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Register a new userOn $L$-packets and depth for $SL_2(K)$ and its inner form
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We consider the group $SL_2(K)$, where $K$ is a local non-archimedean field of characteristic two. We prove that the depth of any irreducible representation of $SL_2 (K)$ is larger than the depth of the corresponding Langlands parameter, with equality if and only if the L-parameter is essentially tame. We also work out a classification of all $L$-packets for $SL_2 (K)$ and for its non-split inner form, and we provide explicit formulae for the depths of their $L$-parameters.
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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.
This paper begins the project of defining Arthur packets of all unipotent representations for the $p$-adic exceptional group $G_2$. Here we treat the most interesting case by defining and computing Arthur packets with component group $S_3$. We also show that the distributions attached to these packets are stable, subject to a hypothesis. This is done using a self-contained microlocal analysis of simple equivariant perverse sheaves on the moduli space of homogeneous cubics in two variables. In forthcoming work we will treat the remaining unipotent representations and their endoscopic classification and strengthen our result on stability.
Let $SL_2$ be the rank one simple algebraic group defined over an algebraically closed field $k$ of characteristic $p>0$. The paper presents a new method for computing the dimension of the cohomology spaces $text{H}^n(SL_2,V(m))$ for Weyl $SL_2$-modules $V(m)$. We provide a closed formula for $text{dim}text{H}^n(SL_2,V(m))$ when $nle 2p-3$ and show that this dimension is bounded by the $(n+1)$-th Fibonacci number. This formula is then used to compute $text{dim}text{H}^n(SL_2, V(m))$ for $n=1, 2,$ or $3$. For $n>2p-3$, an exponential bound, only depending on $n$, is obtained for $text{dim}text{H}^n(SL_2,V(m))$. Analogous results are also established for the extension spaces $text{Ext}^n_{SL_2}(V(m_2),V(m_1))$ between Weyl modules $V(m_1)$ and $V(m_2)$. In particular, we determine the degree three extensions for all Weyl modules of $SL_2$. As a byproduct, our results and techniques give explicit upper bounds for the dimensions of the cohomology of the Specht modules of symmetric groups, the cohomology of simple modules of $SL_2$, and the finite group of Lie type $SL_2(p^s)$.
We investigate various ways to define an analogue of BGG category $mathcal{O}$ for the non-semi-simple Takiff extension of the Lie algebra $mathfrak{sl}_2$. We describe Gabriel quivers for blocks of these analogues of category $mathcal{O}$ and prove extension fullness of one of them in the category of all modules.
Let $K$ be a non-archimedean local field. In the local Langlands correspondence for tori over $K$, we prove an asymptotic result for the depths.
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