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Register a new userReductive groups, epsilon factors and Weil indices
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Authors
Robert E. Kottwitz
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The paper proves an identity involving Weil indices and epsilon factors for a local field. The starting point is a pair consisting of a reductive group and a maximal torus.
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We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive groups to the split reductive case and the pseudo-split pseudo-reductive commutative case. Moreover, we give the first results on the latter, including a rather complete description of the rank one case.
We introduce graded Hecke algebras H based on a (possibly disconnected) complex reductive group G and a cuspidal local system L on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and investigated by Lusztig for connected G. We develop the representation theory of the algebras H. obtaining complete and canonical parametrizations of the irreducible, the irreducible tempered and the discrete series representations. All the modules are constructed in terms of perverse sheaves and equivariant homology, relying on work of Lusztig. The parameters come directly from the data (G,M,L) and they are closely related to Langlands parameters. Our main motivation for considering these graded Hecke algebras is that the space of irreducible H-representations is canonically in bijection with a certain set of logarithms of enhanced L-parameters. Therefore we expect these algebras to play a role in the local Langlands program. We will make their relation with the local Langlands correspondence, which goes via affine Hecke algebras, precise in a sequel to this paper.
Let $F$ be either $mathbb{R}$ or a finite extension of $mathbb{Q}_p$, and let $G$ be a finite central extension of the group of $F$-points of a reductive group defined over $F$. Also let $pi$ be a smooth representation of $G$ (Frechet of moderate growth if $F=mathbb{R}$). For each nilpotent orbit $mathcal{O}$ we consider a certain Whittaker quotient $pi_{mathcal{O}}$ of $pi$. We define the Whittaker support WS$(pi)$ to be the set of maximal $mathcal{O}$ among those for which $pi_{mathcal{O}} eq 0$. In this paper we prove that all $mathcal{O}inmathrm{WS}(pi)$ are quasi-admissible nilpotent orbits, generalizing some of the results in [Moe96,JLS16]. If $F$ is $p$-adic and $pi$ is quasi-cuspidal then we show that all $mathcal{O}inmathrm{WS}(pi)$ are $F$-distinguished, i.e. do not intersect the Lie algebra of any proper Levi subgroup of $G$ defined over $F$. We also give an adaptation of our argument to automorphic representations, generalizing some results from [GRS03,Shen16,JLS16,Cai] and confirming some conjectures from [Ginz06]. Our methods are a synergy of the methods of the above-mentioned papers, and of our preceding paper [GGS17].
Geometric symmetry induces symmetries of function spaces, and the latter yields a clue to global analysis via representation theory. In this note we summarize recent developments on the general theory about how geometric conditions affect representation theoretic properties on function spaces, with focus on multiplicities and spectrum.
Let G be a connected reductive group over an algebraic closure of a finite field Fq. In this paper it is proved that the infinite dimensional Steinberg module of kG defined by N. Xi in 2014 is irreducible when k is a field of positive characteristic and char k is not char Fq. For certain special linear groups, we show that the Steinberg modules of the groups are not quasi-finite with respect to some natural quasi-finite sequences of the groups.
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