Relativistic spinors are characterized by 2 semiinteger positive numbers, corresponding to the nonunitary, finite dimensional representations of the SL(2, C) group.

If a spinor of a non-relativistic particle is characterized by a number [itex]j[/itex] (which is integer or half-integer), then the wave function has [itex]2j+1[/itex] components, that means it is of form

We basically go, in group theory language, from the double cover of SO(3) = SU(2) (Galilei case) to the double cover of [itex] SO_{o} (1,3) [/itex] = SL(2,C).

In term of Lie algebras (= stands for isomorphisms),

so(3)=su(2), finite dimensional irreds described by one parameter.

so(1,3) = sl(2,c) = su(2) [itex] \oplus [/itex] su(2), thus in the special relativistic case the numbers of parameters describing the irreds is double (2).

If I want to have a transformation group that transforms some objects under rotations, the group will need to be parametrized with three variables. If I then want to extend the transformation to be applied with (relativistic) boosts too, the amount of parameters must be increased to six. Is this, what the two indices are all about?

That doesn't make fully sense. The amount of parameters in rotations and boosts are always going to be three in both. How is this related to the [itex]j[/itex], which is related to the amount of components in spinor?

That almost sounds as if a particle could transform like spin-1/2 particle in rotations, and like spin-1 particle in boosts. Wouldn't make any sense!

I think you are confusing the parameters needed for the group manifold, and the parameters labeling the irreducible representations.

As bigubau said, the irreducible representations of the Lorentz group can be labeled by a pair of "[tex]SU(2)[/tex] labels": [tex](i,j)[/tex]. For a given [tex](i,j)[/tex], the dimension of the representation is (2j+1)(2i+1). A right and left handed Weyl spinor are labeled as [tex](0, 1/2)[/tex] and [tex](1/2, 0)[/tex], respectively. While a Dirac spinor, which is reducible, is labeled as [tex](1/2,0)\oplus (0,1/2)[/tex] and has 4 components.