On lambda operations on mixed motives. (Q2874243)

\noindent The author studies lambda operations on mixed motives. This structure is induced from the natural tensor structure on the category \(D:=DM_{\text{gm}}(\mathrm{Spec}(F),{\mathbb Q}).\) Let NEWLINE\[NEWLINE{\zeta}(X,t)={\sum}_{n\geq 0}{\text{cl(Sym}}^{n}(X))t^{n}NEWLINE\]NEWLINE be the power series with coefficient in \(K_{0}(D),\) where the class of a motive \(X\) in \(K_{0}\) is is denoted as \({\text{cl}}(X)\). \(K_{0}(D)\) is defined to be the Grothendieck ring of the additive category \(D\) modulo the subgroup generated by \({\text{cl}}(Z)-{\text{cl}}(X))-{\text{cl}}(Y)\) for distinguished triangles \(X\rightarrow Z\rightarrow Y\rightarrow X[1]\). \textit{M. Kapranov} [``The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups'', \url{arXiv:math/0001005}] considered the zeta function of a motive \(X\). Denote \({\lambda}(X):={\zeta}(X,-t)^{-1}.\) The assignment \({\text{cl}}(X)\rightarrow {\lambda}(X)\) defines a \({\lambda}\)-structure on \(K_{0}(D).\) The author considers the Adams operations on this \({\lambda}\)-ring in the equivariant context i.e. there is an action of a finite group \(G\) on a motive \(X.\)