Heisenberg idempotents on unipotent groups

Deshpande, Tanmay

doi:10.4310/mrl.2010.v17.n3.a4

Cited by 10 publications

(22 citation statements)

References 3 publications

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“…So X is invertible, and since X ⊗ X ∼ = X , we see that X ∼ = 1, proving the lemma. [15,Lemma 1.4], the -equivariantization M 0 is equivalent to M. By [15,Theorem 1.3], M 0 is also a fusion category, so we see that e L is the only simple object of M 0 (indeed, every simple object of M 0 can be realized as a direct summand of a simple object of M 0 ). On the other hand, all simple objects of M 0 are described in [15, §4.1], and that description implies that U is connected and that ϕ L is an isomorphism.…”

Section: Remarks 84 (1)mentioning

confidence: 97%

“…One can show that the converse of Proposition 8.1(c) holds as well (see [15]), but we do not need this fact.…”

Section: Remark 82mentioning

confidence: 99%

“…11 The notion of a Heisenberg idempotent is the geometric analogue of the notion of a Heisenberg representation of a finite group, introduced in [10]. 12 The results of [15] use the construction of the arrow 1 −→ e L presented in Sect. 8.1.2, but are otherwise independent of the current section.…”

Section: Proof Of Proposition 81(b)mentioning

confidence: 99%

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Characters of unipotent groups over finite fields

Boyarchenko

2010

Sel. Math. New Ser.

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Let G be a connected unipotent group over a finite field F q . In this article, we propose a definition of L-packets of complex irreducible representations of the finite group G(F q ) and give an explicit description of L-packets in terms of the so-called admissible pairs for G. We then apply our results to show that if the centralizer of every geometric point of G is connected, then the dimension of every complex irreducible representation of G(F q ) is a power of q, confirming a conjecture of Drinfeld. This paper is the first in a series of three papers exploring the relationship between representations of a group of the form G(F q ) (where G is a unipotent algebraic group over F q ), the geometry of G, and the theory of character sheaves.

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Section: Remarks 84 (1)mentioning

confidence: 97%

“…One can show that the converse of Proposition 8.1(c) holds as well (see [15]), but we do not need this fact.…”

Section: Remark 82mentioning

confidence: 99%

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Characters of unipotent groups over finite fields

Boyarchenko

2010

Sel. Math. New Ser.

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“…This associativity constraint is not quite canonical, however this does not matter in the proof of the next lemma. Moreover, note that by [De1] each eD U (U a) is the bounded derived category of a finite semisimple abelian category and D is equivalent to the bounded derived category of a fusion category.…”

Section: Tanmay Deshpandementioning

confidence: 99%

Character sheaves on neutrally solvable groups

Deshpande

2017

Represent. Theory

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Abstract. Let G be an algebraic group over an algebraically closed field k of characteristic p > 0. In this paper we develop the theory of character sheaves on groups G such that their neutral connected components G • are solvable algebraic groups. For such algebraic groups G (which we call neutrally solvable) we will define the set CS(G) of character sheaves on G as certain special (isomorphism classes of) objects in the category D G (G) of G-equivariant Qcomplexes (where we fix a prime = p) on G. We will describe a partition of the set CS(G) into finite sets known as L-packets and we will associate a modular category M L with each L-packet L of character sheaves using a truncated version of convolution of character sheaves. In the case where k = F q and G is equipped with an F q -Frobenius F we will study the relationship between F -stable character sheaves on G and the irreducible characters of (all pure inner forms of) G F . In particular, we will prove that the notion of almost characters (introduced by T. Shoji using Shintani descent) is well defined for neutrally solvable groups and that these almost characters coincide with the "trace of Frobenius" functions associated with F -stable character sheaves. We will also prove that the matrix relating the irreducible characters and almost characters is block diagonal where the blocks on the diagonal are parametrized by F -stable L-packets. Moreover, we will prove that the block in this transition matrix corresponding to any F -stable L-packet L can be described as the crossed S-matrix associated with the auto-equivalence of the modular category M L induced by F .

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“…are (infinite) spherical braided Z-crossed category with trivial components eD G (G) and M G,e respectively with a (rigid) duality functor which we denote by (·) ∨ (cf. [De1,§2.3]) and a natural isomorphism (·) ∨∨ ∼ = id of monoidal functors. In particular, for each M ∈ eD F G (G) we have a twist θ F,e M defined as the composition…”

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confidence: 99%

Shintani descent for algebraic groups and almost characters of unipotent groups

Deshpande

2016

Compositio Math.

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In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field Fq. For this, it is essential to treat all the pure inner Fq-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by T. Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the "trace of Frobenius" functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko-Drinfeld's theory of character sheaves on neutrally unipotent are in fact positive integral confirming a conjecture due to Drinfeld.

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Heisenberg idempotents on unipotent groups

Cited by 10 publications

References 3 publications

Characters of unipotent groups over finite fields

Characters of unipotent groups over finite fields

Character sheaves on neutrally solvable groups

Shintani descent for algebraic groups and almost characters of unipotent groups

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