arXiv subjects

Johannes Girsch

Publications and source records attributed to Johannes Girsch.

7 recordsLinked to original sources

arXiv2026

On classical doubling method gamma factors for certain depth zero representations

Piatetski-Shapiro--Rallis discovered an integral representation construction, known as the doubling method, for the tensor product $L$-function of a cuspidal automorphic representation of $G \times \mathrm{GL}_1$, where $G$ is a classical group. Lapid--Rallis defined and studied the counterpart local factors. In this article, following Lapid--Rallis, we define and study an analogous doubling method gamma factor associated to irreducible representations of classical finite groups of Lie type. We prove that this gamma factor is multiplicative and use results of Yost-Wolff--Zelingher to give explicit formulas for it in terms of the Deligne--Lusztig data of the representation in the non-conjugate-dual character case. Finally, we relate our construction to the local construction of Lapid--Rallis via certain depth zero supercuspidal representations of classical groups.

math.RT↗

arXiv2025

On tori periods of Weil representations of unitary groups

We determine the restriction of Weil representations of unitary groups to maximal tori. In the local case, we show that the Weil representation contains a pair of compatible characters if and only if a root number condition holds. In the global case, we show that a torus period corresponding to a maximal anisotropic torus of the global theta lift of a character does not vanish if and only if the local condition is satisfied everywhere and a central value of an $L$-function does not vanish. Our proof makes use of the seesaw argument and of the well-known theta lifting results from $\operatorname{U}\left(1\right)$ to $\operatorname{U}\left(1\right)$. Our results are used in other papers to construct Arthur packets for $G_2$.

math.RT↗

arXiv2025

On families of degenerate representations of GL_n(F)

We consider the stratification of the category of smooth representations of $\mbox{GL}_n(F)$ (for $F$ a $p$-adic field) induced by degenerate Whittaker models. We show that, remarkably, the successive quotient categories in this stratification turn out to be module categories over commutative rings. In fact these rings turn out to be infinite products of smooth algebras over the ground field. We further obtain explict descriptions of these rings in terms of the Zelevinsky classification; these descriptions closely resemble the Bernstein-Deligne description of the Bernstein center.

math.RT↗

arXiv2025

Newforms in Cuspidal Representations

We consider newform vectors in cuspidal representations of $p$-adic general linear groups. We extend the theory from the complex setting to include~$\ell$-modular representations with~$\ell\neq p$, and prove that the conductor is compatible with congruences modulo~$\ell$ for (ramified) supercuspidal~$\ell$-modular representations and for depth zero cuspidals. In the complex and modular setting, we prove explicit formulae for depth zero and minimax cuspidal representations of integral depth, in Bushnell-Kutzko and Whittaker models.

math.NT↗

arXiv2024

The Twisted Doubling Method in Algebraic Families

We define the twisted doubling zeta integrals of Cai-Friedberg-Ginzburg-Kaplan in the setting of algebraic families. We then prove a rationality result and a functional equation for these zeta integrals. This allows us to define an unnormalized $γ$-factor associated to certain families of representation of a classical group times a general linear group.

math.RT↗

arXiv2021

The Doubling Method in Algebraic Families

We define the doubling zeta integral for smooth families of representations of classical groups. Following this we prove a rationality result for these zeta integrals and show that they satisfy a functional equation. Moreover, we show that there exists an apropriate normalizing factor which allows us to construct $γ$-factors for smooth families out of the functional equation. We prove that under certain hypothesis, specializing this $γ$-factor at a point of the family yields the $γ$-factor defined by Piateski-Shapiro and Rallis.

math.NT↗

arXiv2015

Class invariants for certain non-holomorphic modular functions

Inspired by prior work of Bruinier and Ono and Mertens and Rolen, we study class polynomials for non-holomorphic modular functions arising from modular forms of negative weight. In particular, we give general conditions for the irreducibility of class polynomials. This allows us to easily generate infintely many new class invariants.

math.NT↗