Notation for $k$-vector fields on a smooth manifold? Contravariant version of $\Omega^k(M)$? I’m fairly new to differential geometry. It seems that $\Omega^k(M)$ is the common notation used for differential $k$-form fields on a manifold, $M$. Such that if $\omega \in \Omega^k(M)$ then $\omega \colon x \mapsto \omega_x \in \bigwedge^k T^*_x M$.
Question: But what about some alternating $(k,0)$-tensor field, $\lambda: x \mapsto \lambda_x \in \bigwedge^k T_x M $? I would express this as $\lambda \in (???)$. What is the tangent-bundle analogue of $\Omega^k(M)$?
Example: Let $(M,\omega)$ be a real symplectic manifold with canonical coordinates $(q^i,p_i)$ for $i=1,\dots , n$. The canonical symplectic form is then a differential $2$-form field, $\omega = \text{d}q^i \wedge \text{d}p_i\in \Omega^2(M)$. We can often find the “inverse” of this $2$-form which is then given by $\omega^{-1} = -\partial_{q^i} \wedge \partial_{p_i} $ such that $\omega^{-1}_{x} \in \bigwedge^2 T_x M$.  To what space does the bivector field $\omega^{-1}$ belong and what is the notation for it? Somehow, I have never come across it.
 A: I am posting my comment above as an answer here and adding some more details.

In Poisson geometry this is called the space of multivector fields (or $k$-vector fields) and is sometimes denoted as $\mathfrak X^\bullet(M):=\Gamma(\wedge^\bullet TM)$. In particular, a Poisson structure is a 2-vector field $\pi\in \mathfrak X^2(M)$ obeying some structural equation.
In fact, your example with the inverse of the symplectic form is something foundational in Poisson geometry, as it is the basic example of a Poisson structure on a manifold; every symplectic manifold is Poisson but the converse is not true; the Poisson structure is given by $\pi=\omega^{-1}$.
A: A differential $k$-form on a smooth manifold $M$ is a section of $\bigwedge^kT^*M$, that is $\Omega^k(M) = \Gamma(M, \bigwedge^kT^*M)$. What you're describing is a section of $\bigwedge^kTM$, which is called a polyvector field. The space of polyvector fields is therefore $\Gamma(M, \bigwedge^kTM)$, but I am not aware of any other notation for this space.
