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Manish Mishra

Publications and source records attributed to Manish Mishra.

At least 19 recordsLinked to original sources
arXiv2026

A Pinned Local Langlands Correspondence for Depth-Zero Supercuspidal Representations

We construct a pinning-normalized local Langlands correspondence for depth-zero supercuspidal representations of a connected reductive group over a non-archimedean local field. After fixing a pinned splitting of the quasi-split inner form, we obtain a canonical bijection between irreducible depth-zero supercuspidal representations and relevant cuspidal enhanced depth-zero Langlands parameters. The construction is organized around the two pieces naturally present in a depth-zero type: a tame toral part and a finite cuspidal representation of a parahoric quotient. The toral part is matched using the local Langlands correspondence for maximally unramified elliptic tori and normalized \(L\)-embeddings. The finite cuspidal part is compared with the parameter side by a pinned Jordan decomposition for the relevant finite reductive quotients. Since these quotients may be disconnected, the finite comparison must retain the Clifford-theoretic data that records the possible extension ambiguity. On the connected unipotent part we use the correspondence of Feng--Opdam--Solleveld for supercuspidal unipotent representations. Combining the toral, unipotent, and Clifford-theoretic pieces gives the enhanced parameter attached to a depth-zero supercuspidal representation, and the inverse map is obtained by reversing the same construction. The correspondence is canonical relative to the fixed pinned normalization. It is compatible with the tame inertial parameter attached to the depth-zero character, with weakly unramified twists, and with central characters via the torus correspondence. Under the DeBacker--Reeder logarithm hypothesis, the dimension-weighted packet distributions attached to the resulting packets are stable.

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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.

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arXiv2025

Depth Preservation and Close-Field Transfer in the Local Langlands Correspondence

We introduce a revised notion of depth for Langlands parameters for tori defined over a nonarchimedean local field \(F\) that restores depth preservation under the local Langlands correspondence (LLC). We leverage that preservation to derive structural results that, taken together, yield a canonical transfer of broad harmonic-analytic results from characteristic \(0\) to characteristic \(p\). When \(F\) has suitably large positive characteristic, we prove a block-by-block equivalence: each Bernstein block of \(G(F)\) is equivalent to a corresponding block for some \(G'(F')\) with \(F'\) of characteristic \(0\) \(\ell\)-close to \(F\); using this, we show that a LLC in characteristic \(0\) corresponds canonically to a LLC in characteristic \(p\). For regular supercuspidals we give a direct, more structured construction via Kaletha. Along the way we recover and extend results on \(\ell\)-close fields -- introducing a depth-transfer function generalizing the normalized Hasse--Herbrand function, proving truncated isomorphisms for arbitrary tori and parahorics, establishing a depth and supercuspidality preserving Kazhdan-type Hecke-algebra isomorphism for arbitrary maximal parahorics of arbitrary connected reductive groups; and a generalized Cartan decomposition for arbitrary maximal parahorics -- thereby subsuming several earlier results in the literature. Collectively, the results let one work in characteristic \(0\) without loss of generality for a wide swath of harmonic analysis on \(p\)-adic groups.

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arXiv2024

Structure of Hecke algebras arising from types

Let $G$ denote a connected reductive group over a nonarchimedean local field $F$ of residue characteristic $p$, and let $\mathcal{C}$ denote an algebraically closed field of characteristic $\ell \neq p$. If $\rho$ is an irreducible, smooth $\mathcal{C}$-representation of a compact, open subgroup $K$ of $G(F)$, then the pair $(K,\rho)$ gives rise to a Hecke algebra $\mathcal{H}(G(F),(K, \rho))$. For a large class of pairs $(K,\rho)$, we show that $\mathcal{H}(G(F),(K, \rho))$ is a semi-direct product of an affine Hecke algebra with explicit parameters with a twisted group algebra, and that it is isomorphic to $\mathcal{H}(G^0(F),(K^0, \rho^0))$ for some reductive subgroup $G^0 \subset G$ with compact, open subgroup $K^0$ and depth-zero representation $\rho^0$ of $K^0$. The class of pairs that we consider includes all depth-zero types. In describing their Hecke algebras, we thus recover a result of Morris as a special case. In a second paper, we will show that our class also contains all the types constructed by Kim and Yu, and hence we obtain as a corollary that arbitrary Bernstein blocks are equivalent to depth-zero Bernstein blocks under minor tameness assumptions. The pairs to which our results apply are described in an axiomatic way so that the results can be applied to other constructions of types by only verifying that the relevant axioms are satisfied. The Hecke algebra isomorphisms are given in an explicit manner and are support preserving.

