Hyperplane arrangements associated
to symplectic quotient singularities

Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverte www.centre-mersenne.org Ann. Inst. Fourier, Grenoble Article à paraître Mis en ligne le 7 juin 2021. FIXED POINTS IN SMOOTH CALOGERO-MOSER SPACES by Cédric BONNAFÉ & Ruslan MAKSIMAU (*)Abstract. -We prove that every irreducible component of the fixed point variety under the action of µ d in a smooth Calogero-Moser space is isomorphic to a Calogero-Moser space associated with another reflection group.Résumé. -Nous montrons que toute composante irréductible de la variété des points fixes sous l'action de µ d dans un espace de Calogero-Moser lisse est isomorphe à un espace de Calogero-Moser associé à un autre groupe de réflexions.

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“…The next result follows from [2], but we provide here a different proof (which works only in type G(d, 1, n)).…”

Section: Quiver Varieties Vs Calogero-moser Spacesmentioning

confidence: 94%

Fixed points in smooth Calogero–Moser spaces

Bonnafé¹,

Maksimau²

2021

Annales De l'Institut Fourier

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“…Generic parameters. Even though we dropped this from the notation, the representation theory of H c , and thus the decomposition matrices in Definition 2.5, depend on the parameter c. In [ What is important for us is that for the complex reflection groups G(ℓ, 1, n) the two types of generic parameters coincide by [Thi17, Cor 3.22] and the hyperplane arrangement of non-generic parameter is explicitly known, see [BST18]. We furthermore have the following important property of simple modules at generic parameters.…”

Section: Definition 22 ([Eg02]mentioning

confidence: 99%

“…It is shown in[Thi16;Thi18] that both types of generic parameters form a non-empty Zariski open subset of the parmater space and that block-generic parameters are decomposition-generic. Moreover, it follows from[BST18] that the set of block-generic parameters is the complement of a finite hyperplane arrangement.…”

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confidence: 99%

Wreath Macdonald polynomials at q=t as characters of rational Cherednik algebras

Mathiä¹,

Thiel²

2021

Preprint

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Using the theory of Macdonald [Mac95], Gordon [Gor03] showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg [EG02] associated to the symmetric group Sn are given by plethystically transformed Macdonald polynomials specialized at q = t. We generalize this to restricted rational Cherednik algebras of wreath product groups C ℓ ≀ Sn and prove that the corresponding characters are given by a specialization of the wreath Macdonald polynomials defined by Haiman in [Hai03].

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“…The representation theory of H c is closely tied to the geometry of Z c and plays a key role in the question of whether (V × V * )/W admits a symplectic (equivalently, crepant) resolution, see [EtGi,GiKa,Nam1,BST1], and in determining the chamber decomposition of the movable cone of a Q-factorial terminalization (i.e. a relative minimal model) of (V × V * )/W , see [Nam2,BST2].…”

Section: Introductionmentioning

confidence: 99%

“…The Calogero-Moser hyperplanes are now known for all exceptional complex reflection groups except G 16 −G 19 , G 21 , and G 32 (see Table 5.C). Because of [Nam2,BST2] we thus know in all these cases the chamber decomposition of the movable cone of a Q-factorial terminalization (i.e. a relative minimal model) of the associated symplectic singularity (V ×V * )/W and we know the number of non-isomorphic relative minimal models.…”

Section: Introductionmentioning

confidence: 99%

Computational aspects of Calogero-Moser spaces

Bonnafé¹,

Thiel²

2021

Preprint

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We present a series of algorithms for computing geometric and representationtheoretic invariants of Calogero-Moser spaces and rational Cherednik algebras associated to complex reflection groups. Especially, we are concerned with Calogero-Moser families (which correspond to the C × -fixed points of the Calogero-Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig's constructible characters based on a Galois covering of the Calogero-Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package (CHAMP) by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a Q-factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity.

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Hyperplane arrangements associated to symplectic quotient singularities

Cited by 8 publications

References 23 publications

Fixed points in smooth Calogero–Moser spaces

Fixed points in smooth Calogero–Moser spaces

Wreath Macdonald polynomials at q=t as characters of rational Cherednik algebras

Computational aspects of Calogero-Moser spaces

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