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## Generalizations of 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 .
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.
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.
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)
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
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.
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.
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.
Radial part
calculations for affine sl2 and the Heun-KZB heat
equation.
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.
The local structure
theorem for real spherical varieties
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.
Cusp forms for
reductive symmetric spaces. For a real
reductive Lie group with the discrete
part of the spectral decomposition of the space of square integrable functions on
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.
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.
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.
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.
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.
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.
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.
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. [Back] |