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arXiv2024

Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms

Let $F$ be a nonarchimedean local field of residual characteristic $p$. Let $G$ denote a connected reductive group over $F$ that splits over a tamely ramified extension of $F$. Let $(K ,\rho)$ be a type as constructed by Kim and Yu. We show that there exists a twisted Levi subgroup $G^0 \subset G$ and a type $(K^0, \rho^0)$ for $G^0$ such that the corresponding Hecke algebras $\mathcal{H}(G(F), (K, \rho))$ and $\mathcal{H}(G^0(F), (K^0, \rho^0))$ are isomorphic. If $p$ does not divide the order of the absolute Weyl group of $G$, then every Bernstein block is equivalent to modules over such a Hecke algebra. Hence, under this assumption on $p$, our result implies that every Bernstein block is equivalent to a depth-zero Bernstein block. This allows one to reduce many problems about (the category of) smooth, complex representations of $p$-adic groups to analogous problems about (the category of) depth-zero representations. Our isomorphism of Hecke algebras is very explicit and also includes an explicit description of the Hecke algebras as semi-direct products of an affine Hecke with a twisted group algebra. Moreover, we work with arbitrary algebraically closed fields of characteristic different from $p$ as our coefficient field. This paper relies on a prior axiomatic result about the structure of Hecke algebras by the same authors and a key ingredient consists of extending the quadratic character of Fintzen--Kaletha--Spice to the support of the Hecke algebra, which might be of independent interest.

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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.

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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.

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arXiv2021

A note on sign of a self-dual representation

D. Prasad showed that the sign of a self-dual representation of a finite or $p$-adic reductive group is often detected by a central element. We study the extension of his results to some more general situations and make some observations about the consequences of his results.

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arXiv2021

Principal series component of Gelfand-Graev representation

Let $G$ be a connected reductive group defined over a non-archimedean local field $F$. Let $B$ be a minimal $F$-parabolic subgroup with Levi factor $T$ and unipotent radical $U$. Let $ψ$ be a non-degenerate character of $U(F)$ and $λ$ a character of $T(F)$. Let $(K,ρ)$ be a Bushnell-Kutzko type associated to the Bernstein block of $G(F)$ determined by the pair $(T,λ)$. We study the $ρ$-isotypical component $(c\text{-ind}_{U(F)}^{G(F)}ψ)^ρ$ of the induced space $c\text{-ind}_{U(F)}^{G(F)}ψ$ of functions compactly supported mod $U(F)$. We show that $(c\text{-ind}_{U(F)}^{G(F)}ψ)^ρ$ is cyclic module for the Hecke algebra $\mathcal{H}(G,ρ)$ associated to the pair $(K,ρ)$. When $T$ is split, we describe it more explicitly in terms of $\mathcal{H}(G,ρ)$. We make assumptions on the residue characteristic of $F$ and later also on the characteristic of $F$ and the center of $G$ depending on the pair $(T,λ)$. Our results generalize the main result of Chan and Savin in \cite{CS18} who treated the case of $λ=1$ for $T$ split.

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arXiv2021

Regular Bernstein blocks

For a connected reductive group $G$ defined over a non-archimedean local field $F$, we consider the Bernstein blocks in the category of smooth representations of $G(F)$. Bernstein blocks whose cuspidal support involves a regular supercuspidal representation are called $\textit{regular}$ Bernstein blocks. Most Bernstein blocks are regular when the residual characteristic of $F$ is not too small. Under mild hypotheses on the residual characteristic, we show that the Bernstein center of a regular Bernstein block of $G(F)$ is isomorphic to the Bernstein center of a regular depth-zero Bernstein block of $G^{0}(F)$, where $G^{0}$ is a certain twisted Levi subgroup of $G$. In some cases, we show that the blocks themselves are equivalent, and as a consequence we prove the ABPS Conjecture in some new cases.

