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This is a question arising from Gaitsgory's Geometric Representation Theory notes.

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$, with fixed Cartan and Borel subalgebras $\mathfrak{h} \subseteq \mathfrak{b}$. On page 14 of these notes, Gaitsgory writes:

Let $\tau$ be the Cartan involution on $\mathfrak{g}$. This is the unique automorphism of $\mathfrak{g}$, which acts as $-1$ on $\mathfrak{h}$ and maps $\mathfrak{b}$ to $\mathfrak{b^-}$.

Why is such an involution unique? For instance, consider $\mathfrak{sl}_2$ with standard generators $e,f,h$ satisfying the defining relations $[h,e] = 2e, [h,f] = -2f, [e,f] = h$. Then let $\varphi: \mathfrak{sl}_2 \to \mathfrak{sl_2}$ defined by $\varphi(e) = cf$, $\varphi(f) = de$, and $\varphi(h) = -1$. If $cd=1$, then $\varphi$ has order two, and $\varphi$ respects the defining relations, e.g. $$-h = \varphi([e,f]) = [ \varphi(e), \varphi(f)] = cd [f,e] = -h.$$ Further, $\varphi(\mathfrak{b}) = \varphi(\mathbb{C}e + \mathbb{C}h) = \mathbb{C}f + \mathbb{C}h = \mathfrak{b}^-$.

If we require that $\tau$ respect the module structure of $\mathfrak{g}$ over itself as well as being a Lie algebra automorphism, then I believe $\tau$ is unique. But is this what Gaitsgory means?

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I have concluded that there is an ambiguity, but it is not a problem. The purpose of introducing $\tau$ is to define the contragradient dual $M^\vee$ of a module $M$ in category $\mathcal{O}$ by first taking the maximal submodule of the dual upon which $\mathfrak{h}$ acts freely, and then twisting the action by $\tau$. Fixing $\tau$ throughout gives a consistent functorial construction of the contragradient dual, and all theorems about duality in category $\mathcal{O}$ go through with any choice of $\tau$ with the stated properties.

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