Dirac operators and representation theory

Grant no. 4176 funded by the Croatian Science Foundation.

Summary of the project:

Dirac operators were introduced into representation theory by Parthasarathy in the 1970s. In the 1990s, Vogan conjectured a strengthening of Parthasarathy's Dirac inequality, which relates the cohomology of the Dirac operator to the infinitesimal character of the representation. This conjecture was proved in a joint paper with Huang . In further research, in part joint with various collaborators (Barbasch, Huang, Kang, Mehdi, Protsak, Renard, Somberg, Zhu), we related Dirac cohomology to other kinds of cohomology, calculated it explicitly for many interesting representations, and studied algebraic Dirac induction, a construction of representations with given Dirac cohomology. The proposed project will continue this research in several directions:

 

1)     the study of representations obtained by Dirac induction, in particular the discrete series representations (doctoral and postdoctoral research of Ana Prlić).

2)     applying Dirac cohomology to problems of restricting representations. Some instances of this have been studied with Huang-Zhu and with Somberg.

3)     trying to strengthen the Dirac inequality, which should have implications for the study of unitary representations (doctoral research of Karmen Grizelj). Some preliminary work on  the convex hull conjecture of Salamanca-Vogan has been done with Renard.

4)     calculating the Dirac cohomology of  unipotent representations. Some cases of this have been obtained jointly with Barbasch. This could lead to better understanding of characters of unipotent representations, which is related to the Assem conjecture. It would also be interesting to calculate the Dirac cohomology of unitary highest weight modules.

 

Other topics we plan to cover include studying translation principle for Dirac index (joint with Mehdi and Vogan); studying n-cohomology and BGG resolutions using Penrose transforms (joint with Souček; doctoral research of Rafael Mrđen);  studying some abstract homotopy and homology theory, with applications to tensor categories of representations (doctoral and postdoctoral research of Matija Bašić); study of twisted Dirac index and twisted characters (joint with Barbasch and Trapa).

 

Collaborators:

Pavle Pandžić, project leader

Mladen Božičević

Hrvoje Kraljević

Matija Bašić

Ksenija Kitanov

Rafael Mrđen

Ana Prlić

Karmen Grizelj

Josip Grgurić

Research topics of the project are regularly presented and discussed at Representation Theory Seminar at Department of Mathematics, University of Zagreb.

Papers:

1.     M. Bašić, T. Nikolaus, Dendroidal sets as models for connective spectra, J. K-Theory 14 (2014), no. 3, 387-421.

2.     D. Barbasch, P. Pandžić, Dirac cohomology of unipotent representations of Sp(2n,R) and U(p,q), J. Lie Theory 25 (2015), no. 1, 185-213.

3.     A. Prlić, K-invariants in the algebra U(g) \otimes C(p) for the group SU(2,1), Glas. Mat. Ser. III 50(70) (2015), no. 2, 397-414.

4.     P. Pandžić, P. Somberg, Branching problems and sl(2,C)-actions, Arch. Math. (Brno) 51 (2015), no. 5, 315-330.

5.     P. Pandžić, P. Somberg, Higher Dirac cohomology of modules with generalized infinitesimal character, Transform. Groups 21 (2016), no. 3, 803-819.

6.     A. Prlić, Algebraic Dirac Induction for Nonholomorphic Discrete Series of SU(2,1), J. Lie Theory 26 (2016), no. 3, 889-910.

7.     P. Pandžić, Classifying A_q(\lambda) modules by their Dirac cohomology, in: Proceedings of the XI International Workshop "Lie Theory and Its Applications in Physics" (Varna, Bulgaria, June 2015), "Springer Proceedings in Mathematics and Statistics"  Vol. 191, ed. V. Dobrev (Springer, Tokyo-Heidelberg, 2016) pp. 387-398.

8.     S. Mehdi, P. Pandžić, D. Vogan, Translation principle for Dirac index, Amer. J. Math. 139 (2017), no. 6, 1465-1491.

