Reading seminar on discrete series and its geometric realizations
Student No.:50
Time:【Updated】Wed 10:00-11:30
Instructor:Hang Wang  [Tsinghua University]
Place:Conference room 1, floor 1
Starting Date:2012-10-17
Ending Date:2012-12-5



Time changes to14:00-15:30pm on Nov. 21. There is no seminar in the morning


Title: An introduction to trace formula 


Speaker: Kuok Fai Chao周國暉(AMSS)


Date: Nov. 21 (Wed.)


Time: 14:00-15:30pm


Place: Conference Room 1, Floor 1, Jin Chun Yuan West Building




In this talk, we would like to give an introduction to (Selberg) trace formula. Using such trace formula on the space L^2(X), where X is a locally symmetric space, we will get two different part of datum. One is called spectral side and other one is called geometric side. Time permitted, we will reveal some relations between index theory and Selberg trace formula. 


(This talk is part of the reading seminar "Discrete series and its geometric realizations". So, there is no seminar in the morning.)










The L^2-solution spaces of Dirac type operators were used to realize geometrically the discrete series representations for some Lie group G. The higher indices of such Dirac operators live in the K-theory of the group C*-algebra, which is related to the representation of G. The L^2-index (obtained from this higher indices) are related to some trace formulas. The aim of this seminar is to understand some details of the relationship among discrete series, Dirac operators and K-theory of group C*-algebras. The first month of the seminar will be preliminaries and introductions. The second month is to present ideas and details of some papers listed in the reference. Lectures on Oct 24 and Nov 7 will be given by Dr. Kuok Fai Chao from Chinese Academy of Sciences.




The following is a tentative schedule for the seminar:



Oct 17: Dirac operators for homogenous space of Lie group and for some locally symmetric spaces.

Oct 24: Noncommutative Fourier analysis and discrete series for semi-simple Lie group.

Oct 31: Group C*-algebras, higher index and L^2-index.

Nov 7: K-theory of group C*-algebra and representation.

Nov 14: L^2-index formula for homogeneous space of Lie group.

Nov 21: Trace formula.

Nov 28: L^2 index for some locally symmetric spaces.

Dec 5: Connes-Kasparov conjecture.





1. M. Atiyah and W. Schmid: A Geometric Construction of the Discrete Series for Semisimple Lie Groups. Invent. math. 42, 1-62 (1977).

2. D. Barbasch and H. Moscovici: $L^2$-index and the Selberg trace formula. Journal of functional analysis 53, 151-201 (1983).

3. A. Connes and H. Moscovici: $L^2$-index theorem for homogeneous spaces of Lie groups. Ann. Math. Vol. 115, No. 2, 291-330 (1982).

4. Harish-Chandra, Discrete series for semi-simple Lie groups, I any II, Acta. Math. 113 241-318 (1965) and 116 1-111 (1966).

5. H. Jacquet, K. F. Lai, and S. Rallis, A trace formula for symmetric spaces, Duke Math. J. 70 305-372 (1993).

5. V. Lafforgue: Banach KK-theory and the Baum-Connes conjecture. ICM 2002. Vol. II 795-812.

6. R. Parthasarathy: Dirac operators and the discrete series. Ann. Math. Vol. 96, No. 1, 1-30 (1972). 

7. Plymen: The reduced $C^*$-algebra of reductive group I, II. Journal of functional analysis.

8. M. Stern: $L^2$-index theorems on locally symmetric spaces. Invent. math. 96, 231-282 (1989).


Title: Background on representation of Lie group (I)


Speaker: Kuok Fai Chao周國暉 (AMSS)

Date: Oct. 24 (Wed.)


Time: 8:30-10:00 am


Place: Conference Room 1, Floor 1, Jin Chun Yuan West Building




For the propose of supporting this seminar, we would like to give an introduction of representation theory of semisimple Lie group. It includes noncommutative Fourier analysis, Placncherel measure and etc. The main reference of the talks is R. Plymen 's note and A. Knapp 's book "Representation of semisimple group".