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Register a new userSome questions on global distinction for $mathrm{SL}(n)$
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Let $E/F$ be a quadratic extension of number fields and let $pi$ be an $mathrm{SL}_n(mathbb{A}_F)$-distinguished cuspidal automorphic representation of $mathrm{SL}_n(mathbb{A}_E)$. Using an unfolding argument, we prove that an element of the $mathrm{L}$-packet of $pi$ is distinguished if and only if it is $psi$-generic for a non-degenerate character $psi$ of $N_n(mathbb{A}_E)$ trivial on $N_n(E+mathbb{A}_F)$, where $N_n$ is the group of unipotent upper triangular matrices of $mathrm{SL}_n$. We then use this result to analyze the non-vanishing of the period integral on different realizations of a distinguished cuspidal automorphic representation of $mathrm{SL}_n(mathbb{A}_E)$ with multiplicity $> 1$, and show that in general some canonical copies of a distinguished representation inside different $mathrm{L}$-packets can have vanishing period. We also construct examples of everywhere locally distinguished representations of $mathrm{SL}_n(mathbb{A}_E)$ the $mathrm{L}$-packets of which do not contain any distinguished representation.
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We study some closed rigid subspaces of the eigenvarieties, constructed by using the Jacquet-Emerton functor for parabolic non-Borel subgroups. As an application (and motivation), we prove some new results on Breuils locally analytic socle conjecture for $mathrm{GL}_n(mathbb{Q}_p)$.
Let $p$ be a prime number. We prove that the $P=W$ conjecture for $mathrm{SL}_p$ is equivalent to the $P=W$ conjecture for $mathrm{GL}_p$. As a consequence, we verify the $P=W$ conjecture for genus 2 and $mathrm{SL}_p$. For the proof, we compute the perverse filtration and the weight filtration for the variant cohomology associated with the $mathrm{SL}_p$-Hitchin moduli space and the $mathrm{SL}_p$-twisted character variety, relying on Grochenig-Wyss-Zieglers recent proof of the topological mirror conjecture by Hausel-Thaddeus. Finally we discuss obstructions of studying the cohomology of the $mathrm{SL}_n$-Hitchin moduli space via compact hyper-Kahler manifolds.
If $E/F$ is a quadratic extension $p$-adic fields, we first prove that the $mathrm{SL}_n(F)$-distinguished representations inside a distinguished unitary L-packet of $mathrm{SL}_n(E)$ are precisely those admitting a degenerate Whittaker model with respect to a degenerate character of $N(E)/N(F)$. Then we establish a global analogue of this result. For this, let $E/F$ be a quadratic extension of number fields and let $pi$ be an $mathrm{SL}_n(mathbb{A}_F)$-distinguished square integrable automorphic representation of $mathrm{SL}_n(mathbb{A}_E)$. Let $(sigma,d)$ be the unique pair associated to $pi$, where $sigma$ is a cuspidal representation of $mathrm{GL}_r(mathbb{A}_E)$ with $n=dr$. Using an unfolding argument, we prove that an element of the L-packet of $pi$ is distinguished with respect to $mathrm{SL}_n(mathbb{A}_F)$ if and only if it has a degenerate Whittaker model for a degenerate character $psi$ of type $r^d:=(r,dots,r)$ of $N_n(mathbb{A}_E)$ which is trivial on $N_n(E+mathbb{A}_F)$, where $N_n$ is the group of unipotent upper triangular matrices of $mathrm{SL}_n$. As a first application, under the assumptions that $E/F$ splits at infinity and $r$ is odd, we establish a local-global principle for $mathrm{SL}_n(mathbb{A}_F)$-distinction inside the L-packet of $pi$. As a second application we construct examples of distinguished cuspidal automorphic representations $pi$ of $mathrm{SL}_n(mathbb{A}_E)$ such that the period integral vanishes on some canonical copy of $pi$, and of everywhere locally distinguished representations of $mathrm{SL}_n(mathbb{A}_E)$ such that their L-packets do not contain any distinguished representation.
Let $rho_p$ be a $3$-dimensional $p$-adic semi-stable representation of $mathrm{Gal}(overline{mathbb{Q}_p}/mathbb{Q}_p)$ with Hodge-Tate weights $(0,1,2)$ (up to shift) and such that $N^2 e 0$ on $D_{mathrm{st}}(rho_p)$. When $rho_p$ comes from an automorphic representation $pi$ of $G(mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real field $F^+$ which is compact at infinite places and $mathrm{GL}_3$ at $p$-adic places), we show under mild genericity assumptions that the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G(mathbb{A}_{F^+}^infty)$ of arbitrary fixed tame level contain (copies of) a unique admissible finite length locally analytic representation of $mathrm{GL}_3(mathbb{Q}_p)$ which only depends on and completely determines $rho_p$.
We define and study non-abelian Poincare series for the group $G=mathrm{SU} (2,1)$, i.e. Poincare series attached to a Stone-Von Neumann representation of the unipotent subgroup $N$ of $G$. Such Poincare series have in general exponential growth. In this study we use results on abelian and non-abelian Fourier term modules obtained in arXiv:1912.01334. We compute the inner product of truncations of these series and those associated to unitary characters of $N$ with square integrable automorphic forms, in connection with their Fourier expansions. As a consequence, we obtain general completeness results that, in particular, generalize those valid for the classical holomorphic (and antiholomorphic) Poincare series for $mathrm{SL}(2,mathbb{R})$.
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