Regular supercuspidal representations

Tasho Kaletha

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Regular supercuspidal representations

Tasho Kaletha

Representation Theory · Submitted 9 Feb 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups.

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We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property.	result 0.86	0.96	0.64	asserted 1.00
s1	We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups.	contribution 0.49	0.96	0.08	asserted 1.00
s2	Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations.	contribution 0.71	0.96	0.29	asserted 1.00
s3	In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups.	result 0.49	0.89	0.31	asserted 0.99

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The paper

Original abstract

We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups.

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12 arXiv papers cite this in 111 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 73 sentences without a confident label

Builds on it 1 paper

Here we use the same notation as that in the proof of [ref] .

Distinguished regular supercuspidal representations · $(θ,εη)$-symmetric pair · 0.80

Uses its method or tool 2 papers

Using Proposition 3.7.4 [ref] , Kaletha chooses a Howe factorization $μ_-1,…,μ_d$ of $μ$ and then constructs in 3.4.3 [ref] a representation $ρ_T^μ_-1$ (denoted by $κ̃_(S,θ)$ in [ref] ).

The Construction of Regular Supercuspidal Representations · A simplified construction of the representation $ρ_T^μ$ of $TH_x,0$ · 0.95
It exploits the choice of $$ and is inspired by 3.7 [ref] .

Constructing tame supercuspidal representations · Recovering $r⃗$ and $$ · 0.91

Compares with it 3 papers

According to Lemma 3.4.12 [ref] , the representation $ρ_T^μ_-$ induces an irreducible representation of the stabilizer $H_x$ in $H$ of the vertex $x$ in the reduced building of $H$.

The Construction of Regular Supercuspidal Representations · Introduction · 0.93
The later paper [ref] offers a different construction of the Langlands correspondence for a much more general class of representations, and also features a type of “cancellation” in which the particular choice of $L$-embedding in [ref] becomes irrelevant.

Epipelagic Langlands parameters and L-packets for unitary groups · Statement of results · 0.87
It is noted in [ref] that this process is analogous to the notion of a Howe factorization of a character of a maximal torus of $GL_n$ in [ref] .

A Gelfand-Graev Formula and Stable Transfer Factors in the Unramidied Case for $\text{SL}_\ell(F)$ and $\text{GL}_\ell(F)$, $\ell$ an odd Prime · Yu's Construction · 0.94

Cites it as support 1 paper

This is shown in 4.3 of [ref] .

A Gelfand-Graev Formula and Stable Transfer Factors in the Unramidied Case for $\text{SL}_\ell(F)$ and $\text{GL}_\ell(F)$, $\ell$ an odd Prime · An Extension of the Supercuspidal Character Formula · 0.96

Cites it as background 7 papers

We refer to [ref] or Section [ref] for the basic notion and knowledge on regular supercuspidal representations.

Distinguished regular supercuspidal representations · Introduction · 1.00
We are interested in the regular supercuspidal representations of a group $G = (F)$, where $$ is a connected reductive group that is defined over a field $F$ that is a finite extension of a field $_p$ of $p$-adic numbers (with the same restrictions on $p$ as in [ref] ).

The Construction of Regular Supercuspidal Representations · Introduction · 0.94
In his recent work [ref] , Kaletha found much simpler data, called tame regular elliptic pairs, to parameterize regular supercuspidal representations which are a subclass of tame supercuspidal representations.

Theta lifts and distinction for regular supercuspidal representations · Overview · 0.91

What it cites · 19 sentences without a confident label

Uses the method or tool of 2 papers

Using it, we are able to prove the existence and uniqueness of generic constituent in each compound $L$-packet as well as the stability and endoscopic transfer of these $L$-packets (the stability of toral $L$-packets under the additional assumption that $S$ is unramified was already shown in [ref] ).

Stability of character sums for positive-depth, supercuspidal representations · Introduction · 0.92
Alternatively, if one replaces $H^1(u →W,-)$ by $B(-)_bas$ of [ref] , this assumption can be droppped.

B(G) for all local and global fields · Construction of $L$-packets · 0.84

Cites as background 1 paper

The Harish-Chandra character $Θ_π$ of $π$ has been computed in the work of Adler and Spice [ref] and later reinterpreted in the work of DeBacker and Spice [ref] .

Stability of character sums for positive-depth, supercuspidal representations · The character formula · 0.88

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