz

The local structure theorem for real spherical varieties

(joint work with Friedrich Knop and Henrik Schlichtkrull) 

 

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 x_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.

 

 

Job Kuit

Cusp forms for reductive symmetric spaces.
(joint work with Erik van den Ban and Henrik Schlichtkrull)

 

For a real reductive Lie group G, there exists a notion of cusp form, which was introduced by Harish-Chandra. He showed that the space of cusp forms coincides

with the discrete part of the spectral decomposition of the space of square integrable

functions on G. The class of real reductive symmetric spaces contains the real reductive Lie groups. It would be interesting to have a notion of cusp form for this class of spaces, but the generalization of Harish-Chandra’s definition turns out to be somewhat problematic due to the fact that certain integrals are divergent. Some years ago, Flensted-Jensen suggested a definition, that led to the ongoing work with Erik van den Ban and Henrik Schlichtkrull. In this talk I will discuss the problems with convergence of the integrals and propose a modification of Flensted-Jensen’s idea that gives a solution for symmetric spaces of split rank 1.

 

 

Bernhard Mühlherr

Bruhat-Tits buildings of exceptional type $\tilde{C}_2$

 

A Bruhat-Tits building is an affine building whose building at infinity is a Moufang building.

Examples arise from isotropic absolutely simple algebraic groups over complete fields with respect to a discrete valuation.

 

In my talk I will report on joint work with H. Petersson and R. Weiss on Bruhat-Tits buildings related to algebraic groups of exceptional type $F_4, E_6,E_7$ and $E_8$. We determine the local structure of these buildings by seeing them as set of fixed points of a Galois group acting on  a pseudo-split Bruhat-Tits building. 

 

 

Karl-Hermann Neeb

Beyond affine Kac--Moody groups: Loop groups with infinite dimensional

targets and unitary representations.

 

The key point in the classical theory of loop groups is the invariant scalar product on the Lie algebra. Therefore it is natural to exploit the extent to which this theory carries over to the situation where the compact target Lie algebra is replaced by a real Hilbert-Lie algebra with an Ad-invariant scalar product. A typical example is the group of unitary operators g for which g -1 is Hilbert Schmidt. In this talk we explain the classification of the 'irreducible' groups showing up in this context and their unitary representations generalizing the highest weight representations of doubly extended loop groups.

 

 

Takaaki Nomura

Homogeneous convex cones and basic relative invariants

 

The role played by principal minors is significant in analysis on the Riemannian symmetric space of positive-definite real symmetric matrices.

Basic relative invariants associated to homogeneous convex cones are generalizations of the principal minors obtained by focusing on the relative invariance. Starting with the basic theorem due to Ishi, I present several results on the basic relative invariants including the latest ones of Nakashima and Yamasaki in addition to various examples.

 

Bertrand Rémy
A survey on compactifications of buildings

We will review various techniques to construct compactifications of Bruhat-Tits buildings, from the use of the compact (Chabauty) space of closed subgroups in a given locally compact group, to the more sophisticated use of (Berkovich) analytic geometry over non-Archimedean valued fields. If time allows, we will mention some research projects.

 

Guy Rousseau
Hecke algebras associated to affine hovels

To an almost split Kac-Moody group G over a non archimedean field K, one may associate an affine hovel on which G acts strongly transitively; it is a generalization of the Bruhat-Tits building associated to a reductive group G over K. Starting with such a strongly transitive action on an affine hovel, one may build a spherical Hecke algebra and a Iwahori-Hecke algebra. They have properties analogous to the algebras associated to a reductive group over K, with some differences.

 

Gerald Schwarz:
Lifting automorphisms of adjoint representations

Let $\mathfrak g$ be a simple complex Lie algebra and let $G$ be the corresponding adjoint group. Consider the $G-module $V$ which is the direct sum of $r$ copies of $\mathfrak g$. We say that $V$ is \emph{large\/} if $r\geq 2$ and  $r\geq 3$ if $G$ has rank 1. We showed that when $V$ is large any algebraic automorphism $\psi$ of the quotient $Z:=V//G$ lifts to an algebraic  mapping $\Psi\colon V\to V$ which sends the fiber over $z$ to the fiber over $\psi(z)$, $z\in Z$. We also showed that one can choose a biholomorphic lift $\Psi$ such that $\Psi(gv)=\sigma(g)\Psi(v)$, $g\in G$, $v\in V$, where $\sigma$ is an automorphism of $G$. This leaves open the following questions: Can one lift holomorphic automorphisms of $Z$? Which automorphisms lift if  $V$ is not large? We answer the first question in the affirmative and also answer the second question.

 

Petra Schwer
Generalizing Kostant convexity to Euclidean buildings

Euclidean buildings are the analogues of symmetric spaces for semisimple groups defined over local fields. In this talk I will explain the generalization of Kostant's convexity theorem (for symmetric spaces) to thick discrete as well as non-discrete Euclidean buildings.

Kostant showed that the image of a certain orbit of a point x in a symmetric space under a projection onto a maximal flat is a Weyl group invariant convex set. The methods used include combinatorics of buildings, some  metric geometry as well as a character formula for highest weight representations of algebraic groups.

Maarten Solleveld:
Bruhat-Tits buildings and representations of reductive p-adic groups

Let V be an admissible complex representation of a reductive p-adic group G.Harish-Chandra proved that the trace of V is a well-defined, locally constant function on the set of regular semisimple elements in G.
We will discuss an alternative proof of this result, which also works for representations over other coefficient fields. An important role is played by the action of G on its Bruhat-Tits building, in particular by the fixed points of semisimple elements.



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