scholarly journals Generic Newton points and the Newton poset in Iwahori-double cosets

2020 ◽  
Vol 8 ◽  
Author(s):  
Elizabeth Milićević ◽  
Eva Viehmann
Keyword(s):  
Reductive Group ◽  
Loop Group ◽  
Index Set ◽  
Double Cosets ◽  
Large Classes ◽  
Bruhat Graph ◽  
Newton Stratification ◽  
Quantum Bruhat Graph

Abstract We consider the Newton stratification on Iwahori-double cosets in the loop group of a reductive group. We describe a group-theoretic condition on the generic Newton point, called cordiality, under which the Newton poset (that is, the index set for non-empty Newton strata) is saturated and Grothendieck’s conjecture on closures of the Newton strata holds. Finally, we give several large classes of Iwahori-double cosets for which this condition is satisfied by studying certain paths in the associated quantum Bruhat graph.

scholarly journals Closure relations of Newton strata in Iwahori double cosets

2021 ◽  
Author(s):  
Stefania Trentin ◽  
Eva Viehmann
Keyword(s):  
Reductive Group ◽  
Double Cosets ◽  
Closure Relations ◽  
Newton Stratification

AbstractWe consider the Newton stratification on Iwahori double cosets for a connected reductive group. We prove the existence of Newton strata whose closures cannot be expressed as a union of strata, and show how this is implied by the existence of non-equidimensional affine Deligne–Lusztig varieties. We also give an explicit example for a group of type $$A_4$$ A 4 .

scholarly journals Minimal Newton Strata in Iwahori Double Cosets

2019 ◽  
Author(s):  
Eva Viehmann
Keyword(s):  
Finite Subset ◽  
Minimal Element ◽  
Reductive Group ◽  
Double Coset ◽  
Loop Group ◽  
Conjugacy Classes ◽  
Regularity Assumption ◽  
Double Cosets

Abstract The set of Newton strata in a given Iwahori double coset in the loop group of a reductive group $G$ is indexed by a finite subset of the set $B(G)$ of Frobenius-conjugacy classes. For unramified $G$, we show that it has a unique minimal element and determine this element. Under a regularity assumption, we also compute the dimension of the corresponding Newton stratum. We derive corresponding results for affine Deligne–Lusztig varieties.

Regular Elliptic Classes and the Stable Relative Trace Formula

1992 ◽  
Vol 35 (2) ◽  
pp. 230-236
Author(s):  
K. F. Lai
Keyword(s):  
Number Field ◽  
Trace Formula ◽  
Algebraic Number ◽  
Reductive Group ◽  
Algebraic Number Field ◽  
Double Cosets ◽  
Relative Trace Formula

AbstractWe study the relative trace formula of a reductive group over an algebraic number field. Following Langlands we stabilize the geometric side of the relative trace formula contributed by the elliptic regular double cosets.

scholarly journals Maximal Newton polygons via the quantum Bruhat graph

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Elizabeth T. Beazley
Keyword(s):  
Weyl Group ◽  
Quantum Cohomology ◽  
Bruhat Order ◽  
Flag Variety ◽  
Necessary Condition ◽  
Newton Polygons ◽  
Weighted Directed Graph ◽  
Affine Weyl Group ◽  
Bruhat Graph ◽  
Quantum Bruhat Graph

