Balanced representations, the asymptotic Plancherel formula, and Lusztig's conjectures for $\tilde{C}_2$

J. Guilhot; J. Parkinson

Home/Archive/arXiv

arXiv · 1803.11067

Balanced representations, the asymptotic Plancherel formula, and Lusztig's conjectures for $\tilde{C}_2$

J. Guilhot·J. Parkinson

View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove Lusztig's conjectures ${\bf P1}$-${\bf P15}$ for the affine Weyl group of type $\tilde{C}_2$ for all choices of positive weight function. Our approach to computing Lusztig's $\mathbf{a}$-function is based on the notion of a `balanced system of cell representations'. Once this system is established roughly half of the conjectures ${\bf P1}$-${\bf P15}$ follow. Next we establish an `asymptotic Plancherel Theorem' for type $\tilde{C}_2$, from which the remaining conjectures follow. Combined with existing results in the literature this completes the proof of Lusztig's conjectures for all rank $1$ and $2$ affine Weyl groups for all choices of parameters.

Explore related subjects

math.RT

Keep this discovery

BibTeX RIS

J. Guilhot, J. Parkinson. 2018-03-27. Balanced representations, the asymptotic Plancherel formula, and Lusztig's conjectures for $\tilde{C}_2$. https://arxiv.org/abs/1803.11067

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Whittaker supermodules over Lie superalgebras

In this paper, we develop a general theory of quasi-Whittaker supermodules over Lie superalgebras induced from an arbitrary ideal. We determine the quasi-Whittaker vectors in universal supermodules, establish an irreducibility criterion, and classify several families of irreducible supermodules. The odd part produces a new irreducibility phenomenon absent from the Lie algebra setting. As applications, we determine all irreducible quasi-Whittaker supermodules over the $N=1$ super Schr\"odinger algebra and the $N=1$ $\frac{3}{2}$-conformal Galilei superalgebra, and over the complete spectrum-generating superalgebra in a special case.

math.RT↗

arXiv2026

Rankin--Selberg integrals of opposite conductor--one newforms

Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $\Pi_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin--Selberg integral of opposite newforms in $\Pi_{n+1}\times \Pi_n$ explicitly and show that its central value is nonzero. As an application, this implies a case of Disegni--Zhang's conjecture on the nonvanishing of local relative characters.

math.RT↗

arXiv2026

Obstructions to Jacobi-Finiteness of Quivers with Potentials

We show that Jacobi-finite potentials need not exist on finite $2$-acyclic quivers. Our main tool is a matrix-valued Golod--Shafarevich--Vinberg inequality for quotients of completed path algebras by finitely many, possibly nonhomogeneous, topological relations. Applied to cyclic derivatives, it yields a potential-dependent obstruction to the finite-dimensionality of completed Jacobian algebras. We then construct a purely quiver-level criterion excluding every Jacobi-finite potential on a given quiver, and exhibit a family of quivers for which every potential has an infinite-dimensional Jacobian algebra.

math.RT↗