What i understand about strata for the nullcone is this: (from Mumford's "Geometric Invariant Theory" and Hesselink's paper "Desingularizations of Varieties of Nullforms")

**ADDED BY DAVID SPEYER** In this setting, we are studying a reductive group $G$ acting on a vector space $V$. We write $T$ for a maximal torus of $G$ and $v$ for a nonzero vector in $V$.

There is a bilinear form on the set $Hom(G_m, T) \otimes \mathbb{R}$ that is invariant under the Weyl group, this bilinear form defines a norm on the set of one-parameter subgroups. Consider the set of one-parameter subgroups $\lambda : \mathbb{R} \rightarrow G$ satisfying the condition $\lambda(t).v = 0$. Also this may mean that there is a morphism $f : A^{1} \rightarrow V$ with $f(0)=0, f(t) = \lambda(t)(v)$ for $t \neq 0$; define $m(v, \lambda)$ to be the multiplicity of the fibre $f^{-1} (0)$. Then the strata are defined via the $\Gamma$ which associates to each $v$ a set of one-parameter subgroups of "shortest length", and also satisfying that $m(v, \lambda) \geq 1$; and two vectors are int the same strata if they have the same set of one-parameter subgroups linked with them.

**Questions:**
When $G = GL_{k}(\mathbb{C})$, how would you define the norm on the space $Hom(G_m, T) \otimes \mathbb{R}$ standard torus $T$ of diagonal matrices? Is there some classification of all one-parameter multiplicative subgroups of $GL_{k}(\mathbb{C})$? And I don't understand how $m(v, \lambda)$ is defined - how can the multiplicity of the fibre $f^{-1}(0)$ be anything other than $1$? Is there some exception in the case where $ \lambda(t).v=0$ for all $t$, then what is the multiplicity (is it $0$)? How can the multiplicity be, for instance, $2$?