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Alan Roche

Publications and source records attributed to Alan Roche.

4 recordsLinked to original sources
arXiv2019

A note on dual modules and the transpose

It is a classical result in matrix algebra that any square matrix over a field can be conjugated to its transpose by a symmetric matrix. For $F$ a non-Archimedean local field, Tupan used this to give an elementary proof that transpose inverse takes each irreducible smooth representation of ${\rm GL}_n(F)$ to its dual. We re-prove the matrix result and related observations using module-theoretic arguments. In addition, we write down a generalization that applies to central simple algebras with an involution of the first kind. We use this generalization to extend Tupan's method of argument to ${\rm GL}_n(D)$ for $D$ a quaternion division algebra over $F$.

math.RA↗
arXiv2016

A factorization result for classical and similitude groups

For most classical and similitude groups, we show that each element can be written as a product of two transformations that a) preserve or almost preserve the underlying form and b) whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well-known result of M{\oe}glin, Vign\'{e}ras and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representations to its dual.

math.RT↗
arXiv2012

Signs, involutions and Jacquet modules

Let $G$ be a connected reductive $p$-adic group and let $θ$ be an automorphism of $G$ of order at most two. Suppose $π$ is an irreducible smooth representation of $G$ that is taken to its dual by $θ$. The space $V$ of $π$ then carries a non-zero bilinear form $(\mspace{7mu},\mspace{6mu})$, unique up to scaling, with the invariance property $(π(g)v, π({}^θg)w) = (v,w)$, for $g \in G$ and $v, w \in V$. The form is easily seen to be symmetric or skew-symmetric and we set $\varepsilon_θ(π) = \pm1$ accordingly. We use Cassleman's pairing (in commonly observed circumstances) to express $\varepsilon_θ(π)$ in terms of certain Jacquet modules of $π$ and thus, via the Langlands classification, reduce the problem of determining the sign to the case of tempered representations. For the transpose-inverse involution of the general linear group, we show that the associated signs are always one.

math.RT↗