# On an interesting subalgebra of the functions on the cotangent bundle of the flag variety

Setup

Let $G$ be a semisimple algebraic group over a field $k$ of characteristic $p$ where $p = 0$ or $p > 0$ is a good prime for $G$. Fix a Borel subgroup $B \subseteq G$ corresponding to the positive roots. Set $\mathfrak g := \textrm{Lie}(G)$. Let $U \subseteq B$ be the unipotent radical and let $U^- \subseteq G$ be the opposite unipotent radical. Set $\mathfrak n := \textrm{Lie}(U)$ and $\mathfrak n^- = \textrm{Lie}(U^-)$.

Let me first recall the following general fact about sections of bundles on the flag variety $G/B$. Let $V$ be any $B$-module. Then we have a left $G$-action on $k[G] \otimes V$ given by the left action in $k[G]$ and we have a right $B$-action given by $(f \otimes v).b = f.b \otimes b^{-1}.v$. The global sections of the $G$-equivariant bundle $G \times^B V$ on $G/B$ now identify with the $G$-submodule $(k[G]\otimes V)^B \subseteq k[G] \otimes V$ of invariants under the right $B$-action. It is also known that the projection map $(k[G] \otimes V)^B \to k[U^-] \otimes V$ given by restriction of functions in the first coordinate is an inclusion which corresponds geometrically to the open inclusion $(G \times^B V) |_{\mathfrak B} \hookrightarrow G \times^B V$, where $\mathfrak B$ is the big cell $U^-B \subseteq G/B$.

In particular let us take $V = \mathfrak n$ so that we have the cotangent bundle $\mathcal T^* := G \times^B \mathfrak n$ of $G/B$. Denote by $\mathcal N \subseteq \mathfrak g$ the nullcone of nilpotent elements. $\mathcal T^*$ is a resolution of singularities of $\mathcal N$ and there is an isomorphism $k[\mathcal T^*] \cong k[\mathcal N]$. Also, there is a natural $B$-equivariant surjection $\mathcal N \twoheadrightarrow \mathfrak g / \mathfrak b \cong \mathfrak n^-$ induced by the projection $\mathfrak g \twoheadrightarrow \mathfrak g / \mathfrak b$. Thus we obtain a $B$-equivariant inclusion $$k[\mathfrak n^-] \hookrightarrow k[\mathcal T^*] .$$ Putting this all together, we obtain an interesting algebra inclusion $$i: k[\mathfrak n^-] \hookrightarrow k[U^-] \otimes k[\mathfrak n] .$$

Remark

It's possible that the above description is overcomplicated, since the inclusion $i$ is just the comorphism of the variety morphism $U^- \times \mathfrak n \to \mathfrak n^-$ given by the composition of the action morphism $U^- \times \mathfrak n \to \mathfrak g, u \times X \mapsto u.X$ and the projection $\mathfrak g \twoheadrightarrow \mathfrak n^-$. However, when described in this way it's not a priori clear to me that $i$ is a dominant morphism, so perhaps the above description is necessary.

Question

It seems highly likely that someone has considered this algebra inclusion before. If so, I would like a reference. In particular, it would be nice to have some sort of concrete description of the image of the algebra morphism $i$, perhaps in terms of generating eigenfunctions for $k[U^-]$ and $k[\mathfrak n]$.

• Upon further reflection it seems as though this is a complicated way of asking something simple, namely: is there a nice description for the coaction $k[\mathfrak g] \to k[U^-] \otimes k[\mathfrak n]$. I considered deleting the question but I guess I'll leave it up as a potential stone soup question. – Chuck Hague Jul 31 '12 at 16:17