# Automorphic forms, representations, and applications

#### Abstract:

Recent and major breakthrough in the Langlands programme is a proof of existence endoscopic transfer of representations both local and global automorphic from split classical groups to GL(n) proved mainly by Arthur. Our research proposal fits very well with this recent development. First of all, the work of Arthur has its local application in the classification of representations of discrete series that was done by Moeglin and Tadić. Based on those results we plan to pursue the following investigations in the representation theory of p-adic groups. We would like to compute the Jacquet modules of discrete series representations completely.This will help us understand representations parabolically induced from those discrete series (generalized principal series) completely. This will have consequence on the problems of determination of complementary series for classical groups a problem related to the classification of unitary representations. Results of Gan, Savin and Ichino will help us extend some of those results to metaplectic groups. This will have important consequence on the determination of theta correspondence for dual pairs consisting of odd orthogonal and metaplectic groups such as explicit determination of all theta lifts. In the theory of automorphic forms, we would like to understand the construction of Arthur and Moeglin more explicitly. This will construct new series of square integrable representations; some special cases were obtained earlier by Muić, and Muić and Hanzer.This will give locally many interesting unitary representations which are more complicated than that has appeared in earlier works of Speh, Rogawski, Tadić, Muić, Savin, and Hanzer. We will apply those methods to study cohomology and poles of automorphic L-functions.

#### Principal investigator:

Goran Muić, Professor, Zagreb

#### Team members:

Darija Brajković, Ph.D. student, Osijek
Igor Ciganović, post. doc., Zagreb
Neven Grbac, Associate Professor, Rijeka
Marcela Hanzer, Professor, Zagreb
Iva Kodrnja, post. doc., Zagreb
Ivan Matić, Associate Professor, Osijek
Marko Tadić, Professor, Zagreb
Sonja Žunar Ph. D. student, Zagreb
Nevena Jurčević Peček, post. doc., Rijeka

#### Publications:

1. I. Ciganović, N. Grbac, The Zelevinsky classification of unramified representations of the metaplectic group. J. Algebra 454 (2016), 357–399.
2. N. Grbac, F. Shahidi, Endoscopic transfer for unitary groups and holomorphy of Asai L-functions, Pacific J. Math. 276 (2015), no. 1, 185–211.
3. N. Grbac, Analytic properties of automorphic L-functions and Arthur classification, RIMS Kôkyûroku, vol. 1934, pp. 26-39, RIMS, Kyoto, 2015.
4. M. Hanzer, An explicit construction of automorphic representations of symplectic group with given quadratic unipotent Arthur parameter, Monatsh. Math. 177 (2015), no. 2, 235–273.
5. M. Hanzer, Degenerate Eisenstein series for symplectic groups, Glas. Mat. Ser. III 50(70) (2015), no. 2, 289–332.
6. A. David, M. Hanzer, J. Ludwig, The conjectural relation between generalized Shalika models on SO4n(F) and the symplectic linear model on Sp4n(F). A Toy Example, Women in Numbers Europe: Research Directions in Number Theory (2015), Editors: Alina Bucur, Marie-José Bertin, Brooke Feigon, Leila Schneps, pp 87-107.
7. M. Hanzer, Generalized Shalika model on SO_4n(F), symplectic linear model on Sp_4n(F) and theta correspondence, Rad HAZU, Matematicke znanosti, vol. 19 (2015) pp 55-68.
8. M. Hanzer, Non-Siegel Eisenstein series for symplectic groups (pp 1-74), Manuscripta Math, DOI: 10.1007/s00229-017-0927-6.
9. M. Hanzer On the cuspidal support of a generic representation, prihhvaćen za obavljivanje Journal of Lie Theory.
10. I. Matić, On discrete series subrepresentations of generalized principal series, Glasnik Matematički, Vol. 51, No. 1 (2016), 125-152.
11. I. Matić, On Jacquet modules of discrete series: the first inductive step. Journal of Lie theory. 26 (2016), 1, 135-168.
12. I. Matić, M. Tadić, Jacquet modules of representations of segment type, Manuscripta Math. 147 (2015), no. 3-4, 437–476.
13. I. Matić, Strongly positive subquotients in a class of induced representations of classical p-adic groups. Journal of algebra. 444 (2015), 504-526.
14. I. Matić, First occurrence indices of tempered representations of metaplectic groups. Proceedings of the American Mathematical Society. 144 (2016), 7, 3157-3172.
15. I. Matić, Aubert duals of strongly positive discrete series and a class of unitarizable representations, Proceedings of the American Mathematical Society 145/8 (2017), 3561-3570.
16. I. Matić, On Langlands quotients of the generalized principal series isomorphic to their Aubert duals, Pacific Journal of Mathematics 289/2 (2017), 395-415.
17. I. Matić, Composition factors of a class of induced representations of classical p-adic groups, Nagoya Mathematical Journal, Nagoya Mathematical Journal 227 (2017), 16-48.
18. G. Muić, On degrees and birationality of the maps $X_0(N)\rightarrow \mathbb P^2$ constructed via modular forms, Monatsh. Math. 180 (2016), no. 3, 607–629.
19. G. Muić, Fourier coefficients of automorphic forms and integrable discrete series. J. Funct. Anal. 270 (2016), no. 10, 3639–3674.
20. G. Muić, Some Results on the Schwartz Space of $\Gamma\backslash G$, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 21(532) (2017), 29–-52.
21. A. Moy, G. Muić, On Existence of Generic Cusp Forms on Semisimple Algebraic Groups, Trans. Amer. Math. Soc. 370 (2018), no. 7, 4731–4757.
22. G. Muić, Smooth cuspidal automorphic forms and integrable discrete series prihvaćen za objavljivanje u Math. Z.
23. M. Tadić, On the reducibility points beyond the ends of complementary series of p-adic general linear groups. J. Lie Theory 25 (2015), no. 1, 147–183.
24. M. Tadić, Remark on representation theory of general linear groups over a non-archimedean local division algebra, Rad HAZU, Matematicke Znanosti, vol. 19 / 523, 2015, pages 27-53.
25. M. Tadić, Some bounds on unitary duals of classical groups - non-archimeden case, prihvaćen za objavljivanje u Bulletin of the Iranian Mathematical Society.
26. I. Kodrnja, On a simple model of $X_0(N)$, prihvaćen za objavljivanje u Monatshefte f\" ur Mathematik.
27. I. Kodrnja, Eta-quotients and Embeddings of $X_0(N)$ in the Projective Plane, prihvaćen za objavljivanje u Ramanujan Journal.
28. S. Žunar, On the non-vanishing of Poincaré series on the metaplectic group, prihvaćen za objavljivanje u Manuscripta Mathematica.
29. S. Žunar, On Poincaré series of half-integral weight, prihvaćen za objavljivanje u Glasnik Matematicki.
30. I. Ciganović, Composition series of a class of induced representations, a case of one half cuspidal reducibility, prihvaćen za objavljivanje u Pacific J. Math.

