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In the paper "On Minuscule Representations and the Principal SL2" by B.H. Gross (link: here) and some others the terminology "Lefschetz $\operatorname{SL}_2$" is used. I think I am confused and lacking background to understand some of the things in this paper, so I have some basic questions.

My first question is the following. What is a Lefschetz $\operatorname{SL}_2$?

Also, in the paper by Gross the setup is that we have $G$ and algebraic group, $P$ a maximal parabolic subgroup associated with a minuscule coweight. He states that there is a Lefschetz $\operatorname{SL}_2$, call it $H$, which acts on the cohomology $H^*(G/P)$ of the flag variety. I guess you could construct $H$ by observing that the set of weights admits an $\operatorname{SL}_2$-representation. But is $H$ defined as some subgroup of $G$? Or with some other natural construction?

The main thing I am interested in is that there is an isomorphism of $\operatorname{SL}_2$-modules $H^*(G/P) \rightarrow V$, where $V$ is a minuscule representation of the dual group $\widehat{G}$. Here the action of $\operatorname{SL}_2$ on $V$ is the action of the principal $\operatorname{SL}_2$ of $\widehat{G}$. The proof in the paper is by noting that the weights of the two representations are the same. Is there some "natural" isomorphism?

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    $\begingroup$ It's generated by multiplication by a class in degree 2, which came from taking c_1 of a line bundle, which in turn comes from your choice of weight, and the adjoint of this operator under the hodge star. Look up "primitive cohomology," those are the highest weight spaces. in particular, it doesn't come from maps of spaces since those preserve degrees. assuming G is connected, the action of elements of G on cohomology should be trivial since they are homotopic to multiplication by the identity. IMO this structure is more just a feature of Hodge theory, it exists on any compact Kähler manifold. $\endgroup$
    – pupshaw
    Commented Jun 14, 2021 at 14:52
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    $\begingroup$ Since the action of the Lie algebra $\mathfrak{sl}_2$ shifts the weights of cohomology classes, it cannot arise as the induced action on cohomology of an action of algebraic groups of $\textbf{SL}_2$ on $G/P$. $\endgroup$
    – Jason Starr
    Commented Jun 14, 2021 at 14:53
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    $\begingroup$ @pupshaw. You posted just as I was finishing my comment (we both make the same observation). $\endgroup$
    – Jason Starr
    Commented Jun 14, 2021 at 14:53
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    $\begingroup$ There is a natural isomorphism coming from the geometric Satake isomorphism, I believe. But I don't know whether it can be constructed without too much sheaf-theoretic machinery. $\endgroup$
    – Will Sawin
    Commented Jun 14, 2021 at 17:43
  • $\begingroup$ Thanks to everyone for these comments, which are very helpful. $\endgroup$
    – spin
    Commented Jun 15, 2021 at 1:09

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