9.     M. Bašić, T. Nikolaus, Homology of dendroidal sets, Homology, Homotopy Appl. 19 (2017), no. 1, 111-134.

10.  D. Barbasch, P. Pandžić, P. Trapa, Dirac index and twisted characters, arXiv:1606.05425, to appear in Trans. Amer. Math. Soc.

11.  H. Kraljević, K-structure of U(g) for su(n,1) and so(n,1), submitted, arxiv:1611.07900.

12.  P. Pandžić, V. Souček, BGG complexes in singular infinitesimal character for type A, J. Math. Phys. 58 (2017), no. 11, 111512, 16 pp.

13.  P. Pandžić, P. Somberg, Dirac operator and its cohomology for the quantum group U_q(sl_2), J. Math. Phys. 58 (2017), no. 4, 041702, 13 pp.

14.  J.-S. Huang, P. Pandžić, D. Vogan, On classifying unitary modules by their Dirac cohomology,  Sci. China Math. 60 (2017), no. 11, 1937-1962.

15.  A. Prlić, The centralizer of K in U(g)\otimes C(p) for the group SO_e(4,1),  Glas. Mat. Ser. III 52(72) (2017), no. 2, 275-288.

16.  R. Mrđen, Singular BGG complexes for the symplectic case, submitted, arXiv:1711.04752. 

 

Dissertations:

        1. M. Bašić, Stable homotopy theory of dendroidal sets, pdf

        2. R. Mrđen, Singular BGG complexes for the symplectic case, pdf

Conferences organized as part of the project activities:

Representation Theory XIV, IUC, Dubrovnik, 21-27.6.2015.

Representation Theory XV, IUC, Dubrovnik, 18-25.6.2017

Participation in conferences:

1.     P. Pandžić, invited lecture at International Workshop on Representation Theory and Harmonic Analysis, Chern Institute of Mathematical Sciences, Nankai University, Tianjin, China, June 2014.

2.     P. Pandžić, invited lecture at „Hodge theory and unitary representations of reductive Lie groups“, Jacobs University Bremen, Germany, June 2014.

3.     P. Pandžić, invited lecture at „Representations of p-adic groups; A conference dedicated to Marko Tadić on his 60th birthday“, University of Zagreb, Croatia, June 2014.

4.     A. Prlić, participation in „ Complex Geometry and Lie groups”, Torino, Italy, June 2014.

5.     R. Mrđen, participation in „ Complex Geometry and Lie groups”, Torino, Italy, June 2014.

6.     M. Bašić, participation in „Young Topologists' Meeting“, Copenhagen, Denmark, July 2014.

7.     A. Prlić, invited lecture at „Representations of real reductive groups, Dirac operators and related topics“, University of Lorraine - Metz, France, October 2014.

8.     P. Pandžić, invited lecture at „Representations of real reductive groups, Dirac operators and related topics“, University of Lorraine - Metz, France, October 2014.

9.     R. Mrđen, participation in „Representations of real reductive groups, Dirac operators and related topics“, University of Lorraine - Metz, France, October 2014.

10.  P. Pandžić, 3 plenary lectures at „35th Winter School Geometry and Physics“, Srni, Czech Republic, January 2015.

11.  R. Mrđen, participation in „35th Winter School Geometry and Physics“, Srni, Czech Republic, January 2015.

12.  P. Pandžić, invited lecture at „Representation Theory XIV“, Dubrovnik, Croatia, June 2015.

13.  H. Kraljević, invited lecture at „Representation Theory XIV“, Dubrovnik, Croatia, June 2015.

14.  M. Božičević, invited lecture at „Representation Theory XIV“, Dubrovnik, Croatia, June 2015.

15.  A. Prlić, participation in „Representation Theory XIV“, Dubrovnik, Croatia, June 2015.

16.  R. Mrđen, participation in „Representation Theory XIV“, Dubrovnik, Croatia, June 2015.

17.  K. Grizelj, participation in „Representation Theory XIV“, Dubrovnik, Croatia, June 2015.

18.  P. Pandžić, invited lecture at „Lie Theory XI“, Varna, Bulgaria, June 2015.

19.  M. Bašić, participation in „Summer School on Category Theory and Algebraic Topology“, Louvain-la Neuve, Belgium, September 2015.

