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Jeffrey D. Adler

Publications and source records attributed to Jeffrey D. Adler.

At least 19 recordsLinked to original sources
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

On smooth-group actions on reductive groups and spherical buildings

Let $k$ be a field, and suppose that $Γ$ is a smooth $k$-group that acts on a connected, reductive $k$-group $\widetilde G$. Let $G$ denote the maximal smooth, connected subgroup of the group of $Γ$-fixed points in $\widetilde G$. Under fairly general conditions, we show that $G$ is a reductive $k$-group, and that the image of the functorial embedding $\mathscr{S}(G) \longrightarrow \mathscr{S}(\widetilde G)$ of spherical buildings is the set of ``$Γ$-fixed points in $\mathscr{S}(\widetilde G)$'', in a suitable sense. In particular, we do not need to assume that $Γ$ has order relatively prime to the characteristic of $k$ (nor even that $Γ$ is finite), nor that the action of $Γ$ preserves a Borel-torus pair in $\widetilde G$.

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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 $ρ$ is an irreducible, smooth $\mathcal{C}$-representation of a compact, open subgroup $K$ of $G(F)$, then the pair $(K,ρ)$ gives rise to a Hecke algebra $\mathcal{H}(G(F),(K, ρ))$. For a large class of pairs $(K,ρ)$, we show that $\mathcal{H}(G(F),(K, ρ))$ is a semi-direct product of an affine Hecke algebra with explicit parameters with a twisted group algebra, and that it is isomorphic to $\mathcal{H}(G^0(F),(K^0, ρ^0))$ for some reductive subgroup $G^0 \subset G$ with compact, open subgroup $K^0$ and depth-zero representation $ρ^0$ 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 $(K ,ρ)$ 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 $(K^0, ρ^0)$ for $G^0$ such that the corresponding Hecke algebras $\mathcal{H}(G(F), (K, ρ))$ and $\mathcal{H}(G^0(F), (K^0, ρ^0))$ 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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arXiv2023

Lifting representations of finite reductive groups II: Explicit conorms

Let $k$ be a field, $\tilde{G}$ a connected reductive $k$-group, and $Γ$ a finite group. In a previous work, the authors defined what it means for a connected reductive $k$-group $G$ to be "parascopic" for $(\tilde{G},Γ)$. Roughly, this is a simultaneous generalization of several settings. For example, $Γ$ could act on $\tilde{G}$, and $G$ could be the connected part of the group of $Γ$-fixed points in $\tilde{G}$. Or $G$ could be an endoscopic group, a pseudo-Levi subgroup, or an isogenous image of $\tilde{G}$. If $G$ is such a group, and both $\tilde{G}$ and $G$ are $k$-quasisplit, then we constructed a map $\hat{\mathcal{N}}^{\text{st}}$ from the set of stable semisimple conjugacy classes in the dual $G^\wedge(k)$ to the set of such classes in $\tilde{G}^\wedge(k)$. When $k$ is finite, this implies a lifting from packets of representations of $G(k)$ to those of $\tilde{G}(k)$. In order to understand such a lifting better, here we describe two ways in which $\hat{\mathcal{N}}^{\text{st}}$ can be made more explicit. First, we can express our map in the general case in terms of simpler cases. We do so by showing that $\hat{\mathcal{N}}^{\text{st}}$ is compatible with isogenies and with Weil restriction, and also by expressing it as a composition of simpler maps. Second, in many cases we can construct an explicit $k$-morphism $\hat N \colon G^\wedge \longrightarrow \tilde{G}^\wedge$ that agrees with $\hat{\mathcal{N}}^{\text{st}}$. As a consequence, our lifting of representations is seen to coincide with Shintani lifting in some important cases.

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arXiv2021

Regular Bernstein blocks

For a connected reductive group $G$ defined over a non-archimedean local field $F$, we consider the Bernstein blocks in the category of smooth representations of $G(F)$. Bernstein blocks whose cuspidal support involves a regular supercuspidal representation are called $\textit{regular}$ Bernstein blocks. Most Bernstein blocks are regular when the residual characteristic of $F$ is not too small. Under mild hypotheses on the residual characteristic, we show that the Bernstein center of a regular Bernstein block of $G(F)$ is isomorphic to the Bernstein center of a regular depth-zero Bernstein block of $G^{0}(F)$, where $G^{0}$ is a certain twisted Levi subgroup of $G$. In some cases, we show that the blocks themselves are equivalent, and as a consequence we prove the ABPS Conjecture in some new cases.

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arXiv2020

Self-dual cuspidal representations

Let $G$ be a connected reductive group over a finite field $\mathfrak{f}$ of order $q$. When $q$ is small, we make further assumptions on $G$. Then we determine precisely when $G(\mathfrak{f})$ admits irreducible, cuspidal representations that are self-dual, of Deligne-Lusztig type, or both. Finally, we outline some consequences for the existence of self-dual supercuspidal representations of reductive $p$-adic groups.

