Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.37184

Representations of shifted quantum affine algebras and cluster algebras II: Non simply-laced types and cluster expansions

Christof Geiss·David Hernandez·Bernard Leclerc
View original source ↗Read source document ↗Export metadata ↓

Abstract

We extend our previous results on cluster structures in the Grothendieck rings of categories $\mathcal{O}^{\mathrm{sh}}$ for shifted quantum affine algebras to the non-simply laced types. We also show that, for all types, the formal power series expressions of the cluster monomials in terms of classes of positive prefundamental representations can be regarded as limits of cluster expansions, thus establishing their positivity.

Explore related subjects

math.RTmath.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Christof Geiss, David Hernandez, Bernard Leclerc. 2026-10-27. Representations of shifted quantum affine algebras and cluster algebras II: Non simply-laced types and cluster expansions. https://arxiv.org/abs/2610.37184

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

KEEP EXPLORING

Related papers

arXiv2026

Comparing classical tilting modules and $τ$-tilting modules

We characterize support $τ$-tilting modules as classical tilting modules over quotient algebras satisfying a tensor-vanishing condition, and characterize classical tilting modules as $τ$-tilting modules satisfying a ${\rm Tor}^1$-vanishing condition. We use delooping levels to study \emph{Self-orthogonal $τ$-tilting Conjecture}: any self-orthogonal $τ$-tilting module is classical tilting. We confirm the conjecture when the endomorphism algebra of the module has finite global delooping level.

math.RT↗
arXiv2026

Minimal depth $K$-types for wild double covers and Shimura correspondences

Using distinguished pseudo-spherical representations of covering tori constructed in a companion work and Dat's parahoric induction, we construct a distinguished family of representations for an Iwahori subgroup $\bar{I}$ in $\bar{G}$, an $n$-fold Brylinski-Deligne (BD) cover of a split connected reductive $p$-adic group $G$, under explicit hypotheses on the BD datum. No tameness assumption $\mathrm{gcd}(p,n)=1$ is imposed. The construction suggests a conjectural theory of Iwahori-types for BD covers beyond the tame case. We verify this for $2$-fold covers of almost simple simply-laced simply-connected Chevalley groups of rank at least $2$ over $2$-adic fields. In particular, for these non-tame $2$-fold covers we prove: 1) The pair $(\bar{I},σ)$ is a Bushnell-Kutzko type for the Bernstein block analogous to the block of unramified principal series in a linear reductive group. This is a non-tame analogue of the Borel-Casselman theorem. 2) The Hecke algebra $\mathcal{H}(\bar{G},\bar{I},σ^{\vee})$ has an Iwahori-Matsumoto presentation up to normalization. In types $A_{r},D_{2r+1},E_{6},E_{7}$ we obtain the complete presentation and show that $\mathcal{H}(\bar{G},\bar{I},σ^{\vee})$ is isomorphic to the Iwahori-Hecke algebra of an explicit linear group. This determines an equivalence of Bernstein blocks producing a local Shimura correspondence. 3) The representation $σ$ extends to a hyperspecial maximal compact subgroup $\bar{K}$. Among the genuine $\bar{K}$-representations, $σ$ has minimal depth with respect to the principal congruence filtration. It also yields a finite Shimura correspondence, generalizing a result of Savin.

math.RT↗
arXiv2026

Wavefront sets for genuine representations of $\rm GL$-covers of Kazhdan--Patterson or Savin types

First, we consider general Brylinski--Deligne covers of the $p$-adic general linear groups, and discuss the theory of Bernstein--Zelevinsky derivatives. We also recall the Zelevinsky-type classification of the irreducible genuine spectrum for the Kazhdan--Patterson and Savin covers. Following this, for these two special families of covers, we determine the wavefront sets of their irreducible genuine representations, expressed in terms of the iterated degrees of the highest Bernstein--Zelevinsky derivatives. Finally, for Kazhdan--Patterson covers, we reinterpret this result on the wavefront set using a version of the local Langlands correspondence and the covering Barbasch--Vogan duality.

math.RT↗