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arXiv2020

Self-dual cuspidal representations

Let $G$ be a connected reductive group over a finite field $\mathfrak{f}$ of order $q$. When $q$ is small, we make further assumptions on $G$. Then we determine precisely when $G(\mathfrak{f})$ admits irreducible, cuspidal representations that are self-dual, of Deligne-Lusztig type, or both. Finally, we outline some consequences for the existence of self-dual supercuspidal representations of reductive $p$-adic groups.

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arXiv2020

A generalization of the 3d distance theorem

Let $P$ be a positive rational number. Call a function $f:\mathbb{R}\rightarrow\mathbb{R}$ to have $\textit{finite gaps property mod}$ $P$ if the following holds: for any positive irrational $α$ and positive integer $M$, when the values of $f(mα)$, $1\leq m\leq M$, are inserted mod $P$ into the interval $[0,P)$ and arranged in increasing order, the number of distinct gaps between successive terms is bounded by a constant $k_{f}$ which depends only on $f$. In this note, we prove a generalization of the 3d distance theorem of Chung and Graham. As a consequence, we show that a piecewise linear map with rational slopes and having only finitely many non-differentiable points has finite gaps property mod $P$. We also show that if $f$ is distance to the nearest integer function, then it has finite gaps property mod $1$ with $k_f\leq6$.

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arXiv2019

Matching of orbital integrals (transfer) and Roche Hecke algebra isomorphisms

Let $F$ be a non-Archimedan local field, $G$ a connected reductive group defined and split over $F$, and $T$ a maximal $F$-split torus in $G$. Let $χ_0$ be a depth zero character of the maximal compact subgroup $\mathcal{T}$ of $T(F)$. It gives by inflation a character $ρ$ of an Iwahori subgroup $\mathcal{I}$ of $G(F)$ containing $\mathcal{T}$. From Roche, $χ_0$ defines a split endoscopic group $G'$ of $G$, and there is an injective morphism of ${\Bbb C}$-algebras $\mathcal{H}(G(F),ρ) \rightarrow \mathcal{H}(G'(F),1_{\mathcal{I}'})$ where $\mathcal{H}(G(F),ρ)$ is the Hecke algebra of compactly supported $ρ^{-1}$-spherical functions on $G(F)$ and $\mathcal{I}'$ is an Iwahori subgroup of $G'(F)$. This morphism restricts to an injective morphism $ζ: \mathcal{Z}(G(F),ρ)\rightarrow \mathcal{Z}(G'(F),1_{\mathcal{I}'})$ between the centers of the Hecke algebras. We prove here that a certain linear combination of morphisms analogous to $ζ$ realizes the transfer (matching of strongly $G$-regular semisimple orbital integrals). If ${\rm char}(F)=p>0$, our result is unconditional only if $p$ is large enough.

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arXiv2019

A note on depth preservation

We show that for a wildly ramified torus, depth is not preserved in general under local Langlands correspondence for tori.

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arXiv2016

Signs of self-dual depth-zero supercuspidal representations

Let $G$ be a quasi-split tamely ramified connected reductive group defined over a $p$-adic field $F$. We show that if $-1$ is in the $F$-points of the absolute Weyl group of $G$, then self-dual supercuspidal representations of $G(F)$ exist. Now assume further that $G$ is unramified and that the center of $G$ is connected. Let $π$ be a generic self-dual depth-zero regular supercuspidal representation of $G(F)$. We show that the Frobenius-Schur indicator of $π$ is given by the sign by which a certain distinguished element of the center of $G(F)$ of order two acts on $π$.

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arXiv2016

Bernstein center of supercuspidal blocks

Let $\bf{ G}$ be a tamely ramified connected reductive group defined over a non-archimedean local field $k$. We show that the Bernstein center of a tame supercuspidal block of $\bf{ G}(k)$ is isomorphic to the Bernstein center of a depth zero supercuspidal block of $\bf{ G}^{0}(k)$ for some twisted Levi subgroup of $\bf{ G}^{0}$ of $\bf{ G}$.

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arXiv2015

Generic representations in L-packets

We give the details of the construction of a map to restate a conjectural expression about adjoint group action on generic representations in L-packets. We give an application of the construction to give another proof of the classification of the Knapp-Stein R-group associated to a unitary unramified character of a torus. Finally we prove the conjecture for unramified L-packets.

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