Atiyah–Segal completion for the HermitianK-theory of symplectic groups

In the case of algebraic K-theory, the corresponding result for split tori T , due to Totaro [47] for K 0 and Knizel-Neshitov [26] for higher K-groups, was the first important step for establishing more general versions of an Atiyah-Segal style completion theorem for linear algebraic groups, due to Krishna [28], Tabuada-van den Bergh [45] and , and their proofs all rely on a reduction to the case of a split torus.The proof for algebraic K-theory and tori T is easier than Rohrbach's theorem above on Hermitian K-theory because the classifying space BT is a product of copies of P 1 , and algebraic K-theory is orientable in the sense of Levine-Morel and Panin.As of yet, there is no completion theorem for Hermitian K-theory in the case of general linear groups GL n , although [40] contains a partial result in this direction by computing the Hermitian K-theory of even-dimensional Grassmannians.We restrict our attention to the Hermitian K-theory of symmetric forms and symplectic forms in degree zero, which is the Z=2Z-graded Grothendieck-Witt group GW .The results can likely be extended to higher Hermitian K-theory over general base schemes using the machinery of derived completion as in [38], in line with the recent work of Tabuada-van den Bergh [45] and .An extension of the completion theorem for Hermitian K-theory to schemes with nontrivial actions seems more difficult, as all the known proofs in algebraic K-theory rely on the equivariant localization theorem, whose analogue in Hermitian K-theory has not yet been established, and on a suitable geometric equivariant decomposition theorem.We refer to the last subsection for further details.In a series of fundamental articles, Panin and Walter establish the theory of symplectic oriented cohomology theories on smooth algebraic varieties.In this setting, the one-dimensional torus GL 1 is replaced by Sp 2 D SL 2 .They also show that Hermitian K-theory is a symplectic orientable theory, which in particular implies the following computation of BSp 2n over a base field k (see [32;34, Theorem 9.1]): BO ; .BSp 2n / BO ; .Spec k/ b 1 ; : : : ; b n ;where the b i are the Borel classes of [34, Section 8] and the right-hand side is the ring of graded power series over BO ; .Spec k/; compare, eg, [27, Section 6.3].From this point of view, one might argue that the conjectural Atiyah-Segal completion result for Sp 2n and Hermitian K-theory is the correct analogue of the completion theorem for GL n and algebraic K-theory, and should provide computations involving free polynomial rings that are easier than Rohrbach's theorem above.We will show that this is indeed the case and prove the following theorem as Corollary 3.3.13below.Theorem 1.0.2Let k be a field of characteristic not two.There is a canonical map of GW .k/-algebras GW .Rep.Sp 2r // !GW .BSp 2r /; which exhibits GW .BSp 2r / as the completion of GW .Rep.Sp 2r // with respect to IO Sp 2r .The strategy for GW-theory consists in some sense in systematically replacing GL 1 by Sp 2 , and more generally GL n by Sp 2n in all steps of the proof for complex or algebraic K-theory.Algebraic & Geometric Topology, Volume 25 (2025) This approach has already been used in computations for Chow-Witt groups, see, eg, [19], and is motivated in part by the fact that real realization of Sp 2n has U.n/ as a maximal compact subgroup.The techniques used in this proof, which build on and slightly generalize those of Morel-Voevodsky and Panin-Walter, can be used to construct geometric classifying spaces for other groups G and other Grothendieck topologies .A general result about geometric descriptions of classifying spaces B G in motivic homotopy theory is given in Theorem 3.2.17.Readers familiar with the proof of [4] might consider still another strategy for proving general Atiyah-Segal completion theorems for real and Hermitian K-theory, respectively: rather than Sp 2n and its subgroup Sp n 2 , one should work with GL n and its maximal split torus, but both equipped with a suitable involution that up to homotopy becomes complex conjugation under complex realization.There are indeed nice models for these in the algebro-geometric setting, which are explained in the introduction of Section 2. In particular, the results in Section 2.3 apply to these algebraic groups with involution.However, we are not yet able to perform all necessary equivariant GW-computations for these algebraic models of GL n and split tori with involution, and the authors hope to return to this topic in the future.This article, particularly Section 2, contains results about the Real (sometimes quadratic or Hermitian) representation ring of a large class of reductive groups with involution, such as Sp 2n and GL n with involution.This builds on previous work of and Zibrowius [49] and is of independent interest.As an example of how some of the techniques of this paper extend to such Real groups, we have included in Section 3.4 a short construction of a classifying space for the multiplicative group G m with a nontrivial involution.