20.  M. Bašić, invited lecture at „Higher Geometric Structures along the Lower Rhine“, Utrecht, The Netherlands, September 2015.

21.  A. Prlić, communication at „Lie Groups: Representation and Geometry”, Erlangen, Germany, September 2015.

22.  A. Prlić, participation in „Workshop on Representations, Support, and Cohomology”,  Bielefeld, Germany, December 2015.

23.  P. Pandžić, plenary lecture at „36th Winter School Geometry and Physics“, Srni, Czech Republic, January 2016.

24.  R. Mrđen, communication at „36th Winter School Geometry and Physics“, Srni, Czech Republic, January 2016.

25.  P. Pandžić, invited lecture atNew Developments in Representation Theory“, National University of Singapore, March 2016.

26.  H. Kraljević, poster presentation at “27th Nordic Congress of Mathematicians”, Stockholm, Sweden, March 2016.

27.  P. Pandžić, invited lecture at „Journées SL(2,R)“, University of Lorraine - Metz, France, June 2016.

28.  H. Kraljević, participation in „Journées SL(2,R)“, University of Lorraine - Metz, France, June 2016.

29.  A. Prlić, participation in „Journées SL(2,R)“, University of Lorraine - Metz, France, June 2016.

30.  R. Mrđen, participation in „Journées SL(2,R)“, University of Lorraine - Metz, France, June 2016.

31.  K. Grizelj, participation in „Journées SL(2,R)“, University of Lorraine - Metz, France, June 2016.

32.  P. Pandžić, plenary lecture at „6th Croatian Mathematical Congress“, University of Zagreb, Croatia, June 2016.

33.  H. Kraljević, communication at „6th Croatian Mathematical Congress“, University of Zagreb, Croatia, June 2016.

34.  A. Prlić, poster presentation at „6th Croatian Mathematical Congress“, University of Zagreb, Croatia, June 2016.

35.  R. Mrđen, communication at „6th Croatian Mathematical Congress“, University of Zagreb, Croatia, June 2016.

36.  K. Grizelj, participation in „6th Croatian Mathematical Congress“, University of Zagreb, Croatia, June 2016.

37.  R. Mrđen, poster presentation at “Differential Geometry and its Applications”, Brno, Czech Republic, July 2016.

38.  H. Kraljević, participation in “7th European Congress of Mathematicians”, Berlin, Germany, July 2016.

39.  M. Bašić, participation in „Topology Day“, University of Utrecht, The Netherlands,  September 2016.

40.  P. Pandžić, invited lecture at „Harmonic Analysis on Lie Groups and Group Algebras of Locally Compact Groups” TSIMF, Sanya, China, December 2016.

41.  P. Pandžić, invited lecture at „37th Winter School Geometry and Physics“, Srni, Czech Republic, January 2017.

42.  P. Pandžić, participation in „Series of talks in honor of Masaki Kashiwara“, University of Strasbourg, France, April 2017.

43.  R. Mrđen, invited lecture at „Representation Theory XV“, Dubrovnik, Croatia, June 2017.

44.  P. Pandžić, plenary lecture at The 11th International Conference on Clifford Algebras and Their Applications in Mathematical Physics (ICCA11)”, Ghent University, Ghent, Belgium.

 

Long term visiting positions:

1.     P. Pandžić, University of Lorraine-Metz, France, June-July 2014, 3 weeks.

2.     M. Bašić, University of Nijmegen, The Netherlands, several months between July 2014 and September 2015.

3.     P. Pandžić, The Hong Kong University of Science and Technology, September 2014.

4.     P. Pandžić, University of Freiburg, Germany, October 2014 – February 2015.

5.     M. Bašić, Univeristy of Utrecht, The Netherlands, September-October 2016.

6.     P. Pandžić, M.I.T., Cambridge, U.S.A., October 2016.

7.     P. Pandžić, Cornell University, Ithaca, U.S.A., November 2016.

8.     P. Pandžić, The Hong Kong University of Science and Technology, December 2016.

9.     P. Pandžić, University of Lorraine-Metz, France, March 2017.

10.  P. Pandžić, University of Freiburg, Germany, April – September 2017.

 

