# Serre's special groups and polynomials in Lefschetz element

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 $$k=\mathbb C$$)?

• Where do the definition and classification of special groups appear? – LSpice Jun 30 '19 at 14:05
• 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. – Hexhist Jun 30 '19 at 15:07

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).