On the topology of geometric and rational orbits for algebraic group actions over valued fields

Abstract

<jats:title>Abstract</jats:title><jats:p>In this note, we study the relationship between Zariski and relative closedness for actions of (smooth) algebraic groups defined over valued (mainly local) fields of any characteristic. In particular, we use some recent basic results regarding the completely reducible subgroups and cocharacter-closedness due to Bate–Herpel–Röhrle–Tange and Uchiyama to construct some actions of simple algebraic groups <jats:italic>G</jats:italic> of the types <jats:inline-formula id="j_jgth-2019-0123_ineq_9999_w2aab3b7d258b1b6b1aab1c11b1b3Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0123_eq_0058.png" /><jats:tex-math>{D_{4}}</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula id="j_jgth-2019-0123_ineq_9998_w2aab3b7d258b1b6b1aab1c11b1b5Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0123_eq_0062.png" /><jats:tex-math>{E_{6}}</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula id="j_jgth-2019-0123_ineq_9997_w2aab3b7d258b1b6b1aab1c11b1b7Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0123_eq_0063.png" /><jats:tex-math>{E_{7}}</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula id="j_jgth-2019-0123_ineq_9996_w2aab3b7d258b1b6b1aab1c11b1b9Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0123_eq_0064.png" /><jats:tex-math>{E_{8}}</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula id="j_jgth-2019-0123_ineq_9995_w2aab3b7d258b1b6b1aab1c11b1c11Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0123_eq_0104.png" /><jats:tex-math>{G_{2}}</jats:tex-math></jats:alternatives></jats:inline-formula> on an affine variety defined over a local function field <jats:italic>k</jats:italic>, and <jats:inline-formula id="j_jgth-2019-0123_ineq_9994_w2aab3b7d258b1b6b1aab1c11b1c15Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0123_eq_0269.png" /><jats:tex-math>{v\in V(k)}</jats:tex-math></jats:alternatives></jats:inline-formula> such that the geometric orbit <jats:inline-formula id="j_jgth-2019-0123_ineq_9993_w2aab3b7d258b1b6b1aab1c11b1c17Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0123_eq_0084.png" /><jats:tex-math>{G.v}</jats:tex-math></jats:alternatives></jats:inline-formula> is Zariski closed although the corresponding relative orbit <jats:inline-formula id="j_jgth-2019-0123_ineq_9992_w2aab3b7d258b1b6b1aab1c11b1c19Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0123_eq_0077.png" /><jats:tex-math>{G(k).v}</jats:tex-math></jats:alternatives></jats:inline-formula> is not closed in the topology induced from <jats:italic>k</jats:italic>. Besides, by using an interesting result due to Gabber, Gille and Moret-Bailly, we show that this phenomenon does not appear when we consider the action of either a smooth unipotent group or a smooth commutative algebraic group, defined over an admissible valued (e.g., local) field.</jats:p>