Summer School: Microlocal and Geometric
Methods in Representation Theory
Schloss Reisensburg, July 17 –28, 2006
July 13, 2006

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Preliminary Program

CoursesB. Keller: Cluster algebras and quiver representations
T. Kobayashi: Unitary representations, restrictions, and their applications
M. Zworski: Quantum monodromy operator and its applications

Complementary Lectures
The “Complementary Lectures” will be given by the staff of the Research Training Group Metz-Paderborn. Since they in part are meant to help the participants to follow the three courses, the topics of these lectures in general have notbeen fixed so far. Here is one example:
J. Hansen: t.b.a.
This lecture will contain some background material relevant in Zworski’s course.

J. Hilgert: Characterizing Smooth and Analytic Functions on Compact LieGroupsThis lecture will contain some background material used in Kobayashi’s course.

A. Pasquale The Plancherel Formula for Real Hyperboloids
T. Wurzbacher Normal Forms and Moment Maps in Symplectic and KählerGeometry

Research Lectures
The “Research Lectures” will be given by participants who want to present someof the research. Here are the topics announced so far:P. Clare: Hilbert C*-modules and intertwining operators
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T. Hartnick: Flag varieties of infinite-dimensional groups - an axiomatic approach
G. Morando: Tempered holomorphic solutions of D-modules on complex curves
P. Schützdeller: Convexity properties of moment maps of real forms actingon Kählerian manifolds
G. Williamson: Kazhdan-Lusztig Polynomials and Soergel’s Bimodules
S. Wolf: The composition monoid and the composition algebra of a cyclic quiver.

ScheduleFirst week
9:15 - 10:45
11:15 - 12:45
15:00 - 16:00
16:15 - 17:15
17:30 - 18:30

Mon
Kobayashi
Zworski
Kobayashi
C (Hilgert)
R

Tue
Kobayashi
Zworski
Kobayashi
C (Hansen)
R

Wed
C (Pasquale)
ZworskiCC (Wurzbacher)
RThu
Kobayashi
C
Kobayashi
C (Wurzbacher)
R

Fri
Kobayashi
Zworski
Kobayashi
C
R

Second week
9:15 - 10:45
11:15 - 12:45
15:00 - 16:00
16:15 - 17:15
17:30 - 18:30

Mon
KellerZworski
KellerC
R

Tue
KellerC
KellerC
R

Wed
KellerZworski
KellerC (Bierstedt)
R

Thu
KellerC
KellerCR

Fri
KellerZworskiHere “C” means “Complementary Lecture” and “R” means “Research Lecture”.
The program will will be finalized during the first week.

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List of Participants (including lecturers)
1) Alldridge, Alexander (Paderborn; alldridg@math.uni-paderborn.de)2) Bäcklund, Pierre (Uppsala; pierre@math.uu.se)3) Balleier, Carsten (Paderborn; balleier@math.upb.de)4) Benameur, Moulay-Tahar (Metz; benameur@math.univ-metz.fr)5) Bierstedt, Klaus Dieter (Paderborn; klausd@uni-paderborn.de)6) Bohr, Mathieu (Metz, bohr@univ-metz.fr)7) Clare, Pierre (Cachan, pierre.clare@gmail.com)
28) Dichev, Nikolay (Paderborn, dichev@math.upb.de)9) Dicu, Camelia (Cluj-Napoca, cdicu@math.ubbcluj.ro)10) Dostert, Mike (Metz, dostert@math.univ-metz.fr)11) Fuchssteiner, Martin (Darmstadt, fuchssteiner@mathematik.tu-darmstadt.de)12) Graeff, Thomas (Magdeburg, graeff@mathematik.uni-marburg.de)13) Greb, Daniel (Bochum, daniel.greb@ruhr-uni-bochum.de)14) Hansen, Sönke (Paderborn, soenke@math.upb.de)15) Hartnick, Tobias (Darmstadt, tobias.hartnick@gmx.de)16) Hilgert, Joachim (Paderborn, hilgert@math.uni-paderborn.de)17) Hitzelsberger, Petra (Darmstadt, hitzelberger@mathematik.tu-darmstadt.de)18) Johansen, Troels (Paderborn, johansen@math.upb.de)19) Keliny, Sameh (Paderborn, sameh@math.uni-paderborn.de)20) Keller, Bernhard (Paris, keller@math.jussieu.fr)21) Kobayashi, Toshiyuki (Kyoto, toshi@kurims.kyoto-u.ac.jp)22) Krause, Henning (Paderborn, hkrause@math.uni-paderborn.de)23) Kurtdere, Ayse (Bochum, kurtdere@gmx.de)24) Magata, Frederik (Darmstadt, magata@mathematik.tu-darmstadt.de)25) Miebach, Christian (Bochum, christian@cplx.rub.de)26) Merigon, Stephane (Nancy, smerigon@yahoo.fr)27) Moeller, Niels Martin (Aarhus, nmm@imf.au.dk)28) Morando, Giovanni (Paris, morando@math.jussieu.fr)29) Pasquale, Angela (Metz, pasquale@math.univ-metz.fr)30) Peter, Matthias (Paderborn, mape@uni-paderborn.de)31) Pohl, Anke (Paderborn, pohl@math.upb.de)32) Rilke, Florian (Paderborn, rilke@math.upb.de)33) Schnürer, Olaf (Freiburg, schnuerer@pcpool00.mathematik.uni-freiburg.de)34) Schützdeller, Patrick (Bochum, patrick@cplx.rub.de)35) Stötzel, Hendrick (Bochum, henrik.stoetzel@rub.de)336) van der Noort, Vincent (Utrecht, noort@math.uu.nl)37) Wagner, Stefan (Darmstadt, Leonard-euler@gmx.de)38) Webster, Benjamin (Berkeley, bwebste@math.berkeley.edu)
39) Wegener Sven-Ake (Paderborn, wegner@math.upb.de)40) Williamson, Geordie (Freiburg, geordie.williamson@math.uni-freiburg.de)41) Wolf, Stefan (Paderborn, swolf@math.upb.de)42) Wolf, Elke (Paderborn, lichte@math.uni-paderborn.de)43) Wurzbacher, Tilmann (Metz, wurzbacher@math.univ-metz.fr)44) Zworski, Maciej (Berkeley, zworski@Math.Berkeley.EDU)

