arXiv subjects

Ju-Lee Kim

Publications and source records attributed to Ju-Lee Kim.

8 recordsLinked to original sources

arXiv2024

The ($Γ$-asymptotic) wavefront sets: $GL_n$

Let $G$ be a connected reductive $p$-adic group. As verified for unipotent representations, it is expected that there is a close relation between the (Harish-Chandra-Howe) wavefronts sets of irreducible smooth representations and their Langlands parameters in the local Langlands correspondence via the Lusztig-Spaltenstein duality and the Aubert-Zelevinsky duality. In this paper, we define the $Γ$-asymptotic wavefront sets generalizing the notion of wavefront sets via the $Γ$-asymptotic expansions (in the sense of Kim-Murnaghan), and then study the their relation with the Langlands parameters. When $G=GL_n$, it turns out that this reduces to the corresponding relation of unipotent representations of the appropriate twisted Levi subgroups via Hecke algebra isomorphisms. For unipotent representations of $GL_n$, we also describe the Harish-Chandra-Howe (HCH) local character expansions of irreducible smooth representations using Kazhdan-Lusztig theory, and give another computation of the coefficients in the HCH expansion and the wavefront sets.

math.RT↗

arXiv2024

The wavefront set: bounds for the Langlands parameter

For an irreducible smooth representation of a connected reductive $p$-adic group, two important associated invariants are the wavefront set and the (partly conjectural) Langlands parameter. While a wavefront set consists of $p$-adic nilpotent orbits, one constituent of the Langlands parameter is a complex nilpotent orbit in the dual Lie algebra. For unipotent representations in the sense of Lusztig, the corresponding nilpotent orbits on the two sides are related via the Lusztig--Spaltenstein duality, by the work of Ciubotaru--Mason-Brown--Okada. In this paper, we formulate a general upper-bound conjecture and several variants relating the nilpotent orbits that appear in the wavefront set and in the Langlands parameter. We also verify these expectations in some cases, including the depth-zero supercuspidal representations of classical groups and all the irreducible representations of $G_2$.

math.RT↗

arXiv2019

Asymptotic behavior of supercuspidal representations and Sato-Tate equidistribution for families

We establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes. For the tame supercuspidals constructed by J.-K. Yu we prove the limit multiplicity property with error terms. Thereby we obtain a Sato-Tate equidistribution for the Hecke eigenvalues of these families. The main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive $p$-adic group tend to zero uniformly for every noncentral semisimple element.

math.RT↗

arXiv2016

Construction of Tame Types

We construct tame types for connected reductive p-adic groups. We also discuss their exhaustion and equivalence.

math.RT↗

arXiv2016

Asymptotics and local constancy of characters of p-adic groups

In this paper we study quantitative aspects of trace characters $Θ_π$ of reductive $p$-adic groups when the representation $π$ varies. Our approach is based on the local constancy of characters and we survey some other related results. We formulate a conjecture on the behavior of $Θ_π$ relative to the formal degree of $π$, which we are able to prove in the case where $π$ is a tame supercuspidal. The proof builds on J.-K.~Yu's construction and the structure of Moy-Prasad subgroups.

math.RT↗

arXiv2013

On the characters of unipotent representations of a semisimple p-adic group

Let G be a semisimple almost simple algebraic group defined and split over a nonarchimedean local field K and let V be a unipotent representation of G(K) (for example, an Iwahori-spherical representation). We calculate the character of V at compact very regular elements of G(K) at compact very regular elements of G(K).

math.RT↗

arXiv2012

On the Steinberg character of a semisimple p-adic group

Let G be a semisimple almost simple algebraic group defined and split over a nonarchimedean local field K and let S be the Steinberg representation of G(K). Let t be be a very regular semisimple element of G(K). In this paper we give a simple formula (not as an alternating sum) for the value of the character of S at t; we show that this value is plus or minus an integer power of q, the cardinal of the residue field of K. We also give an explicit formula for the character at a split very regular element of any irreducible admissible representation of G(K) with nonzero vectors invariant under an Iwahori subgroup (valid under some restrictions on characteristic).

math.RT↗

arXiv2006

Supercuspidal representations: an exhaustion theorem

Let $G$ be a reductive $p$-adic group. We prove that all supercuspidal representations of $G$ arise through Yu's construction subject to certain hypotheses on $k$ (depending on $G$). As a corollary, under the same hypotheses, we see that any supercuspidal representation is compactly induced from a representation of an open subgroup which is compact modulo the center.

math.RT↗