I study representation theory. My favorite questions are those that are related to number theory, specifically, automorphic forms and the Langlands program. Most of my work so far has revolved around theta correspondences.
Publications and preprints
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- Theta lifts of generic representations: the case of odd orthogonal groups
Glasnik Matematički, vol. 54, no. 2 (2019), pp. 421--462.
We determine the occurrence and explicitly describe the theta lifts on all levels of all the irreducible generic representations of the odd orthogonal group defined over a local nonarchimedean field of characteristic zero.
- Theta lifts of generic representations for dual pairs (Sp_{2n},O(V)) Manuscripta Mathematica, vol. 165 (2021), pp. 291--338.
We determine the occurrence and explicitly describe the theta lifts on all levels of all the irreducible generic representations for the dual pair of groups (Sp_{2n}, O(V)) defined over a local nonarchimedean field F of characteristic 0. As a direct application of our results, we are able to produce a series of non-generic unitarizable representations of these groups.
- Generic representations of metaplectic groups and their theta lifts, with M. Hanzer Mathematische Zeitschrift, vol. 297 (2021), pp. 1421--1465.
In this paper we give the description of generic representa- tions of metaplectic groups over p-adic fields in terms of their Langlands parameters and calculate their theta lifts on all levels for any tower of odd orthogonal groups. We also describe precisely all the occurrences of the failure of the standard module conjecture for metaplectic groups.
- Theta correspondence for p-adic dual pairs of type I, with M. Hanzer Journal für die reine und angewandte Mathematik (Crelle's Journal), vol. 2021, no. 776 (2021), pp. 63--117.
We describe explicitly the Howe correspondence for the symplectic-orthogonal and unitary dual pairs over a nonarchimedean local field of characteristic zero. More specifically, for every irreducible admissible representation of these groups, we find its first occurrence index in the theta correspondence and we describe, in terms of their Langlands parameters, the small theta lifts on all levels.
- The Gelfand--Graev representation of classical groups in terms of Hecke algebras, with G. Savin Canadian Journal of Mathematics, vol. 75, issue 4 (2023), pp. 1343--1368.
Let G be a p-adic classical group. The representations in a given Bernstein component can be viewed as modules for the corresponding Hecke algebra--the endomorphism algebra of a pro-generator of the given component. Using Heiermann's construction of these algebras, we describe the Bernstein components of the Gelfand-Graev representation for G=SO(2n+1), Sp(2n), and O(2n).
- Howe duality for a quasi-split exceptional dual pair, with G. Savin Mathematische Annalen, vol. 389 (2024), pp. 325-364.
We prove Howe duality for the theta correspondence arising from the p-adic dual pair G2 x (PU3 x Z/2Z) inside the adjoint quasi-split group of type E6.
- Similitude exceptional theta correspondences, with W.T. Gan and G. Savin Rad HAZU, vol. 28 (2024), pp. 193--122. Volume in honor of Marko Tadić.
We describe a systematic way of constructing dual pairs of similitude groups. We study the theta correspondences arising in this way and prove that Howe duality holds for the similitude dual pair if and only if it holds for the original reductive dual pair used in our construction. Several examples of exceptional correspondences are discussed.
- Theta correspondence and Arthur packets: on the Adams conjecture, with M. Hanzer Preprint.
The Adams conjecture predicts that the local theta correspondence should respect the Arthur parametrization. In this paper, we revisit the Adams conjecture for the symplectic-even orthogonal dual pair over a nonarchimedean local field of characteristic zero. Our results provide a precise description of all situations in which the conjecture holds.
- Global long root A-packets for G_2, with A. Horawa, S.D. Li-Huerta and N. Sweeting Preprint.
Cuspidal automorphic representations τ of PGL2 correspond to global long root A-parameters for G2. Using an exceptional theta lift between PU3 and G2, we construct the associated global A-packet and prove the Arthur multiplicity formula for these representations when τ is dihedral and satisfies some technical hypotheses. We also prove that this subspace of the discrete automorphic spectrum forms a full near equivalence class. Our construction yields new examples of quaternionic modular forms on G2.
These papers are available in preprint form on my arXiv page.