86 research outputs found
Dimensions of Affine Deligne-Lusztig Varieties in Affine Flag Varieties
No full textAffine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper [5] by Haines, Kottwitz, Reuman, and the first named author, about the question which affine Deligne-Lusztig varieties (for a split group and a basic σ-conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a classical group and the chosen basic σ-conjugacy class is the class of b = 1, we also prove the dimension formula predicted in op. cit. in almost all cases.link_to_subscribed_fulltex
A Deligne complex for Artin monoids
No full textIn this paper we introduce and study some geometric objects associated toArtin monoids. The Deligne complex for an Artin group is a cube complex thatwas introduced by the second author and Davis (1995) to study the K(\pi,1)conjecture for these groups. Using a notion of Artin monoid cosets, weconstruct a version of the Deligne complex for Artin monoids. We show that forany Artin monoid this cube complex is contractible. Furthermore, we study theembedding of the monoid Deligne complex into the Deligne complex for thecorresponding Artin group. We show that for any Artin group this is a locallyisometric embedding. In the case of FC-type Artin groups this result can bestrengthened to a globally isometric embedding, and it follows that the monoidDeligne complex is CAT(0) and its image in the Deligne complex is convex. Wealso consider the Cayley graph of an Artin group, and investigate properties ofthe subgraph spanned by elements of the Artin monoid. Our final results showthat for a finite type Artin group, the monoid Cayley graph embedsisometrically, but not quasi-convexly, into the group Cayley graph.<br
DIMENSIONS OF AFFINE DELIGNE-LUSZTIG VARIETIES IN AFFINE FLAG VARIETIES
No full textAffine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper [5] by Haines, Kottwitz, Reuman, and the first named author, about the question which affine Deligne-Lusztig varieties (for a split group and a basic sigma-conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a classical group and the chosen basic sigma-conjugacy class is the class of b = 1, we also prove the dimension formula predicted in op. cit. in almost all cases
Energy of sections of the Deligne–Hitchin twistor space
No full textWe study a natural functional on the space of holomorphic sections of the Deligne–Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We show that the energy is the residue of the pull-back along the section of a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. As a byproduct, we show the existence of a hyper-Kähler potentials for new components of real holomorphic sections of twistor spaces of hyper-Kähler manifolds with rotating S1-action. Additionally, we prove that for a certain class of real holomorphic sections of the Deligne–Hitchin moduli space, the energy functional is basically given by the Willmore energy of corresponding equivariant conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne–Hitchin moduli space from the space of twistor lines. © 2020, The Author(s)
Motivic integration over wild Deligne-Mumford stacks
No full textWe develop the motivic integration theory over formal Deligne-Mumford stacks
over a power series ring of arbitrary characteristic. This is a generalization
of the corresponding theory for tame and smooth Deligne-Mumford stacks
constructed in earlier papers of the author. As an application, we obtain the
wild motivic McKay correspondence for linear actions of arbitrary finite
groups, which has been known only for cyclic groups of prime order. In
particular, this implies the motivic version of Bhargava's mass formula as a
special case. In fact, we prove a more general result, the invariance of
stringy motives of (stacky) log pairs under crepant morphisms.Comment: 84 pages; Introduction and Section 16 have been revised. Minor
changes in other place
Deligne–Lusztig duality and wonderful compactification
No full text© 2018, Springer International Publishing AG, part of Springer Nature. We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne–Lusztig (or Alvis–Curtis) duality for p-adic groups and homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for G= GL(n) and by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct, we describe the Serre functor for representations of a p-adic group
Deligne–Lusztig duality and wonderful compactification
No full textAbstract
We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne–Lusztig (or Alvis–Curtis) duality for p-adic groups and homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for
G
=
G
L
(
n
)
and by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct, we describe the Serre functor for representations of a p-adic group
Translation by the full twist and Deligne-Lusztig varieties
No full textInternational audienceWe prove several conjectures about the cohomology of Deligne-Lusztig varieties: invariance under conjugation in the braid group, behaviour with respect to translation by the full twist, parity vanishing of the cohomology for the variety associated with the full twist. In the case of split groups of type A, and using previous results of the second author, this implies Broué-Michel's conjecture on the disjointness of the cohomology for the variety associated to any good regular element. That conjecture was inspired by Broué's abelian defect group conjecture and the specific form Broué conjectured for finite groups of Lie type
The Deligne-Mostow 9-ball, and the monster
No full textThe "monstrous proposal" of the first author is that the quotient of a
certain 13-dimensional complex hyperbolic braid group, by the relations that
its natural generators have order 2, is the bimonster" (M x M)semidirect Z/2.
Here M is the monster simple group. We prove that this quotient is either the
bimonster or Z/2. In the process, we give new information about the isomorphism
found by Deligne-Mostow, between the moduli space of 12-tuples in CP1 and a
quotient of the complex 9-ball. Namely, we identify which loops in the 9-ball
quotient correspond to the standard braid generators
Translation by the full twist and Deligne-Lusztig varieties
No full textInternational audienceWe prove several conjectures about the cohomology of Deligne-Lusztig varieties: invariance under conjugation in the braid group, behaviour with respect to translation by the full twist, parity vanishing of the cohomology for the variety associated with the full twist. In the case of split groups of type A, and using previous results of the second author, this implies Broué-Michel's conjecture on the disjointness of the cohomology for the variety associated to any good regular element. That conjecture was inspired by Broué's abelian defect group conjecture and the specific form Broué conjectured for finite groups of Lie type
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