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Abstracts for Research Talks

T. Hartnick: Flag varieties of infinite-dimensional groups - an axiomatic approach
The interplay between groups and geometries is a classical topic ever since thework of Felix Klein. It is a remarkable result by Tits that those geometries
corresponding to groups of Lie type can be constructed and characterized in auniform way - they are so-called spherical buildings. This classical result has
many generalizstions: E.g. the spherical buildings associated with Lie groups
can be characterized by additional topological data. Another recent generalization is the correspondence between so-called twin buildings and certain infinitedimensional groups called ”groups of Kac-Moody type”. Combining these two
approaches one may ask for a theory of ”topological twin buildings”, whichshould then corresond to certain topological groups of Kac-Moody type. The
purpose of this talk is to survey some of the classical results, to propose a notionof topological twin buildings and to sketch a structure theory of these buildings
and their associated groups.

G. Morando: Tempered holomorphic solutions of D-modules on complex curves
First I will introduce my research interests which concern sheaves on sites andD-modules. The former topic has been recently widely developed by M. Kashiwara and P. Schapira (Asterisque 271-284), they realized tempered distributions,tempered holomorphic functions and Whitney C-infinity functions as sheaves on
the subanalytic site. Such a work led to the use of such functional spaces withgrowth conditions in the functorial study of partial differential equations i.e. thestudy of D-modules. In the second part of the talk I will introduce some recentresults I obtained studying tempered holomorphic solutions of D-modules oncomplex curves.

4P. Schützdeller: Convexity properties of moment maps of real forms acting on Kählerianmanifolds
Let U be a compact connected Lie group and let U C be the complexification ofU . Further let X be a compact connected Kähler manifold with a holomorphicU C -action such that the induced U -action is Hamiltonian, i. e the U -action fixes
the symplectic form ω and there exists a moment map µ : X → u∗ . By aclassical convexity result of Kirwan the intersection of the image of µ with apositive Weyl chamber t∗+ in a maximal torus t of u is a convex polytope.
We discuss a generalization of this statement to the following situation. Let G bea real closed subgroup of U C , which is compatible with the Cartan decomposition
U C = U · exp(iu) of U C , i. e. G = K · exp(p) where K is a subgroup of U and p
is a K-stable subspace of iu. Given such a subgroup G we get a K-equivariant
map µip : X → ip∗ defined by µip (x) = µ(x)|ip∗ which is the right analog to themoment map µ for the action of G. Let a ⊂ p be a maximal subalgebra which isautomatically abelian and let a+ be the positive Weyl chamber in a. Then the setia∗+ is a slice for the K-action on ip∗ . We are interested in convexity propertiesof the set µip (Y ) ∩ ia∗+ for closed G-stable subsets Y of X. In particular, we geta convexity result in the following situation.
Let V be a finite dimensional complex U C -representation space. Fixing a U invariant Hermitian inner product on V we get a moment map µP(V ) : P(V ) → u∗
for the induced U -action on the projective space P(V ) and the following holds.
Theorem. Let Y be a closed connected G-stable irreducible semialgebraic subsetof P(V ). Then the set µip (Y ) intersects the positive Weyl chamber ia∗+ in aconvex polytope.

S. Wolf: The composition monoid and the composition algebra of a cyclic quiver.
Reineke first introduced the composition monoid as a tool for understanding
extensions between representations of the Dynkin quivers. He later extended
the definition to include all quivers by considering extensions between families ofrepresentations. Here I consider the special case of the nilpotent representationsof a cyclic quiver and exhibit an isomorphism between the composition monoid
and the specialization of the generic composition algebra at q = 0.

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Additional Information• Reisensburg runs an “Internet-Café”. There, it is also possible to print
files. Moreover, internet LAN-connections are available in each room(bring your laptop). The public part of Schloss Reisensburg is covered
by WLAN, which also extends to part, but not all, of the rooms.
• On Saturday, July 22, we will organize an excursion. On Sunday, July 23,
no official activity is planned.
• The email address of the local manager, Mrs. Claudia Ulm, at Reisensburg
is claudia.ulm@uni-ulm.de. Her phone number is ++49-731-50-38000.
5• When you arrive at the train station of Günzburg, you may have to take ataxi to Reisensburg (there are no buses on Sunday). In particular, if you
travel in a larger group you may want to order a van. Here is the phone
number of the taxi company suggested by Mrs. Ulm: ++49-8221-5584.
• There are a grand piano and several music rooms in Schloss Reisensburg- bring your instrument and sheet music.

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