ترغب بنشر مسار تعليمي؟ اضغط هنا

Local parameters of supercuspidal representations

159   0   0.0 ( 0 )
 نشر من قبل Michael Harris
 تاريخ النشر 2021
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

Let $G$ be a connected reductive group over the non-archime-dean local field $F$ and let $pi$ be a supercuspidal representation of $G(F)$. The local Langlands conjecture posits that to such a $pi$ can be attached a parameter $L(pi)$, which is an equivalence class of homomorphisms from the Weil-Deligne group with values in the Langlands $L$-group ${}^LG$ over an appropriate algebraically closed field $C$ of characteristic $0$. When $F$ is of positive characteristic $p$ then Genestier and Lafforgue have defined a parameter, $L^{ss}(pi)$, which is a homomorphism $W_F ra {}^LG(C)$ that is {it semisimple} in the sense that, if the image of $L^{ss}(pi)$, intersected with the Langlands dual group $hat{G}(C)$, is contained in a parabolic subgroup $P subset hat{G}(C)$, then it is contained in a Levi subgroup of $P$. If the Frobenius eigenvalues of $L^{ss}(pi)$ are pure in an appropriate sense, then the local Langlands conjecture asserts that the image of $L^{ss}(pi)$ is in fact {it irreducible} -- its image is contained in no proper parabolic $P$. In particular, unless $G = GL(1)$, $L^{ss}(pi)$ is ramified: it is non-trivial on the inertia subgroup $I_F subset W_F$. In this paper we prove, at least when $G$ is split and semisimple, that this is the case provided $pi$ can be obtained as the induction of a representation of a compact open subgroup $U subset G(F)$, and provided the constant field of $F$ is of order greater than $3$. Conjecturally every $pi$ is compactly induced in this sense, and the property was recently proved by Fintzen to be true as long as $p$ does not divide the order of the Weyl group of $G$. The proof is an adaptation of the globalization method of cite{GLo} when the base curve is $PP^1$, and a simple application of Delignes Weil II.



قيم البحث

اقرأ أيضاً

162 - Koichi Takase 2021
The formal degree conjecture and the root number conjecture are verified with respect to supercuspidal representations of $Sp_{2n}(F)$ and $L$-parameters associated with tamely ramified extension $K/F$ of degree $2n$. The supercuspidal representation is constructed as a compact induction from an irreducible unitary representation of the hyper special compact group $Sp_{2n}(O_F)$, which is explicitly constructed, based upon the general theory developed by the author, by $K$ and certain character $theta$ of the multiplicative group $K^{times}$. $L$-parameter is constructed by the data ${K,theta}$ by means of the local Langlands correspondence of tori and Langlands-Schelstad procedure.
103 - Wen-Wei Li 2018
It is conjectured by Adams-Vogan and Prasad that under the local Langlands correspondence, the L-parameter of the contragredient representation equals that of the original representation composed with the Chevalley involution of the L-group. We verif y a variant of their prediction for all connected reductive groups over local fields of positive characteristic, in terms of the local Langlands parameterization of Genestier-Lafforgue. We deduce this from a global result for cuspidal automorphic representations over function fields, which is in turn based on a description of the transposes of V. Lafforgues excursion operators.
We prove that the local components of an automorphic representation of an adelic semisimple group have equal rank in the sense defined earlier by the second author. Our theorem is an analogue of the results previously obtained by Howe, Li, Dvorsky--S ahi, and Kobayashi--Savin. Unlike previous works which are based on explicit matrix realizations and existence of parabolic subgroups with abelian unipotent radicals, our proof works uniformly for all of the (classical as well as exceptional) groups under consideration. Our result is an extension of the statement known for several semisimple groups that if at least one local component of an automorphic representation is a minimal representation, then all of its local components are minimal.
We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / QQ_p$ we will construct a bijection [ CL_g : CA^0_g(G_2,K) rightarrow CG^0(G_2,K) ] from the set of generic supercuspidal representations of $G_2(K)$ to the set of irreducible continuous homomorphisms $rho : W_K to G_2(CC)$ with $W_K$ the Weil group of $K$. The construction of the map is simply a matter of assembling arguments that are already in the literature, together with a previously unpublished theorem of G. Savin on exceptional theta correspondences, included as an appendix. The proof that the map is a bijection is arithmetic in nature, and specifically uses automorphy lifting theorems. These can be applied thanks to a recent result of Hundley and Liu on automorphic descent from $GL(7)$ to $G_2$.
We study the dimension of the space of Whittaker functionals for depth zero representations of covering groups. In particular, we determine such dimensions for arbitrary Brylinski-Deligne coverings of the general linear group. The results in the pape r are motivated by and compatible with the work of Howard and the second author, and earlier work by Blondel.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا