Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Signs, involutions and Jacquet modules   Let $G$ be a connected reductive $p$-adic group and let $\theta$ be an
automorphism of $G$ of order at most two. Suppose $\pi$ is an irreducible
smooth representation of $G$ that is taken to its dual by $\theta$. The space
$V$ of $\pi$ then carries a non-zero bilinear form
$(\mspace{7mu},\mspace{6mu})$, unique up to scaling, with the invariance
property $(\pi(g)v, \pi({}^{\theta}g)w) = (v,w)$, for $g \in G$ and $v, w \in
V$. The form is easily seen to be symmetric or skew-symmetric and we set
$\varepsilon_\theta(\pi) = \pm1$ accordingly. We use Cassleman's pairing (in
commonly observed circumstances) to express $\varepsilon_\theta(\pi)$ in terms
of certain Jacquet modules of $\pi$ 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.

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The categories of this paper are: hep-th
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-th