arXiv subjects

Johannes Droschl

Publications and source records attributed to Johannes Droschl.

10 recordsLinked to original sources

arXiv2026

Generalized Jantzen filtration, hyperbolic restriction, and characteristic cycles

In this paper we give a geometric description of the behavior of analytic intertwining operators between parabolically induced representations of $\mathrm{GL}_n(\mathrm{F})$, where $\mathrm{F}$ is a local non-archimedean field, in terms of hyperbolic localization functors of Braden. As a consequence, we can show that the image of an intertwining operator is always semi-simple and give a lower bound on the order of its pole, which is conjectured to be an equality as well as an upper bound in terms of the singular support of certain perverse sheaves. Moreover, we are able to reduce the conjecture of Lapid and Mínguez on the shape of irreducible subrepresentations of induced representations to a computation of characteristic cycles. The theory of mixed Hodge modules plays a crucial role in the proofs.

math.RT↗

arXiv2026

Poles of intertwining operators in terms of irreducible components of Lusztig's characteristic variety in type A

In this paper, we propose a conjectural formula for the order of the poles of intertwining operators in the context of the representation theory of general linear groups over $p$-adic fields. More specifically, we conjecturally relate the order of the pole to the dimension of a Hom-space associated with irreducible components of Lusztig's characteristic variety in type $A$. We verify the conjecture in a wide range of cases.

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arXiv2026

The $\ell$-modular local theta correspondence in type II and partial permutations

In this paper we compute the multiplicities appearing in the ${\overline{\mathbb{F}}_\ell}$-modular theta correspondence in type II over a non-archimedean field $\mathrm{F}$, where $\ell$ is a prime not dividing the residue cardinality of $\mathrm{F}$. Unlike for representations with complex coefficients, highly non-trivial multiplicities can emerge. We show that these multiplicities are precisely governed by the action of symmetric groups on the set of partial permutations, and the ${\overline{\mathbb{F}}_\ell}$-representation of symmetric groups these give rise to. The problem is thus reduced to certain branching problems in the modular representation theory of symmetric groups. In particular, if $d$ is the order of the residue cardinality of $\mathrm{F}$ in ${\overline{\mathbb{F}}_\ell}$, and the rank of the involved general linear groups is bounded above by $ d\ell$, the behavior of the theta correspondence can be predicted via explicit algorithms coming from Pieri's Formula.

math.RT↗

arXiv2026

Critical values of $L$-functions of residual representations of $\mathrm{GL}_4$

In this paper we prove rationality results of critical values for $L$-functions attached to representations in the residual spectrum of $\mathrm{GL}_4(\mathbb{A})$. We use the Jacquet-Langlands correspondence to describe their partial $L$-functions via cuspidal automorphic representations of the group $\mathrm{GL}_2'(\mathbb{A})$ over a quaternion algebra. Using ideas inspired by results of Grobner and Raghuram we are then able to compute the critical values as a Shalika period up to a rational multiple.

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arXiv2026

Lifting banal representations of classical groups

Let $\mathrm{G}$ be a symplectic or a split orthogonal group over a local non-archimedean field $\mathrm{F}$. A prime $\ell$ is called banal with respect to $\mathrm{G}$ if it does not divide the cardinality of the $k$-points of $\mathrm{G}$, where $k$ is the residue field of $\mathrm{F}$. In this paper we show that for every banal prime $\ell$, any smooth irreducible $\overline{\mathbb{F}}_\ell$-representation of $\mathrm{G}(\mathrm{F})$ admits a lift to $\overline{\mathbb{Q}}_\ell$. We also state similar results for more general classical groups of symplectic, orthogonal or unitary type. As an application we prove Howe-duality in the strongly banal case for symplectic-orthogonal or unitary dual pairs.

