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Prashant Arote

Publications and source records attributed to Prashant Arote.

6 recordsLinked to original sources
arXiv2026

Pinned Jordan Decomposition of Characters and Depth-Zero Hecke Algebras

We construct a pinned canonical Jordan decomposition of characters for finite reductive groups in cases where the relevant dual centralizers may be disconnected. For a connected reductive group \(G\) over a finite field, with a fixed pinning, and for a semisimple element \(s\in G^*\), we construct a canonical bijection between the Lusztig series \(\mathcal E(G,s)\) and the unipotent characters of \(C_{G^*}(s)^{F^*}\). This refines Lusztig's orbit-valued Jordan decomposition for groups with disconnected centre, and is characterized by compatibility with Deligne--Lusztig character formulae and Harish--Chandra series. We also treat a class of possibly disconnected reductive groups with abelian component group whose rational components admit pinning-preserving representatives. In this setting the natural result is an enriched disconnected Jordan decomposition: the target records the connected unipotent Jordan datum, the source Clifford class, and the corresponding projective Clifford label. When the transported Clifford classes agree with the ordinary Clifford classes on the dual-centralizer side, this enriched target recovers the usual unipotent characters of the corresponding disconnected dual centralizer. The construction uses pinned-normalized preferred extensions of cuspidal unipotent characters, Clifford theory, relative Weyl group comparison, Malle's matching, and connected and disconnected Howlett--Lehrer theory. As an application, we give a pinned canonical form of the finite-field input in Ohara's comparison of depth-zero Hecke algebra parameters with the unipotent case.

math.RT↗
arXiv2023

Prasad's Conjecture about dualizing involutions

Let $G$ be a connected reductive group defined over a finite field $\mathbb{F}_q$ with corresponding Frobenius $F$. Let $ι_G$ denote the duality involution defined by D. Prasad under the hypothesis $2\mathrm{H}^1(F,Z(G))=0$, where $Z(G)$ denotes the center of $G$. We show that for each irreducible character $ρ$ of $G^F$, the involution $ι_G$ takes $ρ$ to its dual $ρ^{\vee}$ if and only if for a suitable Jordan decomposition of characters, an associated unipotent character $u_ρ$ has Frobenius eigenvalues $\pm$ 1. As a corollary, we obtain that if $G$ has no exceptional factors and satisfies $2\mathrm{H}^1(F,Z(G))=0$, then the duality involution $ι_G$ takes $ρ$ to its dual $ρ^{\vee}$ for each irreducible character $ρ$ of $G^F$. Our results resolve a finite group counterpart of a conjecture of D.~Prasad.

math.RT↗
arXiv2022

Harish-Chandra Induction and Jordan Decomposition of Characters

We show that for any finite connected reductive group, a Jordan decomposition can always be chosen such that it commutes with Harish-Chandra induction. En route, we show that the endomorphism algebra of the Harish-Chandra induction of a cuspidal representation of a Levi subgroup is isomorphic to a unipotent counterpart. These results generalize the well known results for groups with connected center.

math.RT↗
arXiv2020

On $G$-crossed Frobenius $\star$-algebras and fusion rings associated with braided $G$-actions

For a finite group $G$, Turaev introduced the notion of a braided $G$-crossed fusion category. The classification of braided $G$-crossed extensions of braided fusion categories was studied by Etingof, Nikshych and Ostrik in terms of certain group cohomological data. In this paper we will define the notion of a $G$-crossed Frobenius $\star$-algebra and give a classification of (strict) $G$-crossed extensions of a commutative Frobenius $\star$-algebra $R$ equipped with a given action of $G$, in terms of the second group cohomology $H^2(G,R^\times)$. Now suppose that $\mathcal{B}$ is a non-degenerate braided fusion category equipped with a braided action of a finite group $G$. We will see that the associated $G$-graded fusion ring is in fact a (strict) $G$-crossed Frobenius $\star$-algebra. We will describe this $G$-crossed fusion ring in terms of the classification of braided $G$-actions by Etingof, Nikshych, Ostrik and derive a Verlinde formula to compute its fusion coefficients.

math.QA↗
arXiv2019

Cohomology of bimultiplicative local systems on unipotent groups

Let $U_1, U_2$ be connected commutative unipotent algebraic groups defined over an algebraically closed field $k$ of characteristic $p>0$ and let $\mathcal{L}$ be a bimultiplicative $\overline{\mathbb{Q}}_\ell$-local system on $U_1\times U_2$. In this paper we will study the $\overline{\mathbb{Q}}_\ell$-cohomology $H^*_c(U_1\times U_2,\mathcal{L})$, which turns out to be supported in only one degree. We will construct a finite Heisenberg group $Γ$ which naturally acts on $H^*_c(U_1\times U_2,\mathcal{L})$ as an irreducible representation. We will give two explicit realizations of this cohomology and describe the relationship between these two realizations as a finite Fourier transform.

math.RT↗