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arXiv2019

Multiplicity upon restriction to the derived subgroup

We present a conjecture on multiplicity of irreducible representations of a subgroup $H$ contained in the irreducible representations of a group $G$, with $G$ and $H$ having the same derived groups. We point out some consequences of the conjecture, and verification of some of the consequences. We give an explicit example of multiplicity $2$ upon restriction, as well as certain theorems in the context of classical groups where the multiplicity is $1$.

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arXiv2019

Root data with group actions

Suppose $k$ is a field, $G$ is a connected reductive algebraic $k$-group, $T$ is a maximal $k$-torus in $G$, and $Γ$ is a finite group that acts on $(G,T)$. From the above, one obtains a root datum $Ψ$ on which $\text{Gal}(k)\timesΓ$ acts. Provided that $Γ$ preserves a positive system in $Ψ$, not necessarily invariant under $\text{Gal}(k)$, we construct an inverse to this process. That is, given a root datum on which $\text{Gal}(k)\timesΓ$ acts appropriately, we show how to construct a pair $(G,T)$, on which $Γ$ acts as above. Although the pair $(G,T)$ and the action of $Γ$ are canonical only up to an equivalence relation, we construct a particular pair for which $G$ is $k$-quasisplit and $Γ$ fixes a $\text{Gal}(k)$-stable pinning of $G$. Using these choices, we can define a notion of taking "$Γ$-fixed points" at the level of equivalence classes, and this process is compatible with a general "restriction" process for root data with $Γ$-action.

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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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arXiv2015

Lifting representations of finite reductive groups: a character relation

Given a connected reductive group $\tilde{G}$ over a finite field $k$, and a semisimple $k$-automorphism $\varepsilon$ of $\tilde{G}$ of finite order, let $G$ denote the connected part of the group of $\varepsilon$-fixed points. Then there exists a lifting from packets of representations of $G(k)$ to packets for $\tilde{G}(k)$. In the case of Deligne-Lusztig representations, we show that this lifting satisfies a character relation analogous to that of Shintani.

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arXiv2014

Lifting representations of finite reductive groups I: Semisimple conjugacy classes

Suppose that $\tilde{G}$ is a connected reductive group defined over a field $k$, and $Γ$ is a finite group acting via $k$-automorphisms of $\tilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the connected part of the group of $Γ$-fixed points in $\tilde{G}$ is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair $(\tilde{G},Γ)$, and consider any group $G$, not just the $Γ$-fixed points of $\tilde{G}$, satisfying the axioms. (In fact, the axioms do not require $Γ$ to act on all of $\tilde{G}$.) If both $\tilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals $\tilde{G}^*$ and $G^*$. We show the existence of and give an explicit formula for a natural map from semisimple stable conjugacy classes in $G^*(k)$ to those in $\tilde{G}^*(k)$. If $k$ is finite, then our groups are automatically quasisplit, and our result specializes to give a map from semisimple conjugacy classes in $G^*(k)$ to those in $\tilde{G}^*(k)$. Since such classes parametrize packets of irreducible representations of $G(k)$ and $\tilde{G}(k)$, one obtains a mapping of such packets.

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arXiv2010

Supercuspidal characters of $\operatorname{SL}_2$ over a $p$-adic field

The character formulas of Sally and Shalika are an early triumph in $p$-adic harmonic analysis, but, to date, the calculations underlying the formulas have not been available. In this paper, which should be viewed as a precursor of the forthcoming volume by the authors and Alan Roche, we leverage modern technology (for example, the Moy-Prasad theory) to compute explicit character tables. An interesting highlight is the computation of the 'exceptional' supercuspidal characters, i.e., those depth-zero representations not arising by inflation-induction from a Deligne-Lusztig representation of finite $\operatorname{SL}_2$; this provides a concrete application for the recent work of DeBacker and Kazhdan.

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arXiv2009

Supercuspidal characters of reductive p-adic groups

We compute the characters of many supercuspidal representations of reductive p-adic groups. Specifically, we deal with representations that arise via Yu's construction from data satisfying a certain compactness condition. Each character is expressed in terms of a depth-zero character of a smaller group, the (linear) characters appearing in Yu's construction, Fourier transforms of orbital integrals, and certain signs and cardinalities that are described explicitly in terms of the datum associated to the representation and of the element at which the character is evaluated.

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arXiv2009

Depth-zero base change for ramified U(2,1)

We give an explicit description of L-packets and quadratic base change for depth-zero representations of ramified unitary groups in two and three variables. We show that this base change lifting is compatible with a certain lifting of families of representations of finite groups. We conjecture that such a compatibility is valid in much greater generality.

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