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arXiv2026

The generic extension map and modular standard modules

In this paper we study two classes of $\ell$-modular standard modules of the general linear group. The first class is obtained by reducing existing standard modules over $\overline{\mathbb{Q}}_\ell$ to $\overline{\mathbb{F}}_\ell$ with respect to their natural integral structure. The second class is obtained by studying the generic extension map of the cyclical quiver, which was motivated by the construction of certain monomial bases of quantum algebras. In the latter case we also manage to prove a modular version of the Langlands classification, similar to the work of Langlands and Zelevinsky over $\mathbb{C}$. We moreover compute the corresponding $\ell$-modular Rankin-Selberg $L$-functions and check that they agree with the $L$-functions of their $\mathrm{C}$-parameters constructed by Kurinczuk and Matringe.

math.RT↗

arXiv2025

A note on the Howe Duality conjecture for symplectic-orthogonal and unitary pairs

In this short note we expand on recent results on the degenerate principle series $I(s,χ)$ of classical groups associated to $s\in \mathbb{C}$ and a quadratic character $χ$. In particular, we strengthen the result for $s\in \mathbb{R}_{\ge 0}$, which allows us to give as a corollary a new proof of the Howe duality conjecture for symplectic-orthogonal and unitary pairs.

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arXiv2025

The spectrum of the symplectic Grassmannian and $\mathrm{Mat}_{n,m}$

Let $\mathbf{G}$ be a reductive group and $\mathbf{X}$ a spherical $\mathbf{G}$-variety over a local non-archimedean field $\mathbb{F}$. We denote by $S(\mathbf{X}(\mathbb{F}))$ the Schwartz-functions on $\mathbf{X}(\mathbb{F})$. In this paper we offer a new approach on how to obtain bounds on \[\dim_{\mathbb{C}}\mathrm{Hom}_{\mathbf{G}(\mathbb{F})}(S(\mathbf{X}(\mathbb{F})),π)\]for an irreducible smooth representation $π$ of $\mathbf{G}(\mathbb{F})$. Our strategy builds on the theory of $ρ$-derivatives and the Local Structure Theorem for spherical varieties. Currently, we focus on the case of the symplectic Grassmannian and the space of matrices. In particular, we obtain a new proof of Howe duality in type II as well as an explicit description of the local Miyawaki-liftings in the Hilbert-Siegel case. Furthermore, we manage to extend previous results of the author regarding the conservation relation in the theta correspondence to metaplectic covers of symplectic groups. Finally, we use our new proof of Howe duality in type II to relate the order of the poles of Godement-Jacquet $L$-functions to the geometry of the space of matrices and the order of poles of certain intertwining operators.

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arXiv2024

On modular representations of inner forms of $\mathrm{GL}_n$ over a local non-archimedean field

Let $\mathrm{F}$ be a local non-archimedean field of residue characteristic $p$ and $\overline{\mathbb{F}}_\ell$ an algebraic closure of a finite field of characteristic $\ell \neq p$. We extend the results of Lapid and Mínguez concerning $\square$-irreducible representations of inner forms of $\mathrm{GL}_n(\mathrm{F})$ to representations over $\overline{\mathbb{F}}_\ell$. As applications, we compute the Godement-Jacquet $L$-factor for any smooth irreducible representation over $\overline{\mathbb{F}}_\ell$ and show that the local factors of a representation agree with the ones of its $\mathrm{C}$-parameter defined by Kurinczuk and Matringe. Moreover, we reprove that the classification of irreducible representations via multisegments due to Vignéras and Mínguez-Sécherre is indeed exhaustive without using the classification of Ariki and Mathas of simple modules of Hecke algebras. Finally, we characterize the irreducible constituents of certain parabolically induced representations, as was already done by Zelevinsky over $\mathbb{C}$.

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arXiv2024

Proof of a conjecture of Kudla and Rallis on quotients of degenerate principal series

In this paper we prove a conjecture of Kudla and Rallis. Let $χ$ be a unitary character, $s\in \mathbb{C}$ and $W$ a symplectic vector space over a non-archimedean field with symmetry group $G(W)$. Denote by $I(χ,s)$ the degenerate principal series representation of $G(W\oplus W)$. Pulling back $I(χ,s)$ along the natural embedding $G(W)\times G(W)\hookrightarrow G(W\oplus W)$ gives a representation $I_{W,W}(χ,s)$ of $G(W)\times G(W)$. Let $π$ be an irreducible smooth complex representation of $G(W)$. We then prove \[\dim _\mathbb{C}\mathrm{Hom}_{G(W)\times G(W)}(I_{W,W}(χ,s),π\otimes π^\lor)=1.\] We also give analogous statements for $W$ orthogonal or unitary. This gives in particular a new proof of the conservation relation of the local Theta correspondence for symplectic-orthogonal and unitary dual pairs.

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