Automorphic forms, representations, and applications
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.
Goran Muić, Professor, Zagreb
Darija Brajković, Ph.D. student, Osijek
Igor Ciganović, post. doc., Zagreb
Neven Grbac, Associate Professor, Rijeka
Marcela Hanzer, Professor, Zagreb
Iva Kodrnja, Ph.D. student, Zagreb
Ivan Matić, Associate Professor, Osijek
Jurčević Peček, Ph.D. student, Rijeka
- I. Ciganović, N. Grbac,
The Zelevinsky classification of unramified representations of the metaplectic group. J. Algebra 454 (2016), 357–399.
- N. Grbac, F. Shahidi, Endoscopic transfer for unitary groups and holomorphy of Asai L-functions, Pacific J. Math.
276 (2015), no. 1, 185–211.
- N. Grbac, Analytic properties of automorphic L-functions and
Arthur classification, RIMS Kôkyûroku, vol. 1934, pp. 26-39, RIMS, Kyoto, 2015.
- 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.
- M. Hanzer, Degenerate Eisenstein series for symplectic groups,
Glas. Mat. Ser. III 50(70) (2015), no. 2, 289–332.
- 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.
- 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.
- M. Hanzer, Non-Siegel Eisenstein series for symplectic groups (pp
1-74), Manuscripta Math DOI: 10.1007/s00229-017-0927-6.
- M. Hanzer On the cuspidal support of a generic representation,
prihhvacen za obavljivanje Journal of Lie Theory.
- I. Matić, On discrete series subrepresentations of
generalized principal series, Glasnik Matematicki, Vol. 51, No. 1 (2016), 125-152.
- I. Matić, On Jacquet modules of discrete series: the first inductive step. Journal
of Lie theory. 26 (2016), 1, 135-168.
- I. Matić, M. Tadić, Jacquet modules of representations of segment
type, Manuscripta Math. 147 (2015), no. 3-4, 437–476.
- I. Matić, Strongly positive subquotients in a class of induced representations of
classical p-adic groups. Journal of algebra. 444 (2015), 504-526.
- I. Matić, First occurrence indices of tempered representations of metaplectic
groups. Proceedings of the American Mathematical Society. 144 (2016), 7, 3157-3172.
- 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.
- I. Matić, On Langlands quotients of the generalized principal
isomorphic to their Aubert duals, Pacific Journal of Mathematics 289/2
- 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.
- 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.
- G. Muić, Fourier coefficients of automorphic forms and integrable discrete series. J. Funct. Anal. 270 (2016),
no. 10, 3639–3674.
- A. Moy, G. Muić,
On Existence of Generic Cusp Forms on
Semisimple Algebraic Groups
Trans. Amer. Math. Soc. (to appear).
- G. Muić,
Some Results on the Schwartz Space of $\Gamma\backslash G$ (special issue
in honor of professor Sibe Mardesic), Rad HAZU Vol. 21 (2017), 29-52.
- 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.
- 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.
- M. Tadić,
Some bounds on unitary duals of classical
groups - non-archimeden case, prihvacen za objavljivanje u Bulletin of
the Iranian Mathematical Society.
- Nevena Jurčević Peček:
Teorija reprezentacija Hermitskih kvaternionskih grupa nad p-adskim
poljima (in Croatian), Department of Mathematics, University of
Zagreb, 2015, 118 pages.
- Iva Kodrnja: Modeli modularnih krivulja,
modularne forme i eta-kvocijenti (in Croatian), Department of
Mathematics, University of
Zagreb, 2016, 126 pages.
- G. Muic: On existence of cuspidal automorphic forms for semisimple
algebraic groups defined over number field (Erwin Schroedinger Institute,
- G. Muic: On existence of cuspidal automorphic forms for semisimple
Lie groups (Department of Mathematics, University of Utah, Salt Lake City
- M. Tadic: Square-integrable representations of classical p-adic
groups and their Jacquet modules (Weizmann Institute, Rehovot, 2015)
- M. Hanzer: Degenerate Eisenstein series for symplectic group
(Erwin Schroedinger Institute, Vienna 2015)
- N. Grbac: Eisenstein series for unitary groups in view of
cohomological applications (Erwin Schroedinger Institute, Vienna 2015)
- G. Muic: Explicit Whittaker models for integrable discrete series and
applications (Department of Mathematics, The Hong Kong University of
Science and Technology, Hong Kong 2016).
- M. Hanzer: Non-Siegel degenerate Eisenstein series (Department of
Mathematics, The University of Singapoure 2016)
- M. Tadic: Some bounds on unitary duals of classical p-adic groups
(Department of Mathematics, University of Utah, Salt Lake City
- M. Hanzer: Degenerate Eisenstein series for GLn and some
applications in number theory (Department of Mathematics, University of
- M. Tadic: On classification of irreducible unitary representations of
classical p-adic groups (Weizmann Institute, Rehovot, 2015)