### Ivan Matić

Head of Algebra, Analysis and Number Theory Research Group

School of Applied Mathematics and Informatics

Josip Juraj Strossmayer University of Osijek

Josip Juraj Strossmayer University of Osijek

### Research Interests

Representation theory of *p*-adic groups

Langlands program

Theta correspondence

### Degrees

PhD in theoretical mathematics, Department of Mathematics, University of Zagreb , 2010.

BSc in Mathematics and Computer Science Education, Department of Mathematics, University of Osijek, 2010.

BSc in Mathematics, Department of Mathematics, University of Zagreb, 2004.

### Publications

Journal Publications

- Y. Kim, I. Matić, Discrete series of odd general spin groups, Monatshefte für Mathematik
**203**/2 (2024), 419-473We obtain a Mœglin–Tadić type classification of the non-cuspidal discrete series of odd general spin groups over non-archimedean local fields of characteristic zero. Our approach presents a simplified, uniform, and slightly different construction of a bijective correspondence between the set of isomorphism classes of non-cuspidal discrete series representations and the set of so called admissible triples, a notion obtained by readily extending the analogous notion due to Mœglin–Tadić from the case of classical groups to that of odd general spin groups. We use almost exclusively algebraic methods, one of which replaces subtle theory on intertwining operators for odd general spin groups by calculation of Jacquet modules. In this way we provide a slightly different proof of the classification of discrete series representations for classical groups, which contains some more concrete information on the admissible triples. We expect that our classification has an advantage of being rather directly applicable to several other reductive p-adic groups, including even general spin groups and similitude classical groups. - I. Matić, On reducibility of representations induced from the essentially Speh representations and discrete series, Rad HAZU, Matematičke znanosti.
**28**(2024), 283-325Let π stand for an essentially Speh representation of the form L(δ([νa ρ, νa+k ρ]), ..., δ([νa+n-1 ρ, νa+k+n-1} ρ])), where ρ is an irreducible cuspidal representation of the general linear group over a non-archimedean local field or its separable quadratic extension, a ≤ 0, 2a + k > 0, and n ≥ 1. Let σ denote a discrete series representation of either symplectic, special odd-orthogonal, or unitary group. We determine when the induced representation π ⋊ σ reduces. - I. Matić, On representations induced from the Zelevinsky segment and a tempered representation in the half-integral case, Journal of Algebra and Its Applications
**22**/11 (2023), 1-55Let $G_n$ denote either the group $SO(2n+1, F)$ or $Sp(2n, F)$ over a non-archimedean local field of characteristic zero. We determine the reducibility criteria for a parabolically induced representation of the form $langle [ nu^{a} rho, nu^{b} rho] rangle rtimes tau$, where $langle [ nu^{a} rho, nu^{b} rho] rangle$ denotes the Zelevinsky segment representation of the general linear group attached to the segment $[ nu^{a} rho, nu^{b} rho]$, with $a$ half-integral, and $tau$ denotes an irreducible tempered representation of $G_n$. - B. Bošnjak, I. Matić, Discrete series and the essentially Speh representations, Journal of Algebra
**611**(2022), 65-81Let $pi$ denote an essentially Speh representation of the general linear group over a non-archimedean local field or its separable quadratic extension, and let $sigma_c$ denote an irreducible cuspidal representation of either symplectic, special odd-orthogonal, or unitary group. We determine when the induced representation $pi rtimes sigma_c$ contains a discrete series subquotient. We also identify all discrete series subquotients. - I. Matić, Irreducibility criteria for the generalized principal series of unitary groups, Proceedings of the American Mathematical Society
**150**/11 (2022), 5009-5021We present an algebraic proof of the irreducibility criteria for the generalized principal series of unitary groups over a non-archimedean local field. - I. Matić, Reducibility of representations induced from the Zelevinsky segment and discrete series, Manuscripta Mathematica
**164**/3-4 (2021), 349-374Let G_n denote either the group SO(2n+1, F) or Sp(2n, F) over a non-archimedean local field. We determine the reducibility criteria for a parabolically induced representation of the form $zs rtimes s$, where $zs$ stands for a Zelevinsky segment representation of the general linear group and $s$ stands for a discrete series representation of G_n, in terms of the M{oe}glin-Tadić classification. - I. Matić, Representations induced from the Zelevinsky segment and discrete series in the half-integral case, Forum Mathematicum
**33**/1 (2021), 193-212Let $G_n$ denote either the group $SO(2n+1, F)$ or $Sp(2n, F)$ over a non-archimedean local field of characteristic different than two. We study parabolically induced representations of the form $langle Delta rangle rt sigma$, where $langle Delta rangle$ denotes the Zelevinsky segment representation of the general linear group attached to the segment $Delta$, and $sigma$ denotes a discrete series representation of $G_n$. We determine composition factors of $zs rt sigma$ in the case when $Delta = [ nu^{a} rho, nu^{b} rho]$ where $a$ is half-integral. - B. Liu, Y. Kim, I. Matić, Degenerate principal series for classical and odd GSpin groups in the general case, Representation Theory
**24**(2020), 403-434Let G_n denote either the group SO(2n+1, F), Sp(2n, F), or GSpin(2n+1, F) over a non-archimedean local field of characteristic different from two. We determine all composition factors of degenerate principal series of G_n, using methods based on the Aubert involution and known results on irreducible subquotients of the generalized principal series of a particular