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Register a new userAlgebraically Independent Generators for the Algebra of Invariant Differential Operators on $mathrm{SL}_n(mathbb R)/mathrm{SO}_n(mathbb R)$
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We provide an explicit set of algebraically independent generators for the algebra of invariant differential operators on the Riemannian symmetric space associated with $SL_n(R)$.
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We prove an estimate for spherical functions $phi_lambda(a)$ on $mathrm{SL}(3,mathbb{R})$, establishing uniform decay in the spectral parameter $lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $mathrm{A}$. In the case of $mathrm{SL}(3,mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $lambda$ and $a$ vary.
We show that Fourier coefficients of automorphic forms attached to minimal or next-to-minimal automorphic representations of ${mathrm{SL}}_n(mathbb{A})$ are completely determined by certain highly degenerate Whittaker coefficients. We give an explicit formula for the Fourier expansion, analogously to the Piatetski-Shapiro-Shalika formula. In addition, we derive expressions for Fourier coefficients associated to all maximal parabolic subgroups. These results have potential applications for scattering amplitudes in string theory.
We generalize Bonahon and Wongs $mathrm{SL}_2(mathbb{C})$-quantum trace map to the setting of $mathrm{SL}_3(mathbb{C})$. More precisely, for each non-zero complex number $q$, we associate to every isotopy class of framed oriented links $K$ in a thickened punctured surface $mathfrak{S} times (0, 1)$ a Laurent polynomial $mathrm{Tr}_lambda^q(K) = mathrm{Tr}_lambda^q(K)(X_i^q)$ in $q$-deformations $X_i^q$ of the Fock-Goncharov coordinates $X_i$ for a higher Teichm{u}ller space, depending on the choice of an ideal triangulation $lambda$ of the surface $mathfrak{S}$. Along the way, we propose a definition for a $mathrm{SL}_n(mathbb{C})$-version of this invariant.
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.
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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