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Let $G$ be a semi-simple group over complex number; for simplicity let us assume that it is simply laced. Let $X$ be the orbit of the highest root line in the adjoint representation of $G$. This is a projective variety, such that the corresponding cone is equal to the minimal nilpotent in the Lie algebra of $G$.

What is the cohomology of $X$? E.g. can you write explicitly its Poincare polynomial? Is it, for example, true that its Euler characteristic is equal to $dim(G)-rank(G)$? It would be great it you could give me a reference.

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I think this might be so standard that there is no obvious reference. The cohomology has a basis of Schubert classes. In the adjoint case, Schubert classes are indexed by the Weyl group orbit of the highest weight of the adjoint module. This is the set of long roots, or in the simply laced case, of all roots, implying, in particular, the claim on the Euler characteristic.

P.S. If you really would like a printed reference to cite, I found one for you: Fact 2.8(ii) in P.E. Chaput, N. Perrin, On the quantum cohomology of adjoint varieties, Proceedings of the London Mathematical Society, Volume 103, Issue 2, August 2011, Pages 294–330.

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  • $\begingroup$ Ok, thanks, the Euler characteristic formula is indeed obvious, but I wonder if there is a simple formula for the Poincare polynomial. $\endgroup$
    – Alexander Braverman
    Commented Mar 18 at 17:41
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    $\begingroup$ @AlexanderBraverman In arxiv.org/abs/2112.12436 Nicolas Perrin and Maxim Smirnov state, in Lemma 4.6, part 3, a simple formula for the degree of the Schubert class corresponding to the root $\alpha$, leading to a simple formula for the Poincaré polynomial. (They refer to the article cited in my answer where this formula is not proved, but I think it should be easy to prove it directly - I just do not have time for this at the moment.) $\endgroup$
    – Vladimir Dotsenko
    Commented Mar 18 at 18:35
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  1. The Poincaré polynomial of $G/P$ in general is $\sum_w q^{\ell(w)}$ where the sum is over all minimal length coset representatives of $W_G/W_P$ (here $W_G$ is the Weyl group of $G$ and $W_P$ is the Weyl group of the Levi factor of $P$). $q$ is an element of degree 2. I don't know who originally proved this, but it is contained in Section 5 of one of the papers of BGG: https://iopscience.iop.org/article/10.1070/RM1973v028n03ABEH001557/meta

  2. This sum has a very simple formula as a quotient: let $d_1,\dots,d_n$ be the degrees of the basic invariants for $G$ and let $e_1,\dots,e_m$ be those for the Levi subgroup. Then the formula is $\displaystyle \frac{[d_1]\cdots [d_n]}{[e_1]\cdots [e_m]}$ where I use the notation $[p] = (1-q^p)/(1-q)$.

  3. If we apply this to $P$ being the parabolic that preserves the highest root of the adjoint representation here are the formulas for simple groups (I hope I did not make any mistakes):

$SL_{n+1}$: $[n][n+1]$

$SO_{2n+1}$: $\displaystyle \frac{[2n-2][2n]}{[2]}$

$Sp_{2n}$: $[2n]$

$SO_{2n}$: $\displaystyle \frac{[2n-4][2n-2][n]}{[n-2][2]}$

$E_6$: $\displaystyle \frac{[8][9][12]}{[3][4]}$

$E_7$: $\displaystyle \frac{[12][14][18]}{[4][6]}$

$E_8$: $\displaystyle \frac{[20][24][30]}{[6][10]}$

$F_4$: $\displaystyle \frac{[8][12]}{[4]}$

$G_2$: $[6]$

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