arXiv subjects

Georg Linden

Publications and source records attributed to Georg Linden.

4 recordsLinked to original sources

arXiv2025

Classification of Equivariant Line Bundles on the Drinfeld Upper Half Plane

We explicitly determine the group of isomorphism classes of equivariant line bundles on the non-archimedean Drinfeld upper half plane for $\mathrm{GL}_2(F)$, for its subgroups of matrices whose determinant has even (respectively trivial) valuation, and for $\mathrm{GL}_2(\mathcal{O}_F)$. Our results extend a recent classification of torsion equivariant line bundles with connection due to Ardakov and Wadsley, but we use a different approach. A crucial ingredient is a construction due to Van der Put which relates invertible analytic functions on the Drinfeld upper half plane to currents on the Bruhat-Tits tree. Another tool we use is condensed group cohomology.

math.AG↗

arXiv2024

The Functors $\mathcal{F}^G_P$ over Local Fields of Positive Characteristic

Let $G$ be a split connected reductive group over a non-archimedean local field. In the $p$-adic setting, Orlik-Strauch constructed functors from the BGG category $\mathcal{O}$ associated to the Lie algebra of $G$ to the category of locally analytic representation of $G$. We generalize these functors to such groups over non-archimedean local fields of arbitrary characteristic. To this end, we introduce the hyperalgebra of a non-archimedean Lie group $G$, which generalizes its Lie algebra, and consider topological modules over the algebra of locally analytic distributions on $G$ and subalgebras related to this hyperalgebra.

math.RT↗

arXiv2023

Equivariant Vector Bundles on the Drinfeld Upper Half Space over a Local Field of Positive Characteristic

We describe the locally analytic $\mathrm{GL}_d(K)$-representations which arise as the global sections of homogeneous vector bundles on the projective space restricted to the Drinfeld upper half space over a non-archimedean local field $K$. We thereby generalize work of Orlik (2008) for $p$-adic fields to the effect that it becomes applicable to local fields of positive characteristic. Our description of this space of global sections is in terms of a filtration by subrepresentations, and a characterization of the resulting subquotients via adaptations of the functors $\mathcal{F}^G_P$ considered by Orlik-Strauch (2015) and Agrawal-Strauch (2022). For a local field $K$ of positive characteristic, we also determine the locally analytic (resp. continuous) characters of $K^\times$ with values in $K$-Banach algebras which are integral domains (resp. with values in finite extensions of $K$) in an appendix.

math.NT↗

arXiv2018

Compactifications of the Drinfeld half space over a Finite Field

When considered as a Deligne-Lusztig variety, the Drinfeld half space $Ω_V$ over a finite field $k$ has a compactification whose boundary divisor is normal crossing and which can be obtained by successively blowing-up projective space along linear subspaces. Pink and Schieder (2014) have introduced a new compactification of $Ω_V$ whose strata of the boundary are glued together in a way dual to the way they are for the tautological compactification by projective space. We show that by applying an analogous sequence of blow-ups to this new compactification we arrive at the compactification by Deligne and Lusztig as well. Moreover, we compute for each of these three compactifications the stabilizers of $\bar{k}$-valued points under the canonical $\mathrm{PGL}(V)$-action. We find that in each case the stratification can be recovered from the unipotent radicals of these stabilizers.

math.AG↗