Quasi-semisimple actions on reductive groups and buildings

Loren Spice (TCU)

(joint with Adler and Lansky) In the early 21st century, Prasad and Yu proved that, for a field of characteristic exponent $p \geq1$ , the identity component of the group of fixed points of a finite group $\Gamma$ of prime-to-p order acting on a reductive group $\tilde{G}$ is itself a reductive group; and that, for a local field of residual characteristic $p$ , the building of the resulting fixed-point group is the fixed-point set for the action of $\Gamma$ on the building of $\tilde G$ .

Applications by Adler and Lansky to lifting require analogous statements for quasisemisimple actions, which are those for which there are a Borel subgroup $\tilde{B}$ of $\tilde{G}$ , and a maximal torus in $\tilde{B}$ , that are preserved by the action of $\Gamma$ . The fixed-point group remains reductive in this case, at least after smoothing. Unfortunately, the directly analogous statement about buildings is too strong; but, fortunately, the directly analogous statement about buildings is too strong, so that we can explore the interesting ways that the building of the fixed-point group can fail to fill out the full fixed-point set in the building of $\tilde G$ , and conditions that recover the exact analogue of the Prasad–Yu statement.