# Invariants and stablizers for the $PGL(V)$ action on $End(V\otimes V)$

Let $K$ be a field of characteristic zero, and $V$ a finite dimensional vector space over $K$. Consider the action of the algebraic group $G:=PGL(V)$ on the vector space $W:=End_K(V^{\otimes 2})$ by conjugation of the diagonal action.

My question is: what is known about the orbits of $G$ in $W$? what is the dimensions of the stabilizer of the generic point? What about if we restrict $W$ to the linear automorphisms of $V^{\otimes 2}$ ? Is there any specific sort of points for which we get finite stabilizers for example?

Some rough ideas: -$\small{End(V\otimes W)\cong (V\otimes W)^*\otimes (V\otimes W)\cong (V^*\otimes V)\otimes (W^*\otimes W) \cong End(V)\otimes End(W)}$ and this remains true as $G$-modules. So your object is almost $T^2(\mathfrak g):=\mathfrak g \otimes \mathfrak g$ with $\mathfrak g=Lie(G)$. (in fact $(\mathbb K\oplus \mathfrak g)\otimes (\mathbb K\oplus \mathfrak g)\cong \mathbb K\oplus \mathfrak g \oplus \mathfrak g\oplus T^2(\mathfrak g)$ as $G$-modules)

-$T^2(\mathfrak g)=S^2(\mathfrak g)\oplus\Lambda^2(\mathfrak g)$ where elements of $S^2(\mathfrak g)$ are polynomial functions of degree $2$ on $\mathfrak g^*\cong\mathfrak g$, so it is well understood.

-If you want to look at invariants on $T^2(\mathfrak g)$, you will consider $S(T^2(\mathfrak g))\subset T(T^2(\mathfrak g))\subset T(\mathfrak g)$, the tensor algebra on $\mathfrak g$. I think that some people know something in type A for such invariants. Maybe are there some hints in Procesi's paper: http://arxiv.org/abs/1501.05190

-for specific elements, you can consider stabilizers of a pure tensor $x_1\otimes x_2$ with $x_1$ and $x_2$ are semisimple. Then $G^{(x_1\otimes x_2)}=G^{(x_1,x_2)}$ (the second is the simultaneous stabilizer of $x_1$ and $x_2$ in $G$). I can't ensure yet, since I have to leave, but I think that this stabilizer is generically finite, even in this very specific case.

Edit: of course, if $x_1$ an $x_2$ are regular semisimple element lying in different tori, the simultaneous centralizer is trivial.

• We can use another work of Procesi, to describe explicitly all the invariants: They can be described in the following way: if $T:V\otimes V\rightarrow V\otimes V$, then take $Tr(\sigma T^{\otimes n}):V^{\otimes 2n}\rightarrow V^{\otimes 2n}$, where $n$ is some number and $\sigma\in S_{2n}$ acts by permuting the tensor factors. By considering the partial traces $T_1,T_2:V\rightarrow V$ I am also convinced now that the generic stabilizer is finite. However, I will be happy to have a better description of some possible quotient spaces. – Ehud Meir Feb 3 '16 at 16:35
• It smells like a wild problem to me. Indeed, just in $(\mathbb K\otimes\mathfrak g)\oplus(\mathfrak g\otimes \mathbb K)\cong\mathfrak g \oplus\mathfrak g$, you have some closed orbits arising from nilpotent elements. E.g. in $\mathfrak{sl}_2\times\mathfrak{sl}_2$, $(e,f)$ has a closed orbit. The same phenomenon should also arise in $\mathfrak g\otimes\mathfrak g$ – Bulois Michael Feb 4 '16 at 8:23
• what if we restrict somehow the variety? that is: if instead of taking the entire $End(V\otimes V)$ we just take some constructible subset? Are there any nice quotients which one can describe? One of the things I am interested in is the subvariety of all solutions for the Young-Baxter Equation: $(T\otimes 1) (1\otimes T)(T\otimes 1) = (1\otimes T) (T\otimes 1) (1\otimes T)$ – Ehud Meir Feb 4 '16 at 12:02
• Sorry, no idea for your particular equation. But generally, I would say that it is indeed usually possible to describe a quotient in either of the two following cases: -a relatively small closed subvariety where nilpotents do not behave very badly (e.g. commuting variety in $\mathfrak g\times \mathfrak g$) -an open locus which keeps only the very general points – Bulois Michael Feb 4 '16 at 12:54