arXiv subjects

Paul Sobaje

Publications and source records attributed to Paul Sobaje.

25 records · Page 2Linked to original sources

arXiv2016

Varieties of $G_r$-summands in Rational $G$-modules

Let $G$ be a simple simply connected algebraic group over an algebraically closed field $k$ of characteristic $p$, with $r$-th Frobenius kernel $G_r$. Let $M$ be a $G_r$-module and $V$ a rational $G$-module. We put a variety structure on the set of all $G_r$-summands of $V$ that are isomorphic to $M$, and study basic properties of these varieties. We give a few applications of this work to the representation theory of $G$, primarily in providing some sufficient conditions for when a $G_r$-module decomposition of $V$ can be extended to a $G$-module decomposition. In particular we are interested in connections to Donkin's tilting module conjecture, and more generally to the problem of finding a $G$-structure for the projective indecomposable $G_r$-modules. To that end, we show that Donkin's conjecture is equivalent to determining the linearizability or non-linearizability of $G$-actions on certain affine spaces.

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arXiv2015

On Liftings of Projective Indecomposable $G_{(1)}$-Modules

Let $G$ be a simple simply connected algebraic group over an algebraically closed field $k$ of characteristic $p$, with Frobenius kernel $G_{(1)}$. It is known that when $p\ge 2h-2$, where $h$ is the Coxeter number of $G$, the projective indecomposable $G_{(1)}$-modules (PIMs) lift to $G$, and this has been conjectured to hold in all characteristics. In this paper, we explore the lifting problem via extensions of algebraic groups, following the work of Parshall and Scott who in turn build upon ideas due to Donkin. We prove various results which augment this approach, and as an application demonstrate that the PIMs lift to $G_{(1)}H$, for particular closed subgroups $H \le G$ which contain a maximal torus of $G$.

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arXiv2015

Springer Isomorphisms In Characteristic $p$

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p$, and assume that $p$ is a very good prime for $G$. Let $P$ be a parabolic subgroup whose unipotent radical $U_P$ has nilpotence class less than $p$. We show that there exists a particularly nice Springer isomorphism for $G$ which restricts to a certain canonical isomorphism $\text{Lie}(U_P) \xrightarrow{\sim} U_P$ defined by J.-P. Serre. This answers a question raised both by G. McNinch in \cite{M2}, and by J. Carlson \textit{et. al} in \cite{CLN}. For the groups $SL_n, SO_n$, and $Sp_{2n}$, viewed in the usual way as subgroups of $GL_n$ or $GL_{2n}$, such a Springer isomorphism can be given explicitly by the Artin-Hasse exponential series.

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arXiv2014

Exponentiation of commuting nilpotent varieties

Let $H$ be a linear algebraic group over an algebraically closed field of characteristic $p>0$. We prove that any "exponential map" for $H$ induces a bijection between the variety of $r$-tuples of commuting $[p]$-nilpotent elements in $Lie(H)$ and the variety of height $r$ infinitesimal one-parameter subgroups of $H$. In particular, we show that for a connected reductive group $G$ in pretty good characteristic, there is a canonical exponential map for $G$ and hence a canonical bijection between the aforementioned varieties, answering in this case questions raised both implicitly and explicitly by Suslin, Friedlander, and Bendel.

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arXiv2012

Support Varieties for Frobenius Kernels of Classical Groups

Let G be a simple classical algebraic group over an algebraically closed field of positive characteristic. We describe the support variety of a simple G-module over the r-th Frobenius kernel of G, in terms of its calculation over the first Frobenius kernel. We then use this result to compute the block varieties of the Frobenius kernels of G.

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arXiv2012

On exponentiation and infinitesimal one-parameter subgroups of reductive groups

Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of characteristic $p>0$, and assume $p$ is good for $G$. Let $P$ be a parabolic subgroup with unipotent radical $U$. For $r \ge 1$, denote by $\mathbb{G}_{a(r)}$ the $r$-th Frobenius kernel of $\mathbb{G}_a$. We prove that if the nilpotence class of $U$ is less than $p$, then any embedding of $\mathbb{G}_{a(r)}$ in $U$ lies inside a one-parameter subgroup of $U$, and there is a canonical way in which to choose such a subgroup. Applying this result, we prove that if $p$ is at least as big as the Coxeter number of $G$, then the cohomological variety of $G_{(r)}$ is homeomorphic to the variety of $r$-tuples of commuting elements in $\mathcal{N}_1(\mathfrak{g})$, the $[p]$-nilpotent cone of $Lie(G)$.

math.GR↗

arXiv2012

Module Invariants and Blocks of Finite Group Schemes

We investigate various topological spaces and varieties which can be associated to a block of a finite group scheme G. These spaces come from the theory of cohomological support varieties for modules, as well as from the representation-theoretic constructions of E. Friedlander and J. Pevtsova.

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