arXiv subjects

Jessica Fintzen

Publications and source records attributed to Jessica Fintzen.

At least 19 recordsLinked to original sources

arXiv2026

Parametrization and reduction to depth zero of $\overline{\mathbb{Z}}[\frac{1}{p}]$-blocks of tame $p$-adic groups

Let $G$ be a reductive group over a non-archimedean local field $F$ of residue characteristic $p$. We consider pairs $(\phi,I)$ consisting of a "wild inertia" Langlands parameter $\phi: P_F \longrightarrow \hat{G}$ whose centralizer $C_{\hat{G}}(\phi)$ is a Levi subgroup of $\hat{G}$, and a cohomological invariant $I$ whose definition is inspired by the theory of endoscopy. Assuming that $p$ is odd and not a torsion prime of $G$ nor of $\hat{G}$, we associate to each such pair $(\phi,I)$ a Serre subcategory $\mathrm{Rep}^{\phi,I}(G(F))$ of the category of smooth $\overline{\mathbb{Z}}[\frac{1}{p}]$-representations of $G(F)$. Then we construct an equivalence between this Serre subcategory and the category of depth-zero $\overline{\mathbb{Z}}[\frac{1}{p}]$-representations of a twisted Levi subgroup $G_{\phi,I}$ of $G$, which is dual to $C_{\hat{G}}(\phi)$. This pattern for reduction to depth zero fits well with the conjectural (categorical) local Langlands correspondence. When $G$ is tamely ramified and $p$ does not divide the order of its Weyl group, then the above Serre subcategories provide the block decomposition of the category of all smooth $\overline{\mathbb{Z}}[\frac{1}{p}]$-representations of $G(F)$. In this case, we thus obtain a reduction-to-depth-zero process for smooth representations of $G(F)$ valued in any algebraically closed field of characteristic different from $p$. When that field has characteristic 0, this recovers some of the recent results of Adler--Fintzen--Mishra--Ohara. When that field is $\overline{\mathbb{F}}_{\ell}$, we use our results together with Zhu's unipotent categorical correspondence to produce a fully faithful embedding of $D\mathrm{Rep}_{\overline{\mathbb{F}}_{\ell}}(\mathrm{GL}_{n}(F))$ into a suitable category of coherent sheaves on the moduli space of $n$-dimensional $\overline{\mathbb{F}}_{\ell}$-representations of the Weil group.

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arXiv2025

Supercuspidal representations: construction, classification, and characters

The building blocks for irreducible smooth representations of p-adic groups are the supercuspidal representations. In these notes that are an expansion of a lecture series given during the IHES summer school 2022 we will explore an explicit exhaustive construction of these supercuspidal representations and their character formulas and observe a striking parallel between a large class of these representations and discrete series representations of real algebraic Lie groups. A key ingredient for the construction of supercuspidal representations is the Bruhat-Tits theory and Moy-Prasad filtration, which we will recall in this survey.

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arXiv2025

An introduction to representations of p-adic groups

An explicit understanding of the (category of all smooth, complex) representations of p-adic groups provides an important tool not just within representation theory. It also has applications to number theory and other areas, and, in particular, it enables progress on various different forms of the Langlands program. In this write-up of the author's ECM 2024 colloquium-style talk, we will introduce p-adic groups and explain how the category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks. We also provide an overview of what we know about the structure of these Bernstein blocks including a sketch of recent results of the author with Adler, Mishra and Ohara that allow to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or are at least easier to achieve. Moreover, we provide an overview of what is known about the construction of supercuspidal representations, which are the building blocks of all smooth representations and whose construction is also the key to obtain the above results about the structure of the whole category of smooth representations. We will, in particular, focus on recent advances which include the work of the author mentioned in the EMS prize citation as well as a hint towards her recent joint work with David Schwein.

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arXiv2025

A depth-zero principal-series block whose Hecke algebra has a non-trivial two-cocycle

Recently the authors have shown that every Hecke algebra associated to a type constructed by Kim and Yu is isomorphic to a Hecke algebra for a depth-zero type. An example in the literature has been suggested as a counterexample to this result. We show that the example is not a counterexample, and exhibit some of its interesting properties, e.g., we show that a principal series, depth-zero type can have a Hecke algebra with non-trivial two-cocyle, a phenomenon that many did not expect could occur.

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arXiv2025

Construction of tame supercuspidal representations in arbitrary residue characteristic

Let F be a nonarchimedean local field whose residue field has at least four elements. Let G be a connected reductive group over F that splits over a tamely ramified field extension of F. We provide a construction of supercuspidal representations of G(F) via compact induction that contains, among others, all the supercuspidal representations constructed by Yu in 2001, but that also works in residual characteristic two. The input for our construction is described uniformly for all residual characteristics and is analogous to Yu's input except that we do not require our input to satisfy the second genericity condition (GE2) that Yu imposes.

