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Bounty Ended with 50 reputation awarded by D. Dona
Make final sentence more emphatic.
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Jeff Adler
Jeff Adler
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See Proposition 7.1.4 in Dat, Orlik, and Rapoport, Period domains over finite and $p$-adic fields, Cambridge tracts in Mathematics, vol. 183. While I don't have this book in front of me, I'm pretty sure that the result asserts the existence of elliptic semisimple elements in all finite reductive groups. Such elements are necessarily regular. Most groups are handled via a uniform argument, but several are treated as special cases. Technically, the proof contains a gap, as the case of $G_2(\mathbb{F}_2)$ was inadvertently omitted. Nonetheless, the result remains true in this case, where one can find a regular element by looking at the Coxeter torus $T$ of $\operatorname{SL}_3$, which we can consider as a subgroup of $G_2$. If memory serves, any Every nontrivial element of $T(\mathbb{F}_2)$ will do the jobis regular in $G_2$.

See Proposition 7.1.4 in Dat, Orlik, and Rapoport, Period domains over finite and $p$-adic fields, Cambridge tracts in Mathematics, vol. 183. While I don't have this book in front of me, I'm pretty sure that the result asserts the existence of elliptic semisimple elements in all finite reductive groups. Such elements are necessarily regular. Most groups are handled via a uniform argument, but several are treated as special cases. Technically, the proof contains a gap, as the case of $G_2(\mathbb{F}_2)$ was inadvertently omitted. Nonetheless, the result remains true in this case, where one can find a regular element by looking at the Coxeter torus $T$ of $\operatorname{SL}_3$, which we can consider as a subgroup of $G_2$. If memory serves, any nontrivial element of $T(\mathbb{F}_2)$ will do the job.

See Proposition 7.1.4 in Dat, Orlik, and Rapoport, Period domains over finite and $p$-adic fields, Cambridge tracts in Mathematics, vol. 183. While I don't have this book in front of me, I'm pretty sure that the result asserts the existence of elliptic semisimple elements in all finite reductive groups. Such elements are necessarily regular. Most groups are handled via a uniform argument, but several are treated as special cases. Technically, the proof contains a gap, as the case of $G_2(\mathbb{F}_2)$ was inadvertently omitted. Nonetheless, the result remains true in this case, where one can find a regular element by looking at the Coxeter torus $T$ of $\operatorname{SL}_3$, which we can consider as a subgroup of $G_2$. Every nontrivial element of $T(\mathbb{F}_2)$ is regular in $G_2$.

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LSpice
LSpice
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See Proposition 7.1.4 in Dat, Orlik, and Rapoport, Period domains over finite and $p$-adic fieldsPeriod domains over finite and $p$-adic fields, Cambridge tracts in Mathematics, vol. 183. While I don't have this book in front of me, I'm pretty sure that the result asserts the existence of elliptic semisimple elements in all finite reductive groups. Such elements are necessarily regular. Most groups are handled via a uniform argument, but several are treated as special cases. Technically, the proof contains a gap, as the case of $G_2(\mathbb{F}_2)$ was inadvertently omitted. Nonetheless, the result remains true in this case, where one can find a regular element by looking at the Coxeter torus $T$ of $SL_3$$\operatorname{SL}_3$, which we can consider as a subgroup of $G_2$. If memory serves, any nontrivial element of $T(\mathbb{F}_2)$ will do the job.

See Proposition 7.1.4 in Dat, Orlik, and Rapoport, Period domains over finite and $p$-adic fields, Cambridge tracts in Mathematics, vol. 183. While I don't have this book in front of me, I'm pretty sure that the result asserts the existence of elliptic semisimple elements in all finite reductive groups. Such elements are necessarily regular. Most groups are handled via a uniform argument, but several are treated as special cases. Technically, the proof contains a gap, as the case of $G_2(\mathbb{F}_2)$ was inadvertently omitted. Nonetheless, the result remains true in this case, where one can find a regular element by looking at the Coxeter torus $T$ of $SL_3$, which we can consider as a subgroup of $G_2$. If memory serves, any nontrivial element of $T(\mathbb{F}_2)$ will do the job.

See Proposition 7.1.4 in Dat, Orlik, and Rapoport, Period domains over finite and $p$-adic fields, Cambridge tracts in Mathematics, vol. 183. While I don't have this book in front of me, I'm pretty sure that the result asserts the existence of elliptic semisimple elements in all finite reductive groups. Such elements are necessarily regular. Most groups are handled via a uniform argument, but several are treated as special cases. Technically, the proof contains a gap, as the case of $G_2(\mathbb{F}_2)$ was inadvertently omitted. Nonetheless, the result remains true in this case, where one can find a regular element by looking at the Coxeter torus $T$ of $\operatorname{SL}_3$, which we can consider as a subgroup of $G_2$. If memory serves, any nontrivial element of $T(\mathbb{F}_2)$ will do the job.

Source Link
Jeff Adler
Jeff Adler
  • 583
  • 4
  • 6

See Proposition 7.1.4 in Dat, Orlik, and Rapoport, Period domains over finite and $p$-adic fields, Cambridge tracts in Mathematics, vol. 183. While I don't have this book in front of me, I'm pretty sure that the result asserts the existence of elliptic semisimple elements in all finite reductive groups. Such elements are necessarily regular. Most groups are handled via a uniform argument, but several are treated as special cases. Technically, the proof contains a gap, as the case of $G_2(\mathbb{F}_2)$ was inadvertently omitted. Nonetheless, the result remains true in this case, where one can find a regular element by looking at the Coxeter torus $T$ of $SL_3$, which we can consider as a subgroup of $G_2$. If memory serves, any nontrivial element of $T(\mathbb{F}_2)$ will do the job.