type. - I. Matić, Aubert duals of discrete series: the first inductive step, Glasnik Matematički
**54**/1 (2019), 133-178Let $G_n$ denote either symplectic or odd special orthogonal group of rank $n$ over a non-archimedean local field $F$. We provide an explicit description of the Aubert duals of irreducible representations of $G_n$ which occur in the first inductive step in the realization of discrete series representations starting from the strongly positive ones. Our results might serve as a pattern for determination of Aubert duals of general discrete series of $G_n$ and should produce an interesting part of the unitary dual of this group. Furthermore, we obtain an explicit form of some representations which are known to be unitarizable. - I. Matić, Aubert duals of strongly positive discrete series and a class of unitarizable representations, Proceedings of the American Mathematical Society
**145**/8 (2017), 3561-3570Let G_n denote either the group Sp(n, F) or SO(2n + 1, F) over a local non-archimedean field F. We explicitly determine the Aubert duals of strongly positive discrete series representations of the group G_n. This enables us to construct a large class of unitarizable representations of this group. - I. Matić, Composition factors of a class of induced representations of classical p-adic groups, Nagoya Mathematical Journal
**227**(2017), 16-48We study induced representations of the form $delta_1 times delta_2 rtimes sigma$, where $delta_1, delta_2$ are irreducible essentially square-integrable representations of general linear group and $sigma$ is a strongly positive discrete series of classical $p$-adic group, which naturally appear in the non-unitary dual. Employing certain conditions on $delta_1$ and $delta_2$, we determine complete composition series of such induced representation. - I. Matić, On Langlands quotients of the generalized principal series isomorphic to their Aubert duals, Pacific Journal of Mathematics
**289**/2 (2017), 395-415We determine under which conditions is the Langlands quotient of an induced representation of the form $delta rt sigma$, where $delta$ is an irreducible essentially square-integrable representation of a general linear group and $sigma$ is a discrete series representation of the classical $p$-adic group, isomorphic to its Aubert dual. - I. Matić, On Jacquet Modules of Discrete Series: the First Inductive Step, Journal of Lie Theory
**26**/1 (2016), 135-168The purpose of this paper is to determine Jacquet modules of discrete series which are obtained by adding a pair of consecutive elements to the Jordan block of an irreducible strongly positive representation such that the $epsilon$-function attains the same value on both elements. Such representations present the first inductive step in the realization of discrete series starting from the strongly positive ones. We are interested in determining Jacquet modules with respect to the maximal standard parabolic subgroups, with an irreducible essentially square-integrable representation on the general linear part. - I. Matić, First occurrence indices of tempered representations of metaplectic groups, Proceedings of the American Mathematical Society
**144**/7 (2016), 3157-3172We explicitly determine the first occurrence indices of tempered representations of metaplectic groups over a non-archimedean local field of characteristic zero with odd residual characteristic. - I. Matić, On discrete series subrepresentations of the generalized principal series, Glasnik Matematički
**51**/1 (2016), 125-152We study a family of the generalized principal series and obtain necessary and sufficient conditions under which the induced representation of studied form contains a discrete series subquotient. Furthermore, we show that if the generalized principal series which belongs to the studied family has a discrete series subquotient, then it has a discrete series subrepresentation. - I. Matić, M. Tadić, On Jacquet modules of representations of segment type, Manuscripta Mathematica
**147**/3 (2015), 437-476Let $G_{n}$ denote either the group $Sp(n, F)$ or $SO(2n+1, F)$ over a local non-archimedean field $F$. We study representations of segment type of group $G_{n}$, which play a fundamental role in the constructions of discrete series, and obtain a complete description of the Jacquet modules of these representations. Also, we provide an alternative way for determination of Jacquet modules of strongly positive discrete series and a description of top Jacquet modules of general discrete series. - I. Matić, Strongly positive representations in an exceptional rank-one reducibility case (an appendix to: 'Strongly positive representations of GSpin_{2n+1} and the Jacquet module method' by Yeansu Kim), Mathematische Zeitschrift
**279**/1-2 (2015), 271-296We obtain some results on the strongly positive discrete series in an exceptional rank-one reducibility case. Such results appear to be important for the classification of strongly positive representations for GSpin groups. - I. Matić, Strongly positive subquotients in a class of induced representations of classical $p$-adic groups, Journal of Algebra
**444**(2015), 504-526We determine under which conditions the induced representation of the form $delta_{1} times delta_{2} rtimes sigma$, where $delta_{1}, delta_{2}$ are irreducible essentially square integrable representations of a general linear group and $sigma$ is a discrete series representation of classical $p$-adic group, contains an irreducible strongly positive subquotient. - I. Matić, Discrete series of metaplectic groups having generic theta lifts, Journal of the Ramanujan Mathematical Society
**29**/2 (2014), 201-219We prove that a discrete series representations of metaplectic group over a non-archimedean local field has a generic theta lift on the split odd orthogonal tower if and only if it is generic. Also, we