Extended Deligne–Lusztig varieties for general and special linear groups

Stasinski, Alexander

doi:10.1016/j.aim.2010.10.010

Cited by 17 publications

(18 citation statements)

References 30 publications

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“…In this paper we develop some of the structure theory of these algebraic groups. These results allow for a smoother treatment of parts of the construction in [18], and are necessary (but not sufficient) for a generalisation of the construction in [19] beyond general and special linear groups. The algebraic groups we consider are extensions of reductive groups by connected unipotent groups, and as such are generally not reductive.…”

Section: Introductionmentioning

confidence: 99%

Reductive group schemes, the Greenberg functor, and associated algebraic groups

Stasinski

2012

Journal of Pure and Applied Algebra

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Let A be an Artinian local ring with algebraically closed residue field k, and let G be an affine smooth group scheme over A. The Greenberg functor F associates to G a linear algebraic group G := (F G)(k) over k, such that G ∼ = G(A). We prove that if G is a reductive group scheme over A, and T is a maximal torus of G, then T is a Cartan subgroup of G, and every Cartan subgroup of G is obtained uniquely in this way. Moreover, we prove that if G is reductive and P is a parabolic subgroup of G, then P is a selfnormalising subgroup of G, and if B and B ′ are two Borel subgroups of G, then the corresponding subgroups B and B ′ are conjugate in G. Errata notesA previous version of the present paper has appeared in J. Pure Appl. Algebra., 216 (2012), 1092-1101. After the publication the author was notified by Cristian D. González-Avilés and Alessandra Bertapelle that the formula on p. 1094, l. 14 in the published version does not hold in general. More precisely, in [8], p. 636 Greenberg defined the local ring scheme A over k (using different notation). However, the formula A(R) = A ⊗ Wm(k) W m (R) does not hold for all k-algebras R. Nevertheless, it does hold when R is a perfect kalgebra. The formula occurs in [2], p. 276, l. -18, but was stated correctly by Loeser and Sebag [14], p. 318. Moreover, Nicaise and Sebag have given counter-examples for non-perfect algebras; see [16], 2.2.In the present paper we have corrected the statements involving the above formula by either removing the formula or adding the hypothesis that R is a perfect k-algebra. The corrections correspond to p. 1094, l. 14 and l. -17, as well as the second line of the proof of Lemma 2.3, in the published version. The formula is actually not necessary for the results of our paper, and can simply be ignored. All that is needed is that A is an affine local ring scheme over k, and that it is isomorphic to some affine space. As mentioned above, these facts were established by Greenberg.Apart from these modifications, the content of the present version remains identical to the published version.

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Section: Introductionmentioning

confidence: 99%

Reductive group schemes, the Greenberg functor, and associated algebraic groups

Stasinski

2012

Journal of Pure and Applied Algebra

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“…As a representation of G F = SL 2 (F p ), the structure of the space S 2 (Γ(p)) + S 2 (Γ(p)) depends on p mod 12, and it can be written as a linear combination of R θ T for various (T, θ) with θ| Z = 1 (hence uniform in the sense of [Lus78, 2.15]), whose coefficients can be chosen to be rational linear polynomials in p: see e.g. [Lus04], [Sta11], and [Che18]. Moreover, Weinstein's formula (2) still holds, and there are also possible candidates of the Steinberg representation, like the ones given in [Lee78] and [Cam07].…”

Section: Comparing the Spacesmentioning

confidence: 99%

A note on cusp forms and representations of $\mathrm{SL}_2(\mathbb{F}_p)$

Chen

2018

Preprint

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Cusp forms are certain holomorphic functions defined on the upper half-plane, and the space of cusp forms for the principal congruence subgroup Γ(p), p a prime, is acted by SL 2 (F p ). Meanwhile, there is a finite field incarnation of the upper half-plane, the Deligne-Lusztig (or Drinfeld) curve, whose cohomology space is also acted by SL 2 (F p ). In this note we study the relation between these two spaces in the weight 2 case.

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“…Representations of open compact subgroups of reductive groups over local fields have received much attention in the past two decades. One approach, taken by Lusztig and Stasinski, is to generalise Deligne-Lusztig theory [12] (which is itself a generalisation of the Harish-Chandra theory) to such groups; see [34,44], and also [6] and [35]. Another approach, taken by Hill [18][19][20][21], consists of a direct Clifford-theoretic analysis of representations according to their restrictions to congruence kernels.…”

Section: Related Constructionsmentioning

confidence: 99%

A Variant of Harish-Chandra Functors

Crisp

Meir

Onn

2017

J. Inst. Math. Jussieu

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ABSTRACT. Harish-Chandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors which apply to a wide range of finite and profinite groups, typical examples being compact open subgroups of reductive groups over non-archimedean local fields. We prove that these generalisations are compatible with two of the tools commonly used to study the (smooth, complex) representations of such groups, namely Clifford theory and the orbit method. As a test case, we examine in detail the induction and restriction of representations from and to the Siegel Levi subgroup of the symplectic group Sp 4 over a finite local principal ideal ring of length two. We obtain in this case a Mackey-type formula for the composition of these induction and restriction functors which is a perfect analogue of the well-known formula for the composition of Harish-Chandra functors. In a different direction, we study representations of the Iwahori subgroup In of GLnpFq, where F is a non-archimedean local field. We establish a bijection between the set of irreducible representations of In and tuples of primitive irreducible representations of smaller Iwahori subgroups, where primitivity is defined by the vanishing of suitable restriction functors.

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Extended Deligne–Lusztig varieties for general and special linear groups

Cited by 17 publications

References 30 publications

Reductive group schemes, the Greenberg functor, and associated algebraic groups

Reductive group schemes, the Greenberg functor, and associated algebraic groups

A note on cusp forms and representations of $\mathrm{SL}_2(\mathbb{F}_p)$

A Variant of Harish-Chandra Functors

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