arXiv subjects

Aranya Lahiri

Publications and source records attributed to Aranya Lahiri.

7 recordsLinked to original sources

arXiv2026

Asymptotics for number of indecomposable components of tensor powers of the natural $\mathrm{SL}_2$-module in odd characteristic

Let $K$ be an algebraically closed field of odd characteristic $p$, let $G= {\rm SL}_2(K)$, and let $V$ be the natural representation of $G$. Let $b_k$ denote the number of $G$-indecomposable factors of $V^{\otimes k}$, counted with multiplicity, and let $δ_p=1-\log_{p^2}\!\bigl(\tfrac{p+1}{2}\bigr)$. Then there exists a smooth, strictly positive, multiplicatively $p^2$-periodic function $ω(t)$ such that $b_k$ is asymptotic to $ω(k)k^{-δ_p}2^k$. We also show that $t^{-δ}ω(t)$ arises as the limiting density of renormalized convolutions of rescaled copies of a positive weight $3/2$ theta function, obtained from the boundary heat flux of the Dirichlet heat kernel on $(0,p)$.

math.RT↗

arXiv2023

Dagger groups and $p$-adic distribution algebras

Let $(G,ω)$ be a $p$-saturated group and $K/\mathbb{Q}_p$ a finite extension. In this paper we introduce the space of $K$-valued overconvergent functions $\mathcal{C}^\dagger(G,K)$. In the process we promote the rigid analytic group attached to $(G,ω)$ in a previous work of the first two authors to a dagger group. A main result of this article is that under certain assumptions (satisfied for example when $G$ is a uniform pro-$p$ group) the distribution algebra $D^\dagger(G,K)$, i.e. the strong dual of $\mathcal{C}^\dagger(G,K)$, is a Fréchet-Stein algebra in the sense of Schneider and Teitelbaum. In the last section we introduce overconvergent representations and show that there is an anti-equivalence of categories between overconvergent $G$-representations of compact type and continuous $D^\dagger(G, K)$-modules on nuclear Fréchet spaces. This is analogous to the anti-equivalence between locally analytic representations and modules over the locally analytic distribution algebra as proved by Schneider and Teitelbaum.

math.RT↗

arXiv2023

Presentation of an Iwasawa algebra:The pro-$p$-Iwahori of reductive groups

In this article we generalize results of Clozel and Ray (for $SL_2$ and $SL_n$ respectively) to give explicit ring-theoretic presentation in terms of a complete set of generators and relations of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of a connected, split, reductive group $\mathbb{G}$ over $\mathbb{Q}_p$.

math.RT↗

arXiv2022

Translation functors for locally analytic representations

Let $G$ be a $p$-adic Lie group with reductive Lie algebra $\mathfrak{g}$. In analogy to the translation functors introduced by Bernstein and Gelfand on categories of $U(\mathfrak{g})$-modules we consider similarly defined functors on the category of coadmissible modules over the locally analytic distribution algebra $D(G)$ on which the center of $U(\mathfrak{g})$ acts locally finite. These functors induce equivalences between certain subcategories of the latter category. Furthermore, these translation functors are naturally related to those on category $\mathcal{O}$ via the functors from category $\mathcal{O}$ to the category of coadmissible modules. We also investigate the effect of the translation functors on locally analytic representations $Π(V)^{\rm la}$ associated by the $p$-adic Langlands correspondence for ${\rm GL}_2(\mathbb{Q}_p)$ to 2-dimensional Galois representations $V$.

math.RT↗

arXiv2022

Rigid vectors in $p$-adic principal series representations

In this paper we view pro-$p$ Iwahori subgroups $I$ as rigid analytic groups $\Bbb{I}$ for large enough $p$. This is done by endowing $I$ with a natural $p$-valuation, and thereby generalizing results of Lazard for $\text{GL}_n$. We work with a general connected reductive split group over some $p$-adic field (with simply connected derived group) and study the $\Bbb{I}$-analytic vectors in principal series representations. Our main result is an irreducibility criterion which generalizes results of Clozel and Ray in the $\text{GL}_n$-case.

math.RT↗

arXiv2020

Resolutions of locally analytic principal series representations of $GL_2$

For a finite field extension $F/\mathbb{Q}_p$ we associate a coefficient system attached on the Bruhat-Tits tree of $G:= {\rm GL}_2(F)$ to a locally analytic representation $V$ of $G$. This is done in analogy to the work of Schneider and Stuhler for smooth representations. This coefficient system furnishes a chain-complex which is shown, in the case of locally analytic principal series representations $V$, to be a resolution of $V$.

math.RT↗

arXiv2019

Rigid Analytic Vectors in Locally Analytic Representations

Let $H$ be a uniform pro-$p$ group. Associated to $H$ are rigid analytic affinoid groups $\bbH_n$, and their "wide open" subgroups $\bbH_n^{\circ}$. Denote by $D^\la(H)= C^\la(H)'_b$ the locally analytic distribution algebra of $H$ and by $\DHnn$ Emerton's ring of $\bbH_n^{\circ}$-rigid analytic distributions on $H$. If $V$ is an admissible locally analytic representation of $H$, and if $V_{\bbH_n^\circ-\an}$ denotes the subspace of $\bbH_n^\circ$-rigid analytic vectors (with its intrinsic topology), then we show that the continuous dual of $V_{\bbH_n^\circ-\an}$ is canonically isomorphic to $\DHnn \ot_{D^\la(H)} V'$. From this we deduce the exactness of the functor $V \rightsquigarrow V_{\bbH_n^\circ-\an}$ on the category of admissible locally analytic representations of $H$.

math.NT↗