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Register a new userOn the Gan-Gross-Prasad problem for finite classical groups
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Zhicheng Wang
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In this paper we study the Gan-Gross-Prasad problem for finite classical groups. Our results provide complete answers for unipotent representations, and we obtain the explicit branching laws for these representations. Moreover, for arbitrary representations, we give a formula to reduce the Gan-Gross-Prasad problem to the restriction problem of Deligne-Lusztig characters.
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In previous work, we study the Gan-Gross-Prasad problem for unipotent representations of finite classical groups. In this paper, we deduce the Gan-Gross-Prasad problem for arbitrary representations from the unipotent representations by Lusztig correspondence.
Through combining the work of Jean-Loup Waldspurger (cite{waldspurger10} and cite{waldspurgertemperedggp}) and Raphael Beuzart-Plessis (cite{beuzart2015local}), we give a proof for the tempered part of the local Gan-Gross-Prasad conjecture (cite{ggporiginal}) for special orthogonal groups over any local fields of characteristic zero, which was already proved by Waldspurger over $p$-adic fields.
We prove an analogue of the celebrated Hall-Higman theorem, which gives a lower bound for the degree of the minimal polynomial of any semisimple element of prime power order $p^{a}$ of a finite classical group in any nontrivial irreducible cross characteristic representation. With a few explicit exceptions, this degree is at least $p^{a-1}(p-1)$.
Let $k$ be an algebraically closed field of characteristic $p > 2$, let $n in mathbb Z_{>0} $, and take $G$ to be one of the classical algebraic groups $mathrm{GL}_n(k)$, $mathrm{SL}_n(k)$, $mathrm{Sp}_n(k)$, $mathrm O_n(k)$ or $mathrm{SO}_n(k)$, with $mathfrak g = operatorname{Lie} G$. We determine the maximal $G$-stable closed subvariety $mathcal V$ of the nilpotent cone $mathcal N$ of $mathfrak g$ such that the $G$-orbits in $mathcal V$ are in bijection with the $G$-orbits of $mathfrak{sl}_2$-triples $(e,h,f)$ with $e,f in mathcal V$. This result determines to what extent the theorems of Jacobson--Morozov and Kostant on $mathfrak{sl}_2$-triples hold for classical algebraic groups over an algebraically closed field of ``small odd characteristic.
The normalizer $N_G(H_G)$ of a maximal torus $H_G$ in a semisimple complex Lie group $G$ does not in general allow a presentation as a semidirect product of $H_G$ and the corresponding Weyl group $W_G$. Meanwhile, splitting holds for classical groups corresponding to the root systems $A_ell$, $B_ell$, $D_ell$. For the remaining classical groups corresponding to the root systems $C_ell$ there still exists an embedding of the Tits extension of $W_G$ into normalizer $N_G(H_G)$. We provide explicit unified construction of the lifts of the Weyl groups into normalizers of maximal tori for classical Lie groups corresponding to the root systems $A_ell$, $B_ell$, $D_ell$ using embeddings into general linear Lie groups. For symplectic series of classical Lie groups we provide an explanation of impossibility of embedding of the Weyl group into the symplectic group. The explicit formula for adjoint action of the lifts of the Weyl groups on $mathfrak{g}={rm Lie}(G)$ are given. Finally some examples of the groups closely associated with classical Lie groups are considered.
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