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Consider Grothendieck's ring of varieties, $K_0(\mathcal{Var}_k)$ over a field $k$, and the Lefschetz element $\mathbb{L}=[\mathbb{A}^1_k]$. Consider also the subring $\mathcal{L}_k\subset K_0(\mathcal{Var}_k)$ consisting of all classes which are polynomials in $\mathbb{L}$, which for convenience I'll call the "Lefschetz ring".

An algebraic group $G$ over $k$ is called special (as defined by Serre and classified by Grothendieck) if any principal $G$-bundle over a $k$-variety is locally trivial in the Zariski topology.

It is known that $GL_n(k)$ is in $\mathcal{L}_k$, for example: $$[GL_2(k)]=(\mathbb{L}^2-1)(\mathbb{L}^2-\mathbb{L}).$$ Also, $\mathbb{G}_a\cong \mathbb{A}^1_k$ and $\mathbb{G}_m\cong k^\times$ are clearly in $\mathcal{L}_k$. Since the examples above are all special groups, (and not having found references with computation of the classes in $K_0(\mathcal{Var}_k)$ of orthogonal groups (which are not special, for example)), I wonder if all special groups have their classes in the Lefschetz ring.

More generally, the question is: is there any known relation between special groups and groups whose classes are in $\mathcal{L}_k$ (if necessary, one could assume $k=\mathbb C$)?

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    $\begingroup$ Where do the definition and classification of special groups appear? $\endgroup$ – LSpice Jun 30 '19 at 14:05
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    $\begingroup$ I added the definition: $G$ is called special if any principal $G$-bundle over a $k$-variety is locally trivial in the Zariski topology. I was following this reference: arxiv.org/abs/1411.2710, although they are considering more general concepts. $\endgroup$ – Hexhist Jun 30 '19 at 15:07
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Seems that class of any split reductive algebraic group, special or not, lies in this ring. Use $[G]=[G/B][B]$ and Bruhat decomposition for $G/B$ (while $B$ is glued from ${\mathbb G}_a$'s and ${\mathbb G}_m$'s).

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    $\begingroup$ And I think that over an algebraically closed field, this should be true for any linear algebraic group whatsoever (since the unipotent radical is also in the ring, and is special.) $\endgroup$ – Daniel Litt Jul 2 '19 at 21:17

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