Atiyah-Segal completion for the Hermitian K-theory of Symplectic Groups
Jens Hornbostel, Herman Rohrbach, Marcus Zibrowius
TL;DR
The paper develops an Atiyah–Segal–type completion theorem in Hermitian K-theory for symplectic groups, showing that for a field k with characteristic not equal to 2, GW^±(BSp_{2r}) is the IO_{Sp_{2r}}-adic completion of GW^±(Rep(Sp_{2r})). Building on motivic homotopy theory and the framework of Morel–Voevodsky and Panin–Walter, the authors replace GL_1 by Sp_2 and systematically study Real split reductive groups, their ±-symmetric representation rings, and the associated Borel classes to obtain computable polynomial generators. They construct classifying spaces via acceptable gadgets and Čech localization, proving a general BG-type theorem that enables motivic approximations to BG and, in turn, the completion results for GW^±-theory. The work also provides a model for B(G_m) with a nontrivial involution and discusses potential generalizations to base schemes with nontrivial group actions, highlighting the broader applicability of the methods to other groups and involutions. Collectively, this advances computable Hermitian K-theory invariants for symplectic groups and sets the stage for further extensions beyond split tori and base schemes.
Abstract
We study equivariant Hermitian K-theory for representations of symplectic groups, especially $\mathrm{SL}_2$. The results are used to establish an Atiyah-Segal completion theorem for Hermitian $K$-theory and symplectic groups.