Locally Symmetric Spaces (08w5056)

Objectives

In September 2003, J. Rohlfs and B. Speh organized a very successful
workshop on locally symmetric spaces at Oberwolfach. The current workshop
is intended to be a sequel to the Oberwolfach workshop with a greater range
of topics. One goal of having this workshop at BIRS with an expanded list
of organizers is to attract a more diverse group of mathematicians and in
particular to encourage the attendance at the workshop of younger North
American mathematicians.

Locally symmetric spaces are important in geometry, analysis, and number
theory, and their study uses techniques from all these areas. We envision
that the talks at the workshop will likewise cover such a wide range of
topics and techniques. There has been substantial progress within each of
these fields and some interactions between mathematicians from these
different areas. Another goal of this workshop is to bring together
specialists with different backgrounds in order to facilitate and increase
these interactions, as well as to stimulate joint research.

The main topics to be considered are:

- Geometry of compactifications of locally symmetric spaces

- Differential geometric and topological properties and invariants

- Applications of locally symmetric spaces to arithmetic problems

- Spectral geometry and analysis on locally symmetric spaces

Several developments in different fields over the past few years make this
an opportune time to have another workshop. We describe some of them. To
fix notation, we let $G$ be a semisimple connected algebraic group defined
over $mathbb Q$, $G(mathbb R)$ the set of real points, and $K$ a maximal
compact subgroup of $G(mathbb R)$. Then $X = G(mathbb R)/K$ is a
symmetric space. Let $Gamma subset G(mathbb Q)$ be a torsion free
subgroup of finite index. Then $Gamma backslash X$ is a locally symmetric
space.

In many cases of interest, $Gamma backslash X$ is noncompact and various
compactifications (often singular) have been introduced to address
different problems in geometry and number theory. The construction and
relationship between various compactifications has become more clear
through the work of several mathematicians, particularly Borel, Goresky,
Ji, and Zucker.

Various types of cohomology associated to these compactifications are
important in number theory, in particular Langlands's program. One goal
here is to show that the Hasse-Weil zeta function of the Baily-Borel
compactification (which is an algebraic variety defined over a number
field) may be expressed as the product of automorphic $L$-functions.
Towards this goal, the relation between the $L^2$-cohomology and the
intersection cohomology of the Baily-Borel compactification was conjectured
by Zucker and was resolved by Looijenga and Saper-Stern. One then needs to
restrict intersection cohomology to a stratum of the Baily-Borel
compactification and calculate the local trace of Frobenius and Hecke
operators on it; in some cases this is considered in the work of Morel in
the 'etale setting.

In order to avoid difficulties associated with the singularities of the
Baily-Borel compactification, Goresky-Harder-MacPherson have established a
relation between its intersection cohomology and the weighted cohomology of
the (less singular) reductive Borel-Serre compactification. This was used
by Goresky-MacPherson to prove a topological trace formula for Hecke
operators. Independently an analogous relation between the intersection
cohomology of the Baily-Borel compactification and the intersection
cohomology of the reductive Borel-Serre compactification was conjectured by
Rapoport and later established by Saper. More recently the relation
between the space of $L^2$-harmonic forms and the intersection cohomology
of the reductive Borel-Serre compactification was determined by Saper in a
general context. The presence of the reductive Borel-Serre compactification
in recent work shows its ubiquity.

Other connections to number theory also play an important role in the study
of locally symmetric spaces. This can be seen for example in the work of
Ash, Bruinier, Clozel, Emerton, Hanamura, Harder, Kudla, Mahnkopf,
Rapoport, Rohlfs, Schwermer and Speh. Conjectures of Langlands, Tate,
Beilinson and others are the driving force behind these developments which
involve Hecke eigen-functions, $L$-functions, special values of
$L$-functions, special points of varieties, modular symbols, mixed motives,
Chow groups, and $K$-theory.

A topological problem concerning locally symmetric spaces which has
received a fair bit of attention is the problem to determine if a cycle
class of a locally symmetric subvariety is a nontrivial cohomology class;
see for example the work of Bergeron, Clozel, Rohlfs-Speh,
Speh-Venkataramana, and Venkataramana. This problem is related on the one
hand to the difficult problem of determining the restriction to a
semisimple subgroup $H$ of an irreducible representations of $G$ as in the
work of Kobayashi-Oda and on the other hand to the arithmetic problem of
period integrals with respect to $H$ of automorphic representations where
special values of $L$-functions also play a role. Another interesting and
important problem, where similar techniques are useful, are nonvanishing
results for cup products of cohomology classes such as those of Bergeron
and Venkataramana.

Important geometric invariants of locally symmetric spaces which have been
considered are the analytic torsion and the related length spectrum of
closed geodesics, as well as the special values of the geometric theta
functions; in particular we note the work of Deitmar, Juhl, and
Rohlfs-Speh. In this context, invariants of non-arithmetic subgroups are
also of great interest as in the work of Bunke, Olbricht, and Leuzinger.

One of the main problems in analysis on locally symmetric spaces is the
study of the Laplace operator on the space $Gamma backslash X$. A very
well-known spectral problem is to obtain a lower bound on the spectrum of
the Laplacian. For $GL_n$, the Ramanujan conjecture is equivalent to such
a bound. For $n=2$, the best bound is due to Shahidi and Kim. In the
general case, substantial work has been done by P. Sarnak and his
collaborators. The techniques used here are $L$-functions and the lifting
of automorphic representations.

Another interesting analytic problem is to understand scattering theory on
symmetric spaces and locally symmetric spaces. This is the subject of
on-going work by Mazzeo-Vasy which uses micro-local analysis and
compactification theory.

A final source of interesting analytic problems is connected to the
Arthur-Selberg trace formula. Particularly notable here is the recent work
of W. M"uller on the spectral side of the Arthur-Selberg trace formula and
his results about Weyl's law for the cuspidal spectrum.

The study of measures on $Gamma backslash X$ and $Gamma backslash G$
which are invariant under subgroups $H$ of $G$ was begun with the work of
M. Ratner and recently culminated in the unique quantum ergodicity theorem
of E. Lindenstrauss.

We expect advances in all the above mentioned areas during the next few
years. A goal of the workshop will give an opportunity for young
researchers to learn more about the different aspects of the field, the
different methodologies, and the many open problems. Another goal will be
to stimulate and capitalize on new developments through interactions of
researchers from different areas. For example, a new emerging idea is the
use of methods from ergodic theory to attack analytic problems on locally
symmetric spaces. For another example, a breakthrough concerning the
fundamental lemma would lead to significant applications of the Selberg
trace formula.