Motives of some group schemes associated to central simple algebras

Consider a central simple algebra $D$ of prime degree $d$ over a field $F$. We can associate to $D$ the group schemes $GL_1(D)$ and $SL_1(D)$ with $E$-points $GL_1(D)(E) = D_E^* = \{ a \in D_E: Nrd(a)$ non-zero $\}$, $SL_1(D)(E) = \{ a \in D_E: Nrd(a) = 1 \}$, where $Nrd$ denotes the reduced norm. Since $D$ is a form of $M_d(F)$, $GL_1(D)$ and $SL_1(D)$ are forms of $GL_n(F)$ and $SL_n(F)$ respectively. It is not hard to prove that the motives of $GL_n(F)$ and $SL_n(F)$ are (mixed) Tate motives. In the case of $D$ being a quaternion algebra, it follows from simple geometric considerations and the result of Voevodsky on the motives of quadrics, that the motives of $GL_1(D)$ and $SL_1(D)$ lie in the triangulated subcategory of $DM^{eff}_{-}$ generated by Tate motives and the motive of Voevodsky-Chech simplicial scheme $XX = \check{C}(SB(D))$. It was conjectured by Suslin that the same statement holds for any prime degree $d$. Using Suslin's idea we show that it is indeed the case for $GL_1(D)$. For the proof we use slice-filtration in the category of motives over the simplicial scheme $XX$. The case of $SL_1(D)$, $d > 2$ is open, however we investigate the case $d = 3$ where $SL_1(D)$ has a nice smooth compactification $X$ and show (using results of Semenov) that the above conjecture in this case is closely related to rationality of some explicitly given algebraic cycle $\delta \in CH^4(X)$.