# Existence of a Lie algebra element orthogonal to the adjoint orbit of another element

Consider a compact, semi-simple, connected Lie group $G$ and its Lie algebra $\mathfrak{g}$. Denote the Killing form by $K$.

Given a single $A \in \mathfrak{g}$ when (i.e. which groups and which $A$) can one find a $B \in \mathfrak{g}$ such that:

$K(B, Ad_{g}(A)) = 0$ $\ \ \forall g \in G$

The case of $SU(n)$ is particularly important. I believe that I've shown it to never be possible for $SU(2)$.

• Ok, let me extend my question. If $g_t$ is an arbitrary smooth curve on $G$ then when (i.e. for which curves) can we find a $B$ s.t. $K(B, Ad_{g_t}(A)) = 0$. – Benjamin Apr 17 '15 at 18:06
• The curve must have $g_0 = e$ in fact. – Benjamin Apr 17 '15 at 18:13

If your group is simple the adjoint representation is irreducible, so a (nonzero) orbit always spans $\mathfrak g$ and hence will not be contained in $B^\perp$ for any (nonzero) $B$. If $G$ is only semi-simple, you get $G(A)\subset B^\perp$ by taking $A$ and $B$ in different simple factors.
(For your extended question: let's again take $G$ simple and $A\ne0$. As the orbit $G(A)$ spans $\mathfrak g$, you can draw on it a curve $g_t(A)$ that still spans, hence is not in $B^\perp$ for any nonzero $B$. On the other hand, you may of course first fix some $B\in A^\perp$ and then draw, in $B^\perp\cap G(A)$, a curve having your desired property.)
• That's very clear thanks. Is there a criterion that can be deduced for which curves, or at least some non trivial curves, which $Ad_{g_t}(A)$ will fail to span the Lie algebra? In fact this issue only matters if $G$ is simple so that's the only case I care about. – Benjamin Apr 20 '15 at 19:24
• If you are at liberty to draw the curve $\{g_t\}$ at will, then I sketched a way. If you already have the curve and need to decide whether $\{g_t(A)\}$ spans, we might need to know more on how your curve is given. – Francois Ziegler Apr 21 '15 at 0:26
• I'm interested in finding a condition for the curve to have $\{g_t(A)\}$ spanning, so the curve is not given. The actual curves I'm interested in are solution to $\frac{d g_t}{dt} = (a + w(t)b)g_t$ for some smooth function $w(t)$. Ultimately I'm hoping for a condition on the function $w$. – Benjamin Apr 21 '15 at 0:29
• Here $a,b$ are some given elements of $\mathfrak{g}$ such that $\{a,b\}_{L.A.} = \mathfrak{g}$. – Benjamin Apr 21 '15 at 0:37