Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2204.13053

Genuine pro-$p$ Iwahori--Hecke algebras, Gelfand--Graev representations, and some applications

Fan Gao·Nadya Gurevich·Edmund Karasiewicz
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the Iwahori-component of the Gelfand-Graev representation of a central cover of a split linear reductive group and utilize our results for three applications. In fact, it is advantageous to begin at the pro-$p$ level. Thus to begin we study the structure of a genuine pro-$p$ Iwahori-Hecke algebra, establishing Iwahori-Matsumoto and Bernstein presentations. With this structure theory we first describe the pro-$p$ part of the Gelfand-Graev representation and then the more subtle Iwahori part. For the first application we relate the Gelfand-Graev representation to the metaplectic representation of Sahi-Stokman-Venkateswaran, which conceptually realizes the Chinta-Gunnells action from the theory of Weyl group multiple Dirichlet series. For the second we compute the Whittaker dimension of the constituents of regular unramified principal series; for the third we do the same for unitary unramified principal series.

Explore related subjects

math.RTmath.NT

Keep this discovery

BibTeXRIS

Fan Gao, Nadya Gurevich, Edmund Karasiewicz. 2022-04-27. Genuine pro-$p$ Iwahori--Hecke algebras, Gelfand--Graev representations, and some applications. https://arxiv.org/abs/2204.13053

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↗