determine the first occurrence indices of such representations and describe the structure of their theta lifts. - I. Matić, Jacquet modules of strongly positive representations of the metaplectic group $widetilde{Sp(n)}$, Transactions of the American Mathematical Society
**365**(2013), 2755-2778Strongly positive discrete series represent a particularly important class of irreducible square-integrable representations of $p$-adic groups. Indeed, these representations are used as basic building blocks in known constructions of general discrete series. In this paper, we explicitly describe Jacquet modules of strongly positive discrete series. The obtained description of Jacquet modules, which relies on the classification of strongly positive discrete series given in our previous work, is valid in both classical and metaplectic case. We expect that our results, besides being interesting by themselves, should be relevant to some potential applications in the theory of automorphic forms, where both representations of metaplectic groups and structure of Jacquet modules play an important part. - I. Matić, The conservation relation for discrete series representations of metaplectic groups, International Mathematics Research Notices
**2013**/22 (2013), 5227-5269Let $F$ denote a non-archimedean local field of characteristic zero with odd residual characteristic and let $widetilde{Sp(n)}$ denote the rank $n$ metaplectic group over $F$. If $r^{pm}(sigma)$ denotes the first occurrence index of the irreducible genuine representation $sigma$ of $widetilde{Sp(n)}$ in the theta correspondence for the dual pair $(widetilde{Sp(n)},O(V^{pm}))$, the conservation relation, conjectured by Kudla and Rallis, states that $r^{+}(sigma)+r^{-}(sigma)=2n$. A purpose of this paper is to prove this conjecture for discrete series which appear as subquotients of generalized principle series where the representation on the metaplectic part is strongly positive. Also, we prove this relation for many tempered but non-square integrable and non-tempered irreducible subquotients of such representations. Assuming the basic assumption, we prove the conservation relation for general discrete series of metaplectic groups. - I. Matić, Theta lifts of strongly positive discrete series: the case of ($widetilde{Sp(n)}$, O(V)), Pacific Journal of Mathematics
**259**/2 (2012), 445-471Let $F$ denote a non-archimedean local field of characteristic zero with odd residual characteristic. Using the results of Gan and Savin, in this paper we determine the first occurrence indices and theta lifts of strongly positive discrete series representations of metaplectic groups over $F$ in terms of our recent classification of this class of representations. Also, we determine the first occurrence indices of some strongly positive representations of odd orthogonal groups. - I. Matić, Strongly positive representations of metaplectic groups, Journal of Algebra
**334**(2011), 255-274In this paper, we obtain the classification of irreducible strongly positive square-integrable genuine representations of metaplectic groups over $p$-adic fields, using purely algebraic approach. Our results parallel those of M{oe}glin and Tadi'{c} for classical groups, but their work relies on certain conjectures. On the other side, our results are complete and there are no additional conditions or hypothesis. The important point to note here is that our results and technics can be used in the case of classical $p$-adic groups in completely analogous manner. - F.M. Brueckler, I. Matić, The power and the limits of the abacus, Mathematica Pannonica
**22**/1 (2011), 25-48The abacus is a well-known calculating tool with a limited number of place-holders for digits of operands and results. Given a number of rods $n$ of the abacus, a chosen basis of the number system and the first operand $a$, this paper deals with the possible values of the other operand $b$ in the four basic arithmetic operations performed with integers on the abacus. For division we identify several subcases, depending on $n$ and the number of digits $delta_B(a)$ of $a$. If $a$ cannot be divided by all $bleq a$, the number $delta_B(a)$ is called critical. For numbers with the minimal critical number of digits $N_n=lfloorfrac{n-4}{3}rfloor+1$ we explicitly determine the values of the maximal divisor $b_{max}$ such that the division $a:b_{max}$ can be performed. - I. Matić, Composition series of the induced representations of SO(5) using intertwining operators, Glasnik Matematički
**45**/1 (2010), 93-107Let $F$ be a p-adic field of characteristic zero. We determine the composition series of the induced representations of $SO(5,F)$. - M. Hanzer, I. Matić, Irreducibility of the unitary principal series of $p$-adic $widetilde{Sp(n)}$, Manuscripta Mathematica
**132**(2010), 539-547Let $F$ be a p-adic field. We prove irreducibility of the unitary principal series of the group $widetilde{Sp(n)}$ over $F$. - M. Hanzer, I. Matić, The unitary dual of $p$-adic $widetilde{Sp(2)}$, Pacific Journal of Mathematics
**248**/1 (2010), 107-137We investigate the composition series of the induced admissible representations of the metaplectic group $widetilde{Sp(2)}$ over a $p$-adic field $F.$ In this way, we determine the non-unitary and unitary duals of $widetilde{Sp(2)}$ modulo cuspidal representations. - I. Matić, The unitary dual of $p$-adic SO(5), Proceedings of the American Mathematical Society
**138**/2 (2010), 759-767Let $F$ be a p-adic field of characteristic zero. We investigate the composition series of the parabolically induced representations of SO(5,F) and determine the non-cuspidal part of the unitary dual of $SO(5,F)$. - I. Matić, Levi subgroups of $p$-adic Spin(2n+1), Mathematical Communications
**14**/2 (2009), 223-233We explicitly describe Levi subgroups of odd spin groups over algebraic closure of a p-adic field.

