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

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.

1.
D. Barbasch, P.
Pandžić, P. Trapa, *Dirac index
and twisted characters*, Trans. Amer. Math. Soc. 371
(2019), no. 3, 1701-1733.

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

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

12. 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.

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

14. 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.

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

16. S. Mehdi, P. Pandžić, D. Vogan, R. Zierau, Computing the associated cycles of certain Harish-Chandra modules, Glas. Mat. 53(73) (2018), no.2, 275-330.

17. S. Mehdi, P. Pandžić, D. Vogan, R. Zierau, Dirac Index and associated cycles of Harish-Chandra module}, submitted, arXiv:1712.04169.

18.
D.
Husadžić, R. Mrđen, Singular
BGG Complexes Over Isotropic 2-Grassmannian, J. Lie Theory
28** **(2018), no. 4, 1149-1164.

**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 at „New
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.

45. P. Pandžić, invited lecture at “Journée: théorie des représentations”, University of Lorraine – Metz, Metz, Francuska, 07.12.2017.

46. P. Pandžić, invited lecture at “Affine, vertex and W-algebras”, INDAM, Rome, Italy, 11-15.12.2017.

47.
P. Pandžić, lecture
at 38th Winter School „Geometry and Physics“, Srní, Czech
Republic, 13.-20.1.2018.

**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
(lecture).

19. P.
Pandžić, Charles University, Prague, Czech Republic, March 2018.

20. P.
Pandžić, Cornell University, Ithaca, U.S.A.,
April 2018.

21. P.
Pandžić, M.I.T., Cambridge, U.S.A., April
2018.

22. P.
Pandžić, École polytechnique,
Palaiseau, France, May 2018.

**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.

27. Dan Barbasch, Cornell University, Ithaca, USA, January
2018.

28. Petr Somberg, Charles University, Prague, Czech Republic,
February 2018.

29. Dan Barbasch, Cornell University, Ithaca, USA, March 2018.

30. Genkai Zhang, Chalmers University, Gothenburg, Sweden, May
2018.