Let $k$ be a field of characteristic zero, $G$ a connected semi-simple algebraic group over $k$ and $B$ a fixed Borel subgroup of $G$ with maximal torus $T$. Also denote by $W$ the Weyl-group of $G$.

Let $X$ be the (complete) flag variety of $G$ dimension $n$, hence we can assume that $X=G/B$. Furthermore let $x \in X$ be the unique fixed point of $B$ under the natural left action of $G$ on $X$. Then denote for $w \in W$ by $X_w=Bwx$ the corresponding schubert cell in $X$ with dimension $l(w)$.

According to p. 51 in Brylinskis paper "Differential operators on the flag varieties" we have that the formal character of $H^{n-l(w)}_{X_w}(X, \mathscr{O})$ is given by $\frac{e^{-w(\rho)-\rho}}{\prod_{\alpha \in R(B)}(1-e^{-\alpha})}$. Here $R(B)$ is the set the roots of $B$, hence the positive roots of $G$ with respect to $B$ and $\rho$ is the half sum of all positive roots of $G$ with respect to $B$. Finally he concludes that $H^{n-l(w)}_{X_w}(X, \mathscr{O})$ has highest weight $-w(\rho)-\rho$.

He refers to Kempfs "The Grothendieck-Cousin Complex of an Induced Representation" for a proof of this result. But when I look it up there, concretely Lemma 12.8, I would rather say that the formal character of $H^{n-l(w)}_{X_w}(X, \mathscr{O})$ is given by $\frac{e^{w(\rho)+\rho}}{\prod_{\alpha \in R(B)}(1-e^{\alpha})}$. So the sign changed everywhere. Hence it would have lowest weight $w(\rho)+\rho$.

Can anyone solve my confusion?

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