arXiv subjects
Publications and source records attributed to Mihir Sheth.
Let $G$ be a $p$-adic reductive group and $R$ be a noetherian Jacobson $\mathbb{Z}[1/p]$-algebra. In this note, we show that every smooth irreducible $R$-linear representation of $G$ is admissible using the finiteness result of Dat, Helm, Kurinczuk and Moss for Hecke algebras over $R$.
In this note, we study the twisted Jacquet modules of sub-quotients of principal series representations of ${\rm GL}_2(D)$ where $D$ is a division algebra over a non-archimedean local field $F$. We begin with a proof of a conjecture due to D. Prasad on twisted Jacquet modules of Speh representations of ${\rm GL}_2(D)$ when $D$ is the quaternionic division algebra. Later, when $D$ is an arbitrary division algebra over $F$, we focus on depth-zero principal series and compute the dimensions of twisted Jacquet modules of generalised Speh representations and investigate their structure explicitly.
Emerton's theory of Jacquet modules for locally analytic representations provides necessary conditions for the existence of integral structures in locally analytic representations. These conditions are also expected to be sufficient for the integrality of generic irreducible locally algebraic representations. In this article, we prove the sufficiency of Emerton's conditions for some tamely ramified locally algebraic representations of $\mathrm{GL}_{2}(D)$ where $D$ is a $p$-adic division algebra.
Let $p>3$ and $F$ be a non-archimedean local field with residue field a proper finite extension of $\mathbb{F}_p$. We construct smooth absolutely irreducible non-admissible representations of $\mathrm{GL}_2(F)$ defined over the residue field of $F$ extending the earlier results of the authors for $F$ unramified over $\mathbb{Q}_{p}$. This construction uses the theory of diagrams of Breuil and Paskunas. By parabolic induction, we obtain smooth absolutely irreducible non-admissible representations of $\mathrm{GL}_n(F)$ for $n>2$.
Let $F$ be a non-archimedean local field of residual characteristic $p>3$ and residue degree $f>1$. We study a certain type of diagram, called \emph{cyclic diagrams}, and use them to show that the universal supersingular modules of $\mathrm{GL}_{2}(F)$ admit infinitely many non-isomorphic irreducible admissible quotients.
We use a Diamond diagram attached to a $2$-dimensional reducible split mod $p$ Galois representation of $\mathrm{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p^{2}})$ to construct a non-admissible smooth irreducible mod $p$ representation of $GL_{2}(\mathbb{Q}_{p^{2}})$ following the approach of Daniel Le.
The Lubin-Tate moduli space $X_{0}^{\text{rig}}$ is a $p$-adic analytic open unit polydisc which parametrizes deformations of a formal group $H_{0}$ of finite height defined over an algebraically closed field of characteristic $p$. It is known that the natural action of the automorphism group $\text{Aut}(H_{0})$ on $X^{\text{rig}}_{0}$ gives rise to locally analytic representations on the topological duals of the spaces $H^{0}(X^{\text{rig}}_{0},(\mathcal{M}^{s}_{0})^{\mathrm{rig}})$ of global sections of certain equivariant vector bundles $(\mathcal{M}^{s}_{0})^{\mathrm{rig}}$ over $X^{\mathrm{rig}}_{0}$. In this article, we show that this result holds in greater generality. On the one hand, we work in the setting of deformations of formal modules over the valuation ring of a finite extension of $\mathbb{Q}_{p}$. On the other hand, we also treat the case of representations arising from the vector bundles $(\mathcal{M}^{s}_{m})^{\mathrm{rig}}$ over the deformation spaces $X^{\mathrm{rig}}_{m}$ with Drinfeld level-$m$-structures. Finally, we determine the space of locally finite vectors in $H^{0}(X^{\text{rig}}_{m},(\mathcal{M}^{s}_{m})^{\mathrm{rig}})$. Essentially, all locally finite vectors arise from the global sections of invertible sheaves over the projective space via pullback along the Gross-Hopkins period map.