 

Short visits:

1.     P. Pandžić, Chalmers University of Technology, Gothenburg, Sweden, July 2014.

2.     M. Bašić, University of Belgrade, Serbia, November 2014 (lecture).

3.     P. Pandžić, Cornell University, Ithaca, U.S.A., September 2015.

4.     P. Pandžić, M.I.T., Cambridge, U.S.A., September 2015 (lecture).

5.     P. Pandžić, University of Oxford, UK, October 2015 (lecture).

6.     R. Mrđen, Charles University, Prague, Czech Republic, October 2015.

7.     P. Pandžić, University of Lorraine-Metz, France, November 2015 (lecture).

8.     P. Pandžić, Charles University, Prague, Czech Republic, November 2015.

9.     M. Bašić, Max Planck Institut, Bonn, Germany, December 2015.

10.  P. Pandžić, Chalmers University, Gothenburg, Sweden, March 2016 (lecture).

11.  P. Pandžić, Aoyama Gakuin University, Tokyo, Japan, April 2016.

12.  P. Pandžić, University of Aarhus, Denmark, May 2016 (lecture).

13.  R. Mrđen, Charles University, Prague, Czech Republic, May 2016.

14.  M. Bašić, University of Utrecht, The Netherlands, July 2016.

15.  P. Pandžić, Charles University, Prague, Czech Republic, August 2016.

16.  P. Pandžić, Chalmers University, Gothenburg, Sweden, August 2016 (lecture).

17.  R. Mrđen, Charles University, Prague, Czech Republic, November 2016.

18.  P. Pandžić, Charles University, Prague, Czech Republic, November 2017.

 

 

Visitors:

1.     Dan Barbasch, Cornell University, Ithaca, USA, August 2014.

2.     Petr Somberg, Charles University, Prague, Czech Republic, October 2014.

3.     Dan Barbasch, Cornell University, Ithaca, USA, October 2014.

4.     Petr Somberg, Charles University, Prague, Czech Republic, March 2015.

5.     Genkai Zhang, Chalmers University, Gothenburg, Sweden, June 2015.

6.     Dan Barbasch, Cornell University, Ithaca, USA, June 2015.

7.     Jing-Song Huang, HKUST, Hong Kong, June 2015.

8.     Petr Somberg, Charles University, Prague, Czech Republic, August 2015.

9.     Dan Barbasch, Cornell University, Ithaca, USA, October 2015.

10.  Salah Mehdi, University of Lorraine-Metz, France, November 2015.

11.  Petr Somberg, Charles University, Prague, Czech Republic, November 2015.

12.  Thomas Nikolaus, Max Planck Institute for Mathematics, Bonn, Germany, March 2016.

13.  Petr Somberg, Charles University, Prague, Czech Republic, March 2016.

14.  Dan Barbasch, Cornell University, Ithaca, USA, May 2016.

15.  Genkai Zhang, Chalmers University, Gothenburg, Sweden, May 2016.

16.  Dan Barbasch, Cornell University, Ithaca, USA, July 2016.

17.  Kyo Nishiyama, Aoyama Gakuin University, Tokyo, Japan, August 2016.

18.  Petr Somberg, Charles University, Prague, Czech Republic, February 2017.

19.  Petr Somberg, Charles University, Prague, Czech Republic, April 2017.

20.  Genkai Zhang, Chalmers University, Gothenburg, Sweden, May 2017.

21.  David Vogan, Massachusetts Institute of Technology, Cambridge, USA, June 2017.

22.  Genkai Zhang, Chalmers University, Gothenburg, Sweden, June 2017.

23.  Bent Oersted, Aarhus University, Denmark, June 2017.

24.  Wolfgang Soergel, University of Freiburg, Germany, June 2017.

25.  Dan Barbasch, Cornell University, Ithaca, USA, June-July 2017.

26.  Vladimir Souček, Charles University, Prague, Czech Republic, November 2017.