International audience This paper discusses a surprising relationship between the quantum cohomology of the variety of complete flags and the partially ordered set of Newton polygons associated to an element in the affine Weyl group. One primary key to establishing this connection is the fact that paths in the quantum Bruhat graph, which is a weighted directed graph with vertices indexed by elements in the finite Weyl group, encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. In addition, using some geometry associated to the poset of Newton polygons, one obtains independent proofs for several combinatorial statements about paths in the quantum Bruhat graph and its symmetries, which were originally proved by Postnikov using the tilted Bruhat order. An important geometric application of this work is an inequality which provides a necessary condition for non-emptiness of certain affine Deligne-Lusztig varieties in the affine flag variety. Cet article étudie une relation surprenante entre la cohomologie quantique de la variété de drapeaux complets et l'ensemble partiellement ordonné de polygones de Newton associé à un élément du groupe de Weyl affine. L’élément clé pour établir cette connexion est le fait que les chemins dans le graphe de Bruhat quantique, qui est un graphe orienté pondéré dont les sommets sont indexés par des éléments du groupe de Weyl fini, encodent des chaînes saturées dans l'ordre de Bruhat fort sur le groupe de Weyl affine. Cette correspondance est aussi fondamentale dans les travaux de Lam et Shimonozo qui établissent l'isomorphisme de Peterson entre la cohomologie quantique de la variété de drapeaux finie et l'homologie de la Grassmannienne affine. De plus, en utilisant la géométrie associée à l'ensemble partiellement ordonné des polygones de Newton, on obtient des preuves indépendantes pour plusieurs assertions combinatoires sur les chemins dans le graphe de Bruhat quantiques et les symétries de ce graphe, qui ont été originellement démontrées par Postnikov en utilisant l'ordre de Bruhat incliné. Une application géométrique importante de ce travail est une inégalité qui donne une condition nécessaire pour que certaines variétés de Deligne-Lusztig affines dans la variété de drapeaux affine soient non-vides.

scholarly journals Uniform tail approximation of homogenous functionals of Gaussian fields

2017 ◽  
Vol 49 (4) ◽  
pp. 1037-1066 ◽  
Author(s):  
Krzysztof Dȩbicki ◽  
Enkelejd Hashorva ◽  
Peng Liu
Keyword(s):  
Gaussian Random Field ◽  
Positive Function ◽  
Additional Parameter ◽  
Gaussian Field ◽  
Gaussian Fields ◽  
Local Behavior ◽  
Index Set ◽  
Large Classes ◽  
Uniform Approximations ◽  
Piterbarg Constants

Abstract Let X(t), t ∈ ℝd, be a centered Gaussian random field with continuous trajectories and set ξu(t) = X(f(u)t), t ∈ ℝd, with f some positive function. Using classical results we can establish the tail asymptotics of ℙ{Γ(ξu) > u} as u → ∞ with Γ(ξu) = supt ∈ [0, T]d ξu(t), T > 0, by requiring that f(u) tends to 0 as u → ∞ with speed controlled by the local behavior of the correlation function of X. Recent research shows that for applications, more general functionals than the supremum should be considered and the Gaussian field can depend also on some additional parameter τu ∈ K say ξu,τu(t), t ∈ ℝd. In this paper we derive uniform approximations of ℙ{Γ(ξu,τu) > u} with respect to τu, in some index set Ku as u → ∞. Our main result has important theoretical implications; two applications are already included in Dȩbicki et al. (2016), (2017). In this paper we present three additional applications. First we derive uniform upper bounds for the probability of double maxima. Second, we extend the Piterbarg–Prisyazhnyuk theorem to some large classes of homogeneous functionals of centered Gaussian fields ξu. Finally, we show the finiteness of generalized Piterbarg constants.

scholarly journals Inverse K-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type

2021 ◽  
Vol 9 ◽  
Author(s):  
Takafumi Kouno ◽  
Satoshi Naito ◽  
Daniel Orr ◽  
Daisuke Sagaki
Keyword(s):  
Flag Manifolds ◽  
Line Bundles ◽  
Affine Hecke Algebra ◽  
Double Affine Hecke Algebra ◽  
Bruhat Graph ◽  
Schubert Classes ◽  
Explicit Determination ◽  
Schubert Class ◽  
Quantum Bruhat Graph

Abstract We prove an explicit inverse Chevalley formula in the equivariant K-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb {Z}\left [q^{\pm 1}\right ]$ -linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars $e^{\lambda }$ , where $\lambda $ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type $E_8$ . The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetric q-Toda operators for minuscule weights in ADE type.

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