#### Disertacije:

1. Nevena Jurčević Peček: Teorija reprezentacija Hermitskih kvaternionskih grupa nad p-adskim poljima (in Croatian), Department of Mathematics, University of Zagreb, 2015, 118 pages.
2. Iva Kodrnja: Modeli modularnih krivulja, modularne forme i eta-kvocijenti (in Croatian), Department of Mathematics, University of Zagreb, 2016, 126 pages.

#### Lectures:

1. G. Muić: Some results on the Schwartz space of $\Gamma \backslash G$ (Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong 2018).
2. G. Muić: Some results on the Schwartz space of $\Gamma \backslash G$ (Department of Mathematics, University of Utah, Salt Lake City 2018)
3. G. Muić: On existence of cuspidal automorphic forms for semisimple algebraic groups defined over number field (Erwin Schroedinger Institute, Vienna 2015)
4. G. Muić: On existence of cuspidal automorphic forms for semisimple Lie groups (Department of Mathematics, University of Utah, Salt Lake City 2015)
5. M. Tadić: Square-integrable representations of classical p-adic groups and their Jacquet modules (Weizmann Institute, Rehovot, 2015)
6. M. Hanzer: Degenerate Eisenstein series for symplectic group (Erwin Schroedinger Institute, Vienna 2015)
7. N. Grbac: Eisenstein series for unitary groups in view of cohomological applications (Erwin Schroedinger Institute, Vienna 2015)
8. G. Muić: Explicit Whittaker models for integrable discrete series and applications (Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong 2016).
9. M. Hanzer: Non-Siegel degenerate Eisenstein series (Department of Mathematics, The University of Singapoure 2016)
10. M. Tadić: Some bounds on unitary duals of classical p-adic groups (Department of Mathematics, University of Utah, Salt Lake City 2016)
11. M. Hanzer: Degenerate Eisenstein series for GLn and some applications in number theory (Department of Mathematics, University of Sarajevo 2017)
12. M. Tadić: On classification of irreducible unitary representations of classical p-adic groups (Weizmann Institute, Rehovot, 2015)

Worshop: On some recent developments in local and global representation theory (May 19, 2018; Zagreb) organized by N. Grbac, M. Hanzer and G. Muić