Refereed Proceedings

- Y. Kim, I. Matić, Classification of Strongly Positive Representations of Even General Unitary Groups, Representations of Reductive p-adic Groups, Pune, India, 2019, 161-174

### Projects

- Composition series of induced representations of classical
*p*-adic groups (Project run in 2015, supported by University of Osijek.) - Bilateral project Croatia – Austria: Cohomology of arithmetic groups and automorphic forms (Project leaders: Neven Grbac and Joachim Schwermer)
- Discrete series in generalized principal series (Project run in 2013/14, supported by University of Osijek.)

- Automorphic forms, representations and applications (Project leader: Goran Muić. Project was funded in 2014 by the Croatian Science Foundation.)
- Unitary representations and automorphic forms (Project leader: Marko Tadić. Project was funded in 2008 by the Croatian Science Foundation.)
- Unitary representations, automorphic and modular forms (Project leader: Marcela Hanzer. Project was funded in 2018 by the Croatian Science Foundation.)
- Spaces of modular and automorphic forms and representation theory within the framework of the Langlands program (Project leader: Neven Grbac. Project was funded in 2023 by the Croatian Science Foundation.)

### Professional Activities

**Editorial Boards**

**Refereeing/Reviewing**

Zentralblatt MATH (since 2010)

Mathematical Reviews (since 2011)

### Teaching

** Učenička matematička natjecanja**

** Matematika 1 (Građevinski i arhitektonski fakultet)**

** Matematika 2 (Građevinski i arhitektonski fakultet)
**

Matematika (Građevinski i arhitektonski fakultet)

**Konzultacije (Office Hours):** Po dogovoru.