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Register a new userOn the Archimedean Local Gamma Factors for Adjoint Representation of $mathrm{GL}_3$, Part I
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Fangyang Tian
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Studying the analytic properties of the partial Langlands $L$-function via Rankin-Selberg method has been proved to be successful in various cases. Yet in few cases is the local theory studied at the archimedean places, which causes a tremendous gap to complete the analytic theory of the complete $L$-function. In this paper, we will establish the meromorphic continuation and the functional equation of the archimedean local integrals associated with D. Ginzburgs global integral for the adjoint representation of $mathrm{GL}_3$. Via the local functional equation, the local gamma factor $Gamma(s,pi,mathrm{Ad},psi)$ can be defined. In a forthcoming paper, we will compute the local gamma factor $Gamma(s,pi,mathrm{Ad},psi)$ explicitly, which fills in some blanks in the archimedean local theory of Ginzburgs global integral.
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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 $rho_p$.
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 $L$ be a finite extension of $mathbb{Q}_p$ and $ngeq 2$. We associate to a crystabelline $n$-dimensional representation of $mathrm{Gal}(overline L/L)$ satisfying mild genericity assumptions a finite length locally $mathbb{Q}_p$-analytic representation of $mathrm{GL}_n(L)$. In the crystalline case and in a global context, using the recent results on the locally analytic socle from [BHS17a] we prove that this representation indeed occurs in spaces of $p$-adic automorphic forms. We then use this latter result in the ordinary case to show that certain ordinary $p$-adic Banach space representations constructed in our previous work appear in spaces of $p$-adic automorphic forms. This gives strong new evidence to our previous conjecture in the $p$-adic case.
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 $p>2$ 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.
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