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arXiv2024

Structure of Hecke algebras arising from types

Let $G$ denote a connected reductive group over a nonarchimedean local field $F$ of residue characteristic $p$, and let $\mathcal{C}$ denote an algebraically closed field of characteristic $\ell \neq p$. If $\rho$ is an irreducible, smooth $\mathcal{C}$-representation of a compact, open subgroup $K$ of $G(F)$, then the pair $(K,\rho)$ gives rise to a Hecke algebra $\mathcal{H}(G(F),(K, \rho))$. For a large class of pairs $(K,\rho)$, we show that $\mathcal{H}(G(F),(K, \rho))$ is a semi-direct product of an affine Hecke algebra with explicit parameters with a twisted group algebra, and that it is isomorphic to $\mathcal{H}(G^0(F),(K^0, \rho^0))$ for some reductive subgroup $G^0 \subset G$ with compact, open subgroup $K^0$ and depth-zero representation $\rho^0$ of $K^0$. The class of pairs that we consider includes all depth-zero types. In describing their Hecke algebras, we thus recover a result of Morris as a special case. In a second paper, we will show that our class also contains all the types constructed by Kim and Yu, and hence we obtain as a corollary that arbitrary Bernstein blocks are equivalent to depth-zero Bernstein blocks under minor tameness assumptions. The pairs to which our results apply are described in an axiomatic way so that the results can be applied to other constructions of types by only verifying that the relevant axioms are satisfied. The Hecke algebra isomorphisms are given in an explicit manner and are support preserving.

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arXiv2024

Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms

Let $F$ be a nonarchimedean local field of residual characteristic $p$. Let $G$ denote a connected reductive group over $F$ that splits over a tamely ramified extension of $F$. Let $(K ,\rho)$ be a type as constructed by Kim and Yu. We show that there exists a twisted Levi subgroup $G^0 \subset G$ and a type $(K^0, \rho^0)$ for $G^0$ such that the corresponding Hecke algebras $\mathcal{H}(G(F), (K, \rho))$ and $\mathcal{H}(G^0(F), (K^0, \rho^0))$ are isomorphic. If $p$ does not divide the order of the absolute Weyl group of $G$, then every Bernstein block is equivalent to modules over such a Hecke algebra. Hence, under this assumption on $p$, our result implies that every Bernstein block is equivalent to a depth-zero Bernstein block. This allows one to reduce many problems about (the category of) smooth, complex representations of $p$-adic groups to analogous problems about (the category of) depth-zero representations. Our isomorphism of Hecke algebras is very explicit and also includes an explicit description of the Hecke algebras as semi-direct products of an affine Hecke with a twisted group algebra. Moreover, we work with arbitrary algebraically closed fields of characteristic different from $p$ as our coefficient field. This paper relies on a prior axiomatic result about the structure of Hecke algebras by the same authors and a key ingredient consists of extending the quadratic character of Fintzen--Kaletha--Spice to the support of the Hecke algebra, which might be of independent interest.

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arXiv2022

Supercuspidal representations in non-defining characteristics

We show that a mod-$\ell$-representation of a p-adic group arising from the analogue of Yu's construction is supercuspidal if and only if it arises from a supercuspidal representation of a finite reductive group. This has been previously shown by Henniart and Vigneras under the assumption that the second adjointness holds, a statement that is not yet available in the literature.

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arXiv2021

A twisted Yu construction, Harish-Chandra characters, and endoscopy

We give a modification of Yu's construction of supercuspidal representations of a connected reductive group over a non-archimedean local field. This modification restores the validity of certain key intertwining property claims made by Yu, which were recently proven to be false for the original construction. This modification is also an essential ingredient in the explicit construction of supercuspidal L-packets. As further applications, we prove the stability and many instances of endoscopic character identities of these supercuspidal L-packets, subject to some conditions on the base field. In particular, for regular supercuspidal parameters we prove all instances of standard endoscopy. In addition, we prove that these supercuspidal L-packets satisfy a certain property, which, together with standard endoscopy, uniquely characterizes the local Langlands correspondence for supercuspidal L-packets (again subject to the above mentioned conditions on the base field). These results are based on a statement of the Harish-Chandra character formula for the supercuspidal representations arising from the twisted Yu construction.

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arXiv2021

Tame cuspidal representations in non-defining characteristics

Let F be a non-archimedean local field of odd residual characteristic p. Let G be a (connected) reductive group that splits over a tamely ramified field extension of F. We show that a construction analogous to Yu's construction of complex supercuspidal representations yields smooth, irreducible, cuspidal representations over an arbitrary algebraically closed field R of characteristic different from p. Moreover, we prove that this construction provides all smooth, irreducible, cuspidal R-representations if p does not divide the order of the Weyl group of G.

