I'm reading Loday's Cyclic Homology to prepare for my oral exam, and am stuck on a particular detail he mentions very casually. Lemma 9.2.5 states: "The module of coinvariants $(gl(V)^{\otimes n})_{gl(V)}$ is equal to the module $(gl(V)^{\otimes n})_{GL(V)}$ (which is also $(gl(V)^{\otimes n})_{SL(V)}$)."
This result seems surprising, and there isn't a proof provided. The proof refers to A. Borel's Linear algebraic groups, but the result does not appear to be in either edition of that book, or at least not that I can see. (I'm not very familiar with algebraic geometry, so that book is not the easiest read for me, but I can't find a similar statement anywhere in it.)
All I have so far is a fairly easy check that the result does in fact hold in the $n=1$ case (by direct computation), and Loday's note that "This is a consequence of the reductivity of $SL(V)$". My advisor is stumped by this as well. Is this a well-known result in a field I'm unfamiliar with? Am I just not digging deep enough into the computations to succeed in proving it directly?