arXiv subjects

Joshua M. Lansky

Publications and source records attributed to Joshua M. Lansky.

8 recordsLinked to original sources

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 $\widetilde G$.

math.RT↗

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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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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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.

math.RT↗

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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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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arXiv2009

Klyachko models of p-adic special linear groups

We study Klyachko models of ${\rm SL}(n, F)$, where $F$ is a nonarchimedean local field. In particular, using results of Klyachko models for ${\rm GL}(n, F)$ due to Heumos, Rallis, Offen and Sayag, we give statements of existence, uniqueness, and disjointness of Klyachko models for admissible representations of ${\rm SL}(n, F)$, where the uniqueness and disjointness are up to specified conjugacy of the inducing character, and the existence is for unitarizable representations in the case $F$ has characteristic 0. We apply these results to relate the size of an $L$-packet containing a given representation of ${\rm SL}(n, F)$ to the type of its Klyachko model, and we describe when a self-dual unitarizable representation of ${\rm SL}(n, F)$ is orthogonal and when it is symplectic.

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arXiv2006

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

We give an explicit description of L-packets and quadratic base change for depth-zero representations of unramified unitary groups in two and three variables. We show that this base change is compatible with unrefined minimal K-types.

math.RT↗