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Register a new userGeneralized Poincare series for $mathrm{SU}(2,1)$
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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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Let $L$ be a finite extension of $mathbb{Q}_p$, and $rho_L$ be an $n$-dimensional semi-stable non crystalline $p$-adic representation of $mathrm{Gal}_L$ with full monodromy rank. Via a study of Breuils (simple) $mathcal{L}$-invariants, we attach to $rho_L$ a locally $mathbb{Q}_p$-analytic representation $Pi(rho_L)$ of $mathrm{GL}_n(L)$, which carries the exact information of the Fontaine-Mazur simple $mathcal{L}$-invariants of $rho_L$. When $rho_L$ comes from an automorphic representation of $G(mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real filed $F^+$ which is compact at infinite places and $mathrm{GL}_n$ at $p$-adic places), we prove under mild hypothesis that $Pi(rho_L)$ is a subrerpresentation of the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G(mathbb{A}_{F^+})$. In other words, we prove the equality of Breuils simple $mathcal{L}$-invariants and Fontaine-Mazur simple $mathcal{L}$-invariants.
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.
Let $p>2$ be a prime number, and $L$ be a finite extension of $mathbb{Q}_p$, we prove Breuils locally analytic socle conjecture for $mathrm{GL}_2(L)$, showing the existence of all the companion points on the definite (patched) eigenvariety. This work relies on infinitesimal R=T results for the patched eigenvariety and the comparison of (partially) de Rham families and (partially) Hodge-Tate families. This method allows in particular to find companion points of non-classical points.
In this article we study the homology of spaces ${rm Hom}(mathbb{Z}^n,G)$ of ordered pairwise commuting $n$-tuples in a Lie group $G$. We give an explicit formula for the Poincare series of these spaces in terms of invariants of the Weyl group of $G$. By work of Bergeron and Silberman, our results also apply to ${rm Hom}(F_n/Gamma_n^m,G)$, where the subgroups $Gamma_n^m$ are the terms in the descending central series of the free group $F_n$. Finally, we show that there is a stable equivalence between the space ${rm Comm}(G)$ studied by Cohen-Stafa and its nilpotent analogues.
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 $mathrm{GL}_n(mathbb{Q}_p)$.
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