# Deciding whether a representation is orthogonal or symplectic

I'm trying to understand the proof of Proposition 7 part (iii) from this paper of Dadok https://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773051-1/S0002-9947-1985-0773051-1.pdf (Statement of the proposition on page 128, proof on page 131), but getting stuck on one line in the the middle.

The set-up is as follows: We have an irreducible self-dual (complex) representation $$\pi_{\lambda}:\mathfrak{g}\to \mathfrak{gl}(V)$$ of highest weight $$\lambda$$ (say $$v_{\lambda}\in V$$ is a heighest weight vector). There is a distinguished set $$\{\beta_1,\dots, \beta_l\}$$ of strongly orthogonal positive roots of $$\mathfrak{g}$$ (their specific definition isn't important for this part), and we are assuming that $$\lambda$$ is in the (real) span of the $$\beta_i$$. We are considering the subalgebra $$\mathfrak{u}\subseteq \mathfrak{g}$$ generated by the rootspaces of the $$\pm \beta_i$$ (note $$\mathfrak{u}\cong \mathfrak{sl}_2(\mathbb{C})^{\oplus l}$$).

Self-duality of $$\pi_{\lambda}$$ implies that there is a non-degenerate invariant bilinear form $$B:V\times V\to \mathbb{C}$$, and to determine whether $$\pi_{\lambda}$$ is orthogonal or symplectic we need to figure out whether $$B$$ is symmetric or skew-symmetric. Now he defines $$U_{\lambda}\subseteq V_{\lambda}$$ as the $$U(\mathfrak{u})$$-submodule generated by the highest weight vector $$v_{\lambda}$$ (the module $$U_{\lambda}$$ will be simple by the theorem of highest weight, using the fact that $$\lambda$$ is in the span of the $$\beta_i$$). The idea of the argument is that he wants to show that one decide whether $$B$$ is symmetric or skew-symmetric by looking at the restriction $$B:U_{\lambda}\times U_{\lambda}\to \mathbb{C}$$. To that end, he writes:

"If $$B:V\to V\to \mathbb{C}$$ is the nondegenerate bilinear form invariant under $$\pi_{\lambda}$$, we see that $$B$$ must remain non-degenerate on $$U_{\lambda}\times U_{\lambda}\to\mathbb{C}$$ since the $$\mathfrak{u}$$-module $$U_{\lambda}$$ appears in $$V_{\lambda}$$ with multiplicity one."

Can anyone help me make sense of this line? I can't even figure out what he means by multiplicity in this context.

Thanks!

• Why is $\mathfrak{u}\cong\mathfrak{sl}_2$? Isn't it generated by multiple root spaces, or just $\pm\beta_i$ for some $i$? – David Hill Mar 22 at 15:50
• Ah, yes thank you, that was a typo! It should have said $u\cong \mathfrak{sl}_2(\mathbb{C})^{\oplus l}$. – itinerantleopard Mar 22 at 19:41