# Cartan involution coming from triangular decomposition.

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?

## 1 Answer

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.