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Register a new userContragredient representations over local fields of positive characteristic
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Wen-Wei Li
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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 verify 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.
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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.
We describe examples motivated by the work of Serre and Abhyankar.
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--Sahi, 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.
The cohomology of the degree-$n$ general linear group over a finite field of characteristic $p$, with coefficients also in characteristic $p$, remains poorly understood. For example, the lowest degree previously known to contain nontrivial elements is exponential in $n$. In this paper, we introduce a new system of characteristic classes for representations over finite fields, and use it to construct a wealth of explicit nontrivial elements in these cohomology groups. In particular we obtain nontrivial elements in degrees linear in $n$. We also construct nontrivial elements in the mod $p$ homology and cohomology of the automorphism groups of free groups, and the general linear groups over the integers. These elements reside in the unstable range where the homology and cohomology remain poorly understood.
We give upper bounds for the level and the Pythagoras number of function fields over fraction fields of integral Henselian excellent local rings. In particular, we show that the Pythagoras number of $mathbb{R}((x_1,dots,x_n))$ is $leq 2^{n-1}$, which answers positively a question of Choi, Dai, Lam and Reznick.
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