| Summary: | <p>Let <em>F</em> be a finite extension of Qp, <em>n</em> ≥ 1, and let Ω be the (n−1)-dimensional Drinfeld symmetric space. In this thesis we study various vector bundles on Ω and its covering spaces that are of significance in the representation theory of GL<em>n</em>(F).</p>
<p>In the first part, we work with any <em>n</em> ≥ 2, and consider a geometrically connected component Σ1 of the first Drinfeld covering of Ω. The main result is that the canonical homomorphism to the Picard group of Σ1 determined by the second Drinfeld covering</p>
<p>(F, +) → Pic(Σ1)[<em>p</em>]</p>
<p>is injective. Here F is the residue field of the unique degree n unramified extension of <em>F</em>. In particular, Pic(Σ1)[p] is non-zero. We also show that when <em>n</em> = 2, all vector bundles on Ω are trivial, which extends the classical result that Pic(Ω) = 0.</p>
<p>In the second part we take <em>n</em> = 2, and study any affinoid open subset Σ1v of Σ1 that lies above a vertex v of the Bruhat-Tits tree for GL2(F). Our main result is that Pic(Σ1v)[p] = 0, which we establish by showing that Pic(Y)[p] = 0 for Y the <em>Drinfeld curve</em> - the Deligne-Lusztig variety of SL2(Fq).</p>
<p>In third part, we work with any <em>n</em> ≥ 1, and show that the action of the group <em>D</em>× on the Drinfeld tower induces an equivalence of categories from finite dimensional smooth representations of <em>D</em>× to <em>G</em>0-finite GLn(<em>F</em>)-equivariant vector bundles with connection on Ω. Here <em>D</em> is the division algebra over <em>F</em> of invariant 1/n and G0 is the subgroup of GL<em>n</em>(<em>F</em>) of elements with norm 1 determinant.</p>
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