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arXiv2021

Congruences of algebraic automorphic forms and supercuspidal representations

Let $G$ be a connected reductive group over a totally real field $F$ which is compact modulo center at archimedean places. We find congruences modulo an arbitrary power of p between the space of arbitrary automorphic forms on $G(\mathbb A_F)$ and that of automorphic forms with supercuspidal components at p, provided that p is larger than the Coxeter number of the absolute Weyl group of $G$. We illustrate how such congruences can be applied in the construction of Galois representations. Our proof is based on type theory for representations of p-adic groups, generalizing the prototypical case of GL(2) in [arXiv:1506.04022, Section 7] to general reductive groups. We exhibit a plethora of new supercuspidal types consisting of arbitrarily small compact open subgroups and characters thereof. We expect these results of independent interest to have further applications. For example, we extend the result by Emerton--Paškūnas on density of supercuspidal points from definite unitary groups to general $G$ as above.

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arXiv2021

On certain sign characters of tori and their extensions to Bruhat-Tits groups

We consider two sign characters defined on a tamely ramified maximal torus T of a twisted Levi subgroup M of a reductive p-adic group G. We show that their product extends to the stabilizer M(F)_x of any point x in the Bruhat-Tits building of T, and give a formula for this extension. This result is used in the passage between zero and positive depth in the explicit construction of supercuspidal L-packets, as well as in forthcoming work on the Harish-Chandra character formula for supercuspidal representations.

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arXiv2020

Types for tame p-adic groups

Let k be a non-archimedean local field with residual characteristic p. Let G be a connected reductive group over k that splits over a tamely ramified field extension of k. Suppose p does not divide the order of the Weyl group of G. Then we show that every smooth irreducible complex representation of G(k) contains an $\mathfrak{s}$-type of the form constructed by Kim and Yu and that every irreducible supercuspidal representation arises from Yu's construction. This improves an earlier result of Kim, which held only in characteristic zero and with a very large and ineffective bound on p. By contrast, our bound on p is explicit and tight, and our result holds in positive characteristic as well. Moreover, our approach is more explicit in extracting an input for Yu's construction from a given representation.

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arXiv2019

On the construction of tame supercuspidal representations

Let F be a non-archimedean local field of odd residual characteristic. Let G be a (connected) reductive group over F that splits over a tamely ramified field extension of F. We revisit Yu's construction of smooth complex representations of G(F) from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in [Yu01], whose proofs relied on a typo in a reference.

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arXiv2019

On the Moy-Prasad filtration

Let K be a maximal unramified extension of a nonarchimedean local field with arbitrary residual characteristic p. Let G be a reductive group over K which splits over a tamely ramified extension of K. We show that the associated Moy-Prasad filtration representations are in a certain sense independent of p. We also establish descriptions of these representations in terms of explicit Weyl modules and as representations occurring in a generalized Vinberg-Levy theory. As an application, we use these results to provide necessary and sufficient conditions for the existence of stable vectors in Moy-Prasad filtration representations, which extend earlier results by Reeder and Yu (which required p to be large) and by Romano and the author (which required G to be absolutely simple and split). This yields new supercuspidal representations. We also treat reductive groups G that are not necessarily split over a tamely ramified field extension.

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arXiv2018

Differential operators and families of automorphic forms on unitary groups of arbitrary signature

In the 1970's, Serre exploited congruences between $q$-expansion coefficients of Eisenstein series to produce $p$-adic families of Eisenstein series and, in turn, $p$-adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to $p$-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on $q$-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack $q$-expansions when the signature is of the form $(a, b)$, $a\neq b$. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a $p$-adic measure taking values in the space of $p$-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit $p$-adic family of Eisenstein series. One application of our results is to the recently completed construction of $p$-adic $L$-functions for unitary groups by the first named author, Harris, Li, and Skinner.

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arXiv2018

On Kostant Sections and Topological Nilpotence

Let G denote a connected, quasi-split reductive group over a field F that is complete with respect to a discrete valuation and that has a perfect residue field. Under mild hypotheses, we produce a subset of the Lie algebra g(F) that picks out a G(F)-conjugacy class in every stable, regular, topologically nilpotent conjugacy class in g(F). This generalizes an earlier result obtained by DeBacker and one of the authors under stronger hypotheses. We then show that if F is p-adic, then the characteristic function of this set behaves well with respect to endoscopic transfer.

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arXiv2018

Tame tori in p-adic groups and good semisimple elements

Let G be a reductive group over a non-archimedean local field k. We provide necessary conditions and sufficient conditions for all tori of G to split over a tamely ramified extension of k. We then show the existence of good semisimple elements in every Moy-Prasad filtration coset of the group G(k) and its Lie algebra, assuming the above sufficient conditions are met.

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