Crossed S-matrices and character sheaves on unipotent groups

2016

Int Math Res Notices

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Let C be a modular tensor category over an algebraically closed field k of characteristic 0. Then there is the ubiquitous notion of the S-matrix S(C ) associated with the modular category. The matrix S(C ) is a symmetric matrix, its entries are cyclotomic integers and the matrix (dim C ) − 1 2 · S(C ) is a unitary matrix. Here dim C ∈ k denotes the categorical dimension of C and it is a totally positive cyclotomic integer. Now suppose that we also have a modular autoequivalence F : C → C . In this paper, we will define and study the notion of a crossed S-matrix associated with the modular autoequivalence F . We will see that the crossed S-matrix occurs as a submatrix of the usual S-matrix of some "bigger" modular category and hence the entries of a crossed S-matrix are also cyclotomic integers. We will prove that the crossed S-matrix (normalized by the factor (dim C ) − 1 2 ) associated with any modular autoequivalence is a unitary matrix. We will also prove that the crossed S-matrix is essentially the "character table" of a certain semisimple commutative Frobenius k-algebra associated with the modular autoequivalence F . The motivation for most of our results comes from the theory of character sheaves on algebraic groups, where we expect that the transition matrices between irreducible characters and character sheaves can be obtained as certain crossed S-matrices. In the character theory of algebraic groups defined over finite fields, there is the notion of Shintani descent of Frobenius stable characters. We will define and study a categorical analogue of this notion of Shintani descent in the setting of modular categories.

“…(ii) The definition of the crossed S-matrix presented here is slightly different from that given in [De1]. Firstly, in op.…”

Section: The Crossed S-matrix Associated With a Modular Autoequivalencementioning

confidence: 98%

Section: Unitarity Of the Crossed S-matrixmentioning

confidence: 99%

Section: Motivation From the Theory Of Character Sheavesmentioning

confidence: 99%

“…The map which sends an object (C, ψ) ∈ C Z to its class in the algebra K(C , F ) is analogous to the sheaf-function correspondence. We refer to [De1,§7] for details.…”

Section: The Frobenius Algebra Associated With a Modular Autoequivalencementioning

confidence: 99%

See 2 more Smart Citations

Modular Categories, Crossed S-matrices, and Shintani Descent

2016

Int Math Res Notices

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“…For this we take k = F p and consider an F q -Frobenius map F : G −→ G where q is some power of p. We would like to study the character theory of the finite group G(F q ) = G F using our theory of character sheaves on G. However, as observed in [B2] and [De3], it is more natural to consider all the pure inner forms G gF at the same time. Here g ∈ G and gF := ad(g) • F : G −→ G is another F qFrobenius map for G. The pure inner forms of G F are parametrized by the finite set H 1 (F, G) of F -twisted conjugacy classes in G. Note that if G is connected, then by Lang's theorem H 1 (F, G) is singleton and we have only one pure inner form.…”

mentioning

confidence: 99%

Character sheaves on neutrally solvable groups

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 .

Shintani descent for algebraic groups